Squaring the circle is the age old problem of constructing a square with the same area(or perimeter) as the circle. Greek mathematicians attempted to solve this ancient riddle using a ruler and compass only. Due to the transcendental nature of π we can only approximate this geometry. Earth, Moon, the Great Pyramid, and Stonehenge all encode this great philosophical quandary.

Stonehenge encodes the squared circle through its bluestones, named so because when it rains they turn blue. If we draw a square around this circle of stones it will have the same perimeter as the sarsen ring of stones on the very outside. The ancients preserved esoteric gnosis in their sacred buildings as a way to ensure that the information would never be lost or occulted by the greed of mortal men.

*“ A tradition which has been credited by many learned men over the centuries is that the Ancients encoded their knowledge of the world in the dimensions of their sacred monuments.”*

*– John Michell*

The Great Pyramid’s height is in relationship to its base sides as a circle’s radius is to its circumference, and thus it ‘squares the circle’. Put another way, the perimeter of the base equals the circumference of a circle whose radius is equal to the height of the pyramid. This is only achieved due to the slope angle being 51 degrees and 51 minutes. (or 51.84 degrees since there are 60 arc minutes in 1 degree) I noticed that in the six days it supposedly took to create heaven and earth, there are exactly 518,400 seconds, which resonates with the decad (the decimal system), and the slope angle. And just recently my friend Dayne Herndon pointed out the fact that 5184 also resonates with the canonical value for the Precession of the Equinoxes of 25,920 years. As 25,920 x 2 = 51,840

What else could the Great Pyramid possibly encode? How about the Prime Meridian as suggested by Carl Munck? This is a future post but for now check out The Code.

How about Pi and Tau?

*“Squaring the circle” is the alchemical process of transferring an airy concept from the mental plane to the physical dimension so that objective conception and birth become a demonstrative reality. – Dr. John Munford*

The Earth-Moon relationship and the Great Pyramid of Giza both encode the secret to the mystery. Both of these correlations are well over 99.9% accurate. This is what Leonardo Da Vinci’s “Vitruvian Man” is all about.

Based on measurements of Leonardo da Vinci’s Vitruvian Man we can see that the sizes of the square and circle aren’t quite the correct size to actually square the area of the circle. (in red) But we can also square the circle with equal perimeters (in blue). Vitruvius’ square (in yellow) is almost exactly halfway between ‘squaring the circle’ with equal area, and ‘squaring the circle’ with equal perimeters. It’s slightly closer to the latter.

The square that Da Vinci used is a consolidation of two distinct solutions to ‘squaring the circle’, or obtaining the unobtainable. Did Leonardo encode the solution of this ancient philosophical mystery by suggesting that man is the mean between two transcendental impossibilities?

*“The workings of the human body are an analogy of the workings of our universe” – Leonardo da Vinci*

**Squaring the Circle Methods**

**Equal Areas:**

Square with area of 4 has width of 2

Circle with area of 4 has width of 2.256758..

9/8 = 1.125

2.256 / 2 = 1.128379

1.125 / 1.128379 = .9970

A 9:8 ratio squares the area of the circle to 99.70% accuracy

In music theory, a 9 to 8 ratio is the whole step, otherwise known as the whole tone, or major second.

**Equal Perimeters:**

Square with perimeter of 4 has width of 1

Circle with circumference of 4 has width of 1.2732396..

Interestingly, our Moon takes 27.32166 days to make 1 revolution about Earth.

(A 99.998% accurate correlation according to NASA)

14/11 = 1.27272727…

1.2732396 / 1 = 1.2732396

1.272727 / 1.2732396 = .9996

A 14:11 ratio squares the circle with equal perimeters to 99.96% accuracy

Earth and Moon solve the first riddle to a very high degree of accuracy. Depending on which measures you use, equatorial, polar and mean diameters, it is at least 99.9% accurate.

*Square the Circle/ Earth-Moon Correlations:*

*(NASA 2014 measurements in miles)*

*Earth-Moon radii / Earth’s radius = Square the Circle with equal perimeters*

*Equatorial radii:*

*5043.175 / 3963.17 = 1.27251039*

*1.27251039 / 1.2732396 = 99.943% accuracy*

*Polar radii:*

*5028.63 / 3949.93 = 1.27309*

*1.27309 / 1.2732396 is 99.988% accuracy*

*Mean radii:*

*5038.7 / 3958.75 = 1.27280 *

*99.965% accuracy*

As you can see you don’t have to cherry pick measurements to achieve a very high degree of accuracy. Earth is a living and breathing organism. Its measurements change over time so this could never be 100% accurate.

“The circle is a symbol of spirit, of heaven, of the unmanifest, the immeasurable and the infinite, while the square is the symbol of the material, the Earth, the measurable and the finite,” …the symbolic essence of the problem was the reconciliation of seemingly opposing principles, and the resolution of dualities – “a sacred, cosmological act.” -Daniel Pinchbeck

The Earth and Moon are the perfect size to solve the riddle. This only works because the Moon is huge. It’s larger than any other of the solar system in proportion to its planet. Some say we have a double planet system.

*“The squaring of the circle is a stage on the way to the unconscious, a point of transition leading to a goal lying as yet unformulated beyond it. It is one of those paths to the centre.” – Carl G. Jung*

Bert Janssen found the squaring of the circle encoded into certain crop circles. His website is definitely worth checking out.

The alchemical marriage between Earth and Moon reveal profound geometrical symbols we use to understand our reality. These relationships are a hint into the mystery of our existence. Perhaps we are more than mere coincidence and product of time and chance. The Divine Universal Architect may have hid these clues in plain sight for all to see and cherish, however it is only the few who possess such esoteric knowledge that have the eyes to see these Secrets in Plain Sight.

*An exercise in contemplative geometry from Robert Lawlor’s Sacred Geometry, 1982.*

*“There are a number of diagrams in the literature of Sacred Geometry all related to the single idea known as ‘Squaring of the Circle’. This is a practice which seeks, with only the usual compass and straight-edge, to construct a square which is virtually equal in perimeter to the circumference of a given circle, or which is virtually equal in area to the area of a given circle. Because the circle is an incommensurable figure based on π, it is impossible to draw a square more than approximately equal to it. Nevertheless the Squaring of the Circle is of great importance to the geometer-cosmologist because for him the circle represents pure, unmanifest spirit-space, while the square represents the manifest and comprehensible world. When a near equality is drawn between the circle and the square, the infinite is able to express its dimensions or qualities through the finite” (p74).*

*The derivation begins with an initial circle (within the square) of radius unity. Along its horizontal diameter are drawn two tangent circles, each with radius one half. Observe that the total circumference of the smaller circles equals the circumference of the initial circle, but the total area of the smaller circles is one half that of the initial circle: “One has become Two” (p73), an image of the primary duality, of yin-yang.*

*Next are drawn two arcs from the ends of the initial circle’s vertical diameter with radius tangent to the far sides of the smaller circles. This radius is φ, the golden ratio, dividing the vertical radius of the initial circle into the golden section of lengths 1/φ and 1/φ2. The two arcs meet to create a vesica that encloses the primary duality — the mouth of Ra, the Word, the vibrating string.*

*Around the initial circle is drawn a tangent square, with side 2, perimeter 8; and, finally, a large circle is drawn with diameter equal to the width of the vesica, 2√φ, giving a circumference of 2π√φ = 7.993, approximately equal to 8. The circle is squared.*

Squaring the circle with Earth and Moon

Unknown Squared Circle: More than meets the Eye

Panagiotis Stefanides says

Interesting:

Ref:

http://www.stefanides.gr/pdf/D=5,083FOUR_1.pdf

http://www.stefanides.gr/Html/theo_circle.htm

http://www.stefanides.gr/Html/QuadCirc.htm

http://www.stefanides.gr/Html/piquad.htm

[email protected] says

Thx Panagiotis, This looks very interesting indeed. 🙂

Robert Aguirre says

Epogdoon!!!

Liddz says

Squaring the circle with the Great pyramid and Earth and moon.

From: Liddz.

The measurements of the Great pyramid can be used to create a circle with a circumference that is equal in measure to the perimeter of a square.

Some researchers claim that the width of the Great pyramid’s square base measures 756 feet while the height of the Great Pyramid measures 481.090909090909091. Other researchers claim that the width of the Great pyramid’s square base measures 759 feet while the height of the Great Pyramid measures 483 feet. Other researchers have claimed that the width of the square base for the Great Egyptian Pyramid has a measure of 755.92 feet while the height of the Great Egyptian Pyramid of Giza has a measure of 481.04 feet.

In cubits the width of the Egyptian Great pyramid’s square base measures 440 cubits meanwhile the height of the great Egyptian pyramid measures 280 cubits.

“Golden ratio claimed to be found with in the measurements of the largest of the 3 Pyramids of Egypt’s Giza Plateau”.

1st version of measurements for the Great Pyramid of Giza”:

One Egyptian cubit is 1.718181818181818 feet, when the width of a Pyramid’s square base is measured to be 756 feet that is 440 cubits and the height of the pyramid is measured to be 481.09090909090909 feet that is 280 cubits. 1.718181818181818 is close to the golden ratio of 1.618181818181818 because if 0.1 is added to 1.618181818181818 the result is 1.718181818181818. 1.718 is also close to 2.718. 2.718 is another mathematical constant known as the E number. Half the width of the Great Egyptian Giza Pyramid’s square base is 220 cubits and 378 feet. The slant height of the largest of the 3 Egyptian Pyramids at Giza is 611.67272727272727 feet that is 356 cubits. 356 cubits divided by 220 cubits results in the Golden ratio approximation of 1.618181818181818818181818181818.

“Alternative measurements for the Great Pyramid of Giza”:

2nd version of measurements for the Great Pyramid of Giza:

One Egyptian Pyramid cubit is 1.725 feet, when the width of a Pyramid’s square base is measured to be 759 feet that is 440 cubits and the height of the pyramid is measured to be 483 feet that is 280 cubits.

1.725 is close to the second approximation of the Golden ratio in the Fibonacci series of numbers when 13 is divided by 8. If the Golden ratio is approximated to 1.625 then that approximation for the Golden ratio is close to 1.725. If 0.1 is added to 1.625 the result is 1.725. Half the width of the Great Egyptian Giza Pyramid’s square base is 220 cubits and 379.5 feet. The slant height of the largest of the 3 Egyptian Pyramids at Giza is 614.1 feet that is 356 cubits. 356 cubits divided by 220 cubits results in the Golden ratio approximation of 1.618181818181818818181818181818.

3rd version of measurements for the Great Pyramid of Giza:

One Egyptian Pyramid cubit is 1.718 feet, when the width of a Pyramid’s square base is measured to be 755.92 feet that is 440 cubits and the height of the pyramid is measured to be 481.04 feet that is 280 cubits. 1.718 is close to the golden ratio of 1.618 because if 0.1 is added to 1.618 the result is 1.718. Remember again that 1.718 is also close to 2.718. 2.718 is another mathematical constant known as the E number. Half the width of the Great Egyptian Giza Pyramid’s square base is 220 cubits and 377.96 feet. The slant height of the largest of the 3 Egyptian Pyramids at Giza is 611.608 feet that is 356 cubits. 356 cubits divided by 220 cubits results in the Golden ratio approximation of 1.618181818181818818181818181818.

If Pi is approximated to 3.1416 and has 2.618 subtracted from it then the result is 0.5236 and 52.36 centimeters is an approximation of the cubit that is used to build the Great Pyramid of Giza. 52.35 centimeters is probably a better approximation for the cubit used to build the Great Pyramid of Giza.

A meter is 100 centimeters and also equal to 39.37 inches. A millimeter is one thousandth of a meter.

1 inch is 2.54 centimeters .12 inches is 1 foot. 5280 feet is 1 mile. 6 miles is 31680 feet.

If Pi is approximated to 3.1415 and has 2.618 subtracted from it then the result is 0.5235 and 0.5235 is an approximation for the Pyramid cubit in meters.

Pi approximated to 3.1415 divided by 6 = 0.5235.

If a circle is created with a radius equal to the full height of the Egyptian Great pyramid then the circumference of that circle will be 1760 cubits and this measure is equal to the perimeter of the Great Egyptian Pyramid’s square base because the width of the Great Egyptian Pyramid’s square base measures 440 cubits according to 1 of the 13 ancient versions of Pi and that ancient version of Pi is the ancient Egyptian version of Pi 22 divided by 7 = 3.142857142857143. The Great Pyramid also contains an isosceles triangle that is made from 2 Kepler scalene triangles. The longest length of a Kepler triangle divided by the shortest length of a Kepler triangle is the Golden ratio of 1.618. If the apothem of the Great Pyramid is divided by half the width of the Great Pyramid’s square base then again the result is the Golden ratio of 1.618. The second shortest length of a Kepler triangle divided by the shortest length of a Kepler triangle is the square root of the Golden ratio and reads 1.27. If the full height of the Great pyramid is divided by half the width of the Great Pyramid’s square base then the result is the square root of the Golden ratio and again that reads 1.27. If the width of the Great pyramid is divided by the height of the Great pyramid the result is half of Pi = 1.57. If half of the perimeter of the Great Pyramid’s square base is divided by the height of the Great pyramid then the result will be 1 of the 13 ancient approximations of Pi = 3.14. The ratio of the width of the great pyramid’s square base divided by the height of the Great Pyramid is 11 divided by 7. The width of the great pyramid’s square base contains 11 equal units of measure while the height of the great Pyramid measures 7 equal units of measure. If half of the Great Pyramid’s square base is subtracted from the full height of the Great pyramid the remaining measure will represent the size of our Earth’s moon when compared to the size of our Earth.

The total surface area of a Pyramid with 11 equal units of measure for the width of the square base and 7 equal units of measure for the height of the Pyramid is 316.8 equal square units of measure. The surface area of a Pyramid with a square base is derived by multiplying half the perimeter of the Pyramid’s square base by the slant height of the Pyramid and then adding the result to the area of square units from the square base of the Pyramid.

(22 is half of this Pyramid’s perimeter of square base)

(8.9 is the measure for the slant height of this Pyramid)

(121 is the number of square equal units in the area of this Pyramid’s square base). “22 x 8.9 = 195.8 + 121 = 316.8.

The surface area of this Pyramid is 316.8 equal square units of measure and the area of this Pyramid’s square base is 121 equal square units of measure. If the surface area of this Pyramid 316.8 equal square units of measure is divided by the area of the square base that’s 121 equal square units of measure the result will is the Golden ratio squared: 2.618181818181818. Also if the surface area of this Pyramid 316.8 equal units of measure is divided into the Golden ratio the larger segment of the division is 195.8 the result of half of this Pyramid’s square perimeter multiplied by the slant height of this Pyramid 8.9, while the smaller segment of the division is the area of this Pyramid’s square base 121 equal square units of measure. 195.8 divided by 121 = 1.618181818181818. 195.8 + 121 = 316.8. Half of this Pyramid’s square base perimeter multiplied by the central slant of this pyramid divided by the area of the square base of this Pyramid = 1.618181818181818. The volume of this Pyramid is 282.333 equal units of measure. The longest edge lengths for the 4 isosceles triangles that make up this Pyramid is 10.464 equal units of measure. The angle for the central slant that is located in the middle of any of the 4 triangular faces of this pyramid is 51.84 degrees. 51.84 degrees is the slant angle of the pyramid with a square base width of 11 equal units of measure while the height of this Pyramid has 7 equal units of measure. If 51.84 is multiplied by 5 equal times the result is 259.2. 259.2 with no decimal points is 25920 the number of years for the Sun to travel around the Zodiac. 25920 divided by 12 is 2160 the numbers of years in an age and also the sum of all the angles of a Cube. 41.98 degrees is the edge angle for a Pyramid with a height of 7 equal units of measure while the Pyramid’s square base has a width of 11 equal units of measure.

Remember again that the ratio of the width of the Great Egyptian Pyramid’s square base divided by the height of the great Egyptian Pyramid is 11 divided by 7. 11 divided by 7 = half of Pi = 1.57.

11 x 40 = 440. The width of the Great Egyptian Pyramid’s base in cubits is 440.

7 x 40 = 280. The height of the Great Egyptian Pyramid in cubits is 280.

The width of the square base of the Great Egyptian Pyramid is 440 equal Egyptian cubits. Half of the perimeter for the great Egyptian Pyramid’s square base is equal 880 cubits. The slant height of the Great Egyptian Pyramid of Giza is 356 cubits.

The width of the square base of the Great Egyptian Pyramid is 440 equal Egyptian cubits. Half of the perimeter for the great Egyptian Pyramid’s square base is equal 880 cubits. The slant height of the Great Egyptian Pyramid of Giza is 356 cubits.

So the area of the square base of the Great Pyramid of Giza is 193600 square Egyptian cubits of measure.

Half of the perimeter of the square base for the Great Egyptian Pyramid multiplied by 356 cubits the apothem of the Great Egyptian Pyramid is 313280 equal cubits of measure.

So the total surface area of the Great Egyptian pyramid is 506880 square cubits of equal measure.

The surface area of the Great Egyptian Pyramid is 506880 equal square cubit units of measure and the area of the Great Egyptian Pyramid’s square base is 193600 equal square cubit units of measure. If the surface area of the Great Egyptian Pyramid 506880 equal square cubit units of measure is divided by the area of the square base that’s 193600 equal square cubit units of measure the result will is the Golden ratio squared: 2.618181818181818. Also if the surface area of the Great Egyptian Pyramid 506880 equal cubit units of measure is divided into the Golden ratio the larger segment of the division is 313280 cubit units of measure the result of half of this Pyramid’s square perimeter multiplied by the slant height of the Great Egyptian Pyramid 356 equal cubit units of measure, while the smaller segment of the division is the area of the Great Egyptian Pyramid’s square base 193600 equal square cubit units of measure. 313280 divided by 193600 = 1.618181818181818. 313280 + 193600 = 506880. Half of the Great Egyptian Pyramid’s square base perimeter multiplied by the central slant of the Great pyramid of Egypt divided by the area of the square base of the Egyptian Great Pyramid = 1.618181818181818.

The volume of the Great pyramid of Giza is 18068333.333 cubit units of measure. The longest edge lengths of the 4 isosceles triangles that make up the Great Pyramid of Giza is 418 cubit units of measure. The angle for the central slant that is located in the middle of any of the 4 triangular faces of the Great Egyptian pyramid of Giza is 51.84 degrees. 51.84 degrees is the slant angle of the Great Egyptian pyramid. If 51.84 is multiplied by 5 equal times the result is 259.2. 259.2 with no decimal points is 25920 the number of years for the Sun to travel around the Zodiac. 25920 divided by 12 is 2160 the numbers of years in an age and also the sum of all the angles of a Cube .The edge angle of the Great Pyramid of Giza is 41.98 degrees.

The width of the great Egyptian Giza pyramid’s square base is 440 cubits and 440 multiplied by 18 is 7920.

The height of the Great Egyptian Giza pyramid is 280 cubits. 280 multiplied by 18 = 5040. 7920 divided by 5040 is half of Pi = 1.571428571428571. 1.571428571428571 multiplied by 2 = Pi approximated to 22 divided by 7 = 3.142857142857143. 7920 and 5040 gain these are measurements for the equatorial diameter of the Earth and also the radius of our Earth combined with the radius of our Earth’s moon.

316.8 is an important number because 316.8 is the circumference of a circle with a diameter of 100.8 when modern Pi is approximated to 22 dived by 7 = 3.142857142857143. 316.8 is the perimeter of a square with a width of 79.2. 316.8 feet is also the measure for the circumference of the largest of the circles that make the floor plan of the megalithic stone monument called Stone Henge in the United Kingdom. 79.2 feet is the diameter of the second largest circle that is involved in the design of the megalithic stone monument known as Stone Henge in the United Kingdom. So if a square is created with a width equal to the diameter of the second largest circle of the floor plan of Stone Henge then the perimeter of that square is equal to the circumference of the largest circle according to Pi approximated to: 22 divided by 7 = 3.142857142857143.

If the decimal points are removed from 316.8 then we get 31680 the circumference of a circle with a radius that is equal to both the radiuses of our Earth of 3960 statute miles and our Earth’s moon of 1080 statute miles. 3960 + 1080 = 5040. So if a circle is created with a radius of 5040 statute miles the circumference of that circle is 31680 statute miles according to pi = 22 divided by 7 = 3.142857142857143. 31680 is the perimeter of a square with a width that is equal to the equatorial diameter of our Earth The equatorial diameter of our Earth is 7920 statute miles. If Pi is approximated to 22 divided by 7 = 3.142857142857143 and then divided in half = 1.571428571428571 and then multiplied by the radius of a circle with a measure equal to 5040 statute miles the result is the equatorial diameter of our planet Earth in statute miles 7920. 360 is the natural measure for the circumference of a circle and 360 multiplied by 22 is 7920. Remember that the perimeter of the Great Egyptian Giza Pyramid’s square base has a measure of 1760 cubits and 1760 multiplied by 18 equal times is 31680.

3168 divided by 6 is also 528.528 hertz is also another important number in relation to Solfeggio. 528 is also divisible by 11. 528 divided by 11 is 48. 5280 feet is also equal to statute 1 mile. 528 divided by 168 is Pi = 3.142857142857143 approximated to 22 divided by 7. 440 yards is 1 quarter of a statute mile and 440 cubits is the measure for the width of the square base of the Great Egyptian Pyramid of Giza.1760 yards is also equal to 1 statute mile. Remember again 1760 cubits is the measure for the perimeter of the square base of the Great Egyptian pyramid of Giza.

Again the width for the square base of the Great Egyptian Pyramid of Giza is 756 feet and the diameter of our Earth’s Sun in Royal Egyptian miles is 756000 while the equatorial diameter of our Earth’s moon in Royal Egyptian miles is 1890 because our Earth’s Sun is 400 times larger than our Earth’s moon. The equatorial diameter of our Earth in Royal Egyptian miles is 6930.

Again the equatorial diameter of our Earth is 7920 statute miles. The equatorial circumference of our Earth is 24881.41 statute miles according to Pi approximated to 3.141592171717172. The circumference of our Earth in Nautical miles is 21600. The polar diameter of our Earth is 6912 Egyptian Royal miles. The polar circumference of our Earth is 21714.688 Egyptian royal miles according to Pi approximated to 3.141592592592593. The height of the Great Egyptian Pyramid of Giza including the sockle that the Great Egyptian Pyramid is built upon is 147.1445089 meters and if this measure is multiplied by 864000 then we also get the polar diameter of our Earth in kilometers: 127132855.6896. The equatorial diameter of our Earth in kilometers is 12756.319 according to Pi approximated to 3.141592805887027, meanwhile the equatorial circumference of our Earth in kilometers is 40075.16 according to Pi approximated to 3.141592805887027.

1 inch is 2.54 centimeters. 12 inches is 1 foot.

100 centimeters is 1 meter. 1 meter is 39.37 inches.

A millimeter is one thousandth of a meter.

1 statute mile is 5280 feet.

1 mile is also 1760 yards.

6 miles is also equal to 31680 feet.

The Royal Egyptian mile has 7 equal units of measure while the statute mile has 8 equal units of measure.

(So the statute mile is longer than the Royal Egyptian mile. So if the statute mile is divided by 8 equal units of measure and then the division is multiplied by 7 equal units of measure the result can be the measure for the Royal Egyptian mile.)

1 Rod is equal to 16.5 feet and 32 rods is equal to 1 mile.

1 yard is equal to 3 feet.

1 nautical mile is equal to 20.25 yards.

1 nautical mile is also equal to 1852 meters.

Liddz says

Circle and square with approximate equal areas.

It is impossible to create a circle and a square with equal areas involving 100% accuracy because if the area of the square is rational then the area of the circle will be irrational and vice versa is also true.

1 of the most accurate methods for creating a circle and a square with equal square areas of measure is to create a circle with a diameter of 13 equal units of measure, while the width of the square has 11.52 equal units of measure.

Modern Pi = 3.141592653589793. Modern pi approximated to 40.84 divided by 13 = 3.141538461538462 is the correct approximation of modern Pi to use for this equation. Modern Pi approximated to 3.141538461538462 multiplied by 6.5 equal units of measure the radius of the circle = 20.42. 20.42 multiplied by the radius of the circle 6.5 equal units of measure = 132.73.

So the area of the circle is 132.73.

The width of the square is 11.52 equal units of measure.

11.52 multiplied by 11.52 or 11.52 squared = 132.7104.

So if the results are reduced to 4 decimal places then the area of the circle and the area of the square can be the same.

The width of the square is 11.52 equal units of measure and 11.52 multiplied by 11.52 = 132.7104. 11.52 squared = 132.7104. So the area of the circle is close to the width of the square according to the modern version of Pi approximated to 3.141538461538462. The diameter of the circle with 13 equal units of measure divided by the width of the square with 11.52 equal units of measure produces the irrational ratio of 1.128472222222222. The radius of the circle with area of 132.7104 square units of measure again is 6.5 equal units of measure so if a rectangle is created with the longest length of the rectangle being 6.5 equal units of measure while the shortest length of the rectangle is 5.76 equal units of measure the diagonal of the rectangle will measure 48.45 degrees when the longest length of the rectangle is vertical. Regarding the creation of a rectangle with the longest length measuring 6.5 equal units of measure while the shortest length measures 5.76 units of measure the diagonal of the rectangle will measure 41.55 degrees when the longest length of the rectangle is horizontal and the shortest length of the rectangle is vertical. Knowing the angle that is formed between the 2 poles of the circle’s diameter and the centre edge of the square that is parallel to the diameter of the circle can make the creation of a circle and square with approximate equal areas quicker and easier.

If a scalene triangle is created with the second longest length being 6.5 equal units of measure while the shortest length of the scalene triangle is 5.76 equal units of measure then the longest length of the scalene triangle also called the hypotenuse is 8.68 equal units of measure. The ratio of a scalene triangle’s hypotenuse that measures 8.68 equal units of measure divided by the shortest length of the scalene triangle 5.75 equal units of measure is 1.507

So if a square and a circle have approximate equal areas then the measure for the diameter divided by the width of the square can produce the ratios 1.128472222222222 or 1.129 plus an infinite amount of decimal places or 1.125 if the diameter of the circle is 9 and the width of the square is 8. 1.130434782608696 is the irrational ratio that can be obtained when the diameter of a circle has 26 equal units of measure and is divided by the width of a square with 23 equal units of measure. 26 divided by 23 = 1.130434782608696.

To make that above statement again clearer if a circle and a square are created with approximate equal areas of measure then when the measure for the diameter of the circle is divided by the measure for the width of the square then the result will be any of the following ratios depending on the amount of accuracy that is used:

1.128 plus an infinite amount of decimal places.

1.129 plus an infinite amount of decimal places.

1.130 plus an infinite amount of decimal places.

1.125 when the diameter of the circle is 9 or a multiple of 9 and the width of the square is 8 or a multiple of 8.

If a scalene triangle is created with the second shortest length being equal to 13 equal units of measure while the shortest length is equal to 11.52 equal units of measure then the longest length is the hypotenuse and that measures17.36 equal units of measure. 17.36 divided by 11.52 also equals 1.507

“Another version for creating a circle and a square with approximate equal areas”.

The following version for creating a circle and a square with approximate equal areas involves the diameter of the circle having 7 equal units of measure while the width of the square has 6.2 equal units of measure plus the 22 divided by 7 approximation of modern Pi. 22 dived by 7 = 3.142857142857143.

So in this case 22 divided by 7 = 3.142857142857143 multiplied by 3.5 the radius of the circle = 11 multiplied by 3.5 the radius of the circle = 38.5.

The width of the square is 6.2 equal units of measure. 6.2 squared = 38.44 equal units of measure. Also 6.2 multiplied by 6.2 = 38.44 equal units of measure. So again the width of the square is 6.2 equal units of measure and a square with a width of 6.2 equal units of measure has an area of 38.44 units of measure. The radius of the circle is 3.5 equal units of measure and a circle with a diameter of 7 equal units of measure has a radius of 3.5 equal units of measure and again a circle with a radius of 3.5 equal units of measure has an area of 38.5 equal units of measure according to the approximation value of modern Pi that is 22 divided by 7 = 3.142857142857143. Regarding the third version for the creation of a circle and a square with equal areas the area of the circle is 0.06 larger than the area of the square and that means that the measure for both the area of the circle and the square are very close.

“Creating a Circle and square with equal areas from existing measurements of a circle and a square with equal perimeters”:

If a circle with a circumference equal to the perimeter of a square has already been created and the new desire is to create a square with an area approximately equal to the area of the existing circle that has a circumference equal to the perimeter of another square then a solution is to obtain the ratio of the diameter of the circle divided by the edge length of the square with a perimeter equal to the circumference of the circle with the appropriate diameter and then apply the result of the circle’s diameter being divided by the edge of the square with a perimeter equal to the circumference of the circle to square root and the result is the ratio of the diameter of the existing circle divided by the edge length of a square with an area approximately equal to the area of the existing circle.

An Example follows:

1. Measurements of a circle and square with equal perimeters derived from Pi approximated to 3.141361256544503: Diameter of circle is 11.46 equal units of measure and edge length of square is 9 equal units of measure. Perimeter of square is 36 and circumference of circle is 36.

The ratio of the diameter of the circle divided by the edge length of the square is 1.273333333333333 and can also be obtained if 4 is divided by Pi approximated to 3.141361256544503 = 1.273333333333333.

2. If the ratio 1.273333333333333 is applied to square root meaning the square root of 1.273333333333333 = 1.128420725320717.

3. The ratio 1.128420725320717 can tell us the measure for the correct edge length of the square with an area approximately equal to the area of a circle with a diameter of 11.46 equal units of measure. A square with an edge length of 10.155786527886455 equal units of measure is approximately equal to the area of a circle with a diameter that has 11.46 equal units of measure.

4. 10.155786527886455 multiplied by 10.155786527886455 = 103.14. 10.155786527886455 squared = 103.14. Pi approximated to 3.141361256544503 multiplied y 5.73 equal units of the measure being the radius of the circle = 18 and 18 multiplied by 5.73 = 103.14. 103.14 is the area of a circle with a diameter of 11.46 equal units of measure according to Pi approximated to 3.141361256544503. Remember that the measure for the edge of the square must be reduced to 4 decimal places so that means that it is impossible for the area of the square and the area of the circle to have exactly the same measure of square units. The edge length of the square reduced to 4 decimal places is 10.15 and 10.15 squared is 103.225. So both the area of the square and the circle are quite close but not exactly the same in measure.

Liddz says

Excerpt from my book:

What is Pi?

Pi is the ratio that is based upon the circumference of a circle divided by the circle’s diameter. Numerically with a digital calculator the modern version of Pi is 3.141592653589793. The author of this book does not like to use the modern version of Pi because the modern version of Pi does not allow the circumference of a circle to be read and identified properly with no decimal points. If the modern version of Pi is used to obtain the circumference of a circle then an infinite amount of decimal places will also be shown with the calculation and this is not good. The modern version of Pi can be obtained if 1759.2918860102842 is divided by 560. Instead of using the modern version of Pi to solely read the circumference of a circle in relation to the diameter of the circle approximations of the modern version of Pi must be used to know the measure for a circle’s circumference in relation to its diameter. Approximations for the modern version of Pi can be discovered when the readings given to us by the modern version of Pi are reduced to 4 or 5 decimal places. The author of this book prefers to use the 13 ancient versions of Pi. All the 13 ancient versions of Pi are approximations of the modern version of Pi. The circumference of a circle can be numerically known exactly 100% if any of the 13 ancient versions of Pi are multiplied accordingly to the order of there appearance below by the measurement for the diameter of circles that have a diameters that are capable of being divided into 14 equal units of measure or 13.75 equal units of measure or 11.3 equal units of measure or 11.46 equal units of measure or 8.9 equal units of measure or 5.09295 equal units of measure or 39.78 or 15.91 or 12.7325 or 21.2. 864000 or 7920 or 12756.319 equal units of measure. There are many more approximations of modern Pi that allow the circumference of a circle when divided by and compared to the diameter of the circle to be known, but the author of this book has chosen to only promote 13 of the most common approximations of Pi.

• 22 divided by 7 is 1 of the approximations of Pi 3.142857142857143 that properly allows the circumference of a circle to be read and identified properly with no infinite amount of decimal places regarding the measure for the circle’s circumference and also the circle’s diameter when the circle’s diameter is capable of being divided into 14 equal units of measure.

• 864 divided by 275 = 3.141818181818182 and this is the second method that the author of this book promotes for finding the measure of a circle’s circumference in relation to it’s diameter with 100% accuracy, this second method of Pi works when the diameter of the circle is capable of being divided into 13.75 equal units of measure.

• 355 divided by 113 = 3.141592920353982 and is the third method of approximating Pi for the purpose of 100% accurately finding the measurement for the circumference of a circle in relation to the measurement for the diameter of the circle that the author of this book is promoting. The third method Pi works when the diameter of a circle is capable of being divided into 11.3 equal units of measure.

• The fourth method of obtaining an ancient version of Pi is 36 divided by 11.46 or 360 divided by 114.6 = 3.141361256544503. The fourth ancient version of Pi works with numbers that are a multiple of 18 or divisible by 18.

• The fifth ancient version of Pi only works if the circumference of a circle is 28 or any multiple of 28 and is also divisible by 28, while the diameter of the circle is 8.9 or is any multiple of 8.9 or is divisible by 8.9. The fifth version of Pi numerically is 3.146067415730337 and is the result of 28 divided by 8.9.

• The sixth ancient version of Pi only works if the circumference of a circle is 16 or any multiple of 16 and is also divisible by 16, while the diameter of the circle is 5.09295 or is any multiple of 5.09295 or is divisible by 5.09295. The sixth ancient version of Pi numerically is 3.141597698779686 and is the result of 16 divided by 5.09295.

• The seventh ancient version of Pi only works if the circumference of a circle is 125 or any multiple of 125 and is also divisible by 125, while the diameter of the circle is 39.78 or is any multiple of 39.78 or is divisible by 39.78. The eighth ancient version of Pi numerically is 3.14228255404726 and can be if 125 is divided by 39.78.

• The eighth ancient version of Pi only works if the circumference of a circle is 50 or any multiple of 50 and is also divisible by 50, while the diameter of the circle is 15.91 or is any multiple of 15.91 or is divisible by 15.91.The eighth ancient version of Pi numerically is 3.142677561282212 and can be obtained if 50 is divided by 15.91.

• The ninth ancient version of Pi only works if the circumference of a circle is 40 or any multiple of 40 and is also divisible by 40, while the diameter of the circle is 12.7325 or is any multiple of 12.7325 or is divisible by 12.735.The ninth ancient version of Pi numerically is 3.141566856469664 and can be obtained if 40 is divided by 12.7325.

• The tenth ancient version of Pi only works if the circumference of a circle is 66.6 or any multiple of 66.6 and is also divisible by 66.6, while the diameter of the circle is 21.2 or is any multiple of 21.2 or is divisible by 21.2.The tenth ancient version of Pi numerically is 3.141509433962264 and can be obtained if 666 is divided by 212 or 66.6 is divided by 21.2.

• The eleventh ancient version of Pi only works if the circumference of a circle is 2714336 or any multiple of 2714336 and is also divisible by 2714336, while the diameter of the circle is 864000 or is any multiple of 864000 or is divisible by 864000.The tenth ancient version of Pi numerically is 3.141592592592593 and can be obtained if 2714336 is divided by 864000.

• The twelfth ancient version of Pi only works if the circumference of a circle is 24881.41 or any multiple of 24881.41 and is also divisible by 24881.41, while the diameter of the circle is 7920 or is any multiple of 7920 or is divisible by 7920.The twelfth ancient version of Pi numerically is 3.141592171717172 and can be obtained if 24881.41 is divided by 7920.

• The thirteenth ancient version of Pi only works if the circumference of a circle is 40075.16 or any multiple of 40075.16 and is also divisible by 40075.16, while the diameter of the circle is 12756.319 or is any multiple of 12756.319 or is divisible by 12756.319.The thirteenth ancient version of Pi numerically is 3.141592805887027, and can be obtained if 40075.16 is divided by 12756.319.

“Using Pi to know the circumference of a circle when only the measure for the diameter is known”:

The circumference of a circle can also be known if Pi is multiplied by the measure for the diameter of the circle.

“Using Pi to know the measure for the diameter of a circle when only the measure for the circumference of the circle is known:”

If the circumference of a circle is divided by Pi then the measure for the circle’s diameter can then be known.

“Calculating Pi by dividing half the circle’s circumference by the radius of the circle”:

The ratio Pi can also be approximated when half of a circles’ circumference is divided by the radius of the circle.

“Tau is the circumference of a circle divided by the radius of the circle”:

The circumference of a circle can also be known if Pi is multiplied twice and then multiplied by the measure for the radius of the circle. The circumference of a circle divided by the radius of a circle is known as Tau and is the result of Pi being multiplied 2 equal times.

“Obtaining a quarter of a circle’s circumference with half of Pi”:

A quarter of a circle’s circumference can be obtained also if half of Pi is multiplied by the radius of the circle. Remember to use the correct approximation of Pi for the measure of the circle’s circumference that you want. To know

“Calculating Pi by using the edges of multiple polygons in a circle”:

One of the numerous methods for geometrically approximating Pi is the method of the Deceased ancient Greek Mathematician Archimedes. Archimedes’ method for approximating Pi includes dividing the circumference of a circle into multiple polygons by multiplying the divisions of the circle’s circumference such as beginning with the division of the circle’s circumference into 6 and then 12 and then 24 and then 48 and then 96. The higher the division the closer the approximation of Pi will be. Applications for PI include calculating the mass of circular and spherical objects.

“Calculating Pi by multiplying the divisions of a circle across the diameter of a circle”:

Alternatively Pi can be obtained Geometrically when the circumference of a circle is divided into any number that is divisible by 360. To obtain the most accuracy the arcs that divide the circle can de divided into 100 equal units of measure each so that fractions can also be obtained. To obtain Pi Geometrically the circumference of a circle can be divided into any number that is divisible by 360 and then multiplying that measure along the diameter of the circle so the measurement of the circle’s circumference can be compared and divided by the diameter of the circle.

“Using Pi to calculate the area of a circle”;

Pi can also be used to find the area of a circle and this formula reads any version of Pi multiplied by the radius of the circle and then the result of Pi being multiplied by the radius of the circle is then multiplied by the radius of the circle. So a circle with a radius of 4.5 has an area of 63.617251235193308 according to the modern version of Pi.

If the area of a circle is already known and the desire is to then know the measure for the radius of the circle the measure for the area of the circle must be divided by the correct version of Pi. The correct version of Pi is determined by the measure of the diameter of the circle.

“Discovering more approximations of modern Pi the many other versions of Pi”:

More approximations of Pi can be discovered by reducing the results that are given from dividing the circumference of a circle by modern Pi to 4 or 5 decimal places. The result will be the radius of the circle squared when the area of the circle is divided by Pi. The value for the radius of the circle squared must then be applied to square root and then the measure for the radius of the circle can then be known. Remember to reduce the decimal places to 2 or 3 or 4 digits if the result involves an infinite amount of decimal places. Also if the desire is to have the result read as a whole number 0.1 can be added to a value that is reduced to 3 decimal digits and ends in 0.99. Remember that the radius of a circle is half of the diameter of a circle.

“Using Pi to find the surface area of a Sphere”:

Pi can also be used to find the surface area of a Sphere. A sphere is a 3-dimensional version of a circle. A circle is 2-dimensional.

To find the surface area of Sphere first create a circle with the same diameter of measure as the Sphere and then discover the area of that circle and the surface are of a Sphere is 4 times larger than the area of a circle with a diameter of equal measure to the diameter of the Sphere.

Alternatively the surface area of a Sphere can be found with the following steps:

1. Know the parts of the equation, Surface Area = 4πr2. …

2. Find the radius of the sphere. …

3. Square the radius by multiplying it by itself. …

4. Multiply this result by 4. …

5. Multiply the results by pi (π). …

6. Remember to add you units to the final answer. …

7. Practice with an example. …

Understand surface area.

“Using Pi to find the volume of a Sphere”:

“The volume of a Sphere can also be found by involving Pi in the equation” by following the steps proceeding”:

1.Write down the equation for calculating the volume of a sphere. This is the equation: V = ⁴⁄₃πr³. …

2.Find the radius. If you’re given the radius, then you can move on to the next step. …

3.Cube the radius. …

4.Multiply the cubed radius by 4/3. …

5.Multiply the equation by π.

Liddz says

• Squaring the circle:

Squaring the circle involves creating a circle with a circumference equal to a square. Also squaring the circle can involve creating a circle and a square with equal areas. Squaring the circle can also include harmonious relationships such as the part of the square that intersects the circle’s circumference can be similar to the radius of the circle or the same as the radius of the circle or equal to half of the square’s edge length. Squaring the circle with the area of the square being equal to the area of the circle cannot be achieved with 100% accuracy because the modern version of Pi 3.141592653589793 does not allow the circumference of a circle in relation the diameter of the circle to be read with out infinite amount of decimal places. The closest we can get to creating a circle and square with equal areas is to create a circle with 9 equal units of measure while the width of the square has 8 equal units of measure or plus the ancient approximation version of Pi that is the result of 22 divided by 7 = 3.142857142857143. Also the measure for the diameter of the circle can be 7 equal units of measure while the width of the square can be 6.2 equal units of measure and the area of the circle is determined by using the 22 divided by 7 = 3.142857142857143 value approximation of modern Pi. Alternatively the width of the square can measure 11.52 equal units of measure while the diameter of the circle measures 13 equal units of measure and the modern version of Pi = 3.141592653589793 is used to obtain the area of the circle that has a radius of 6.5 equal units of measure. Also the modern version of Pi cannot be used to create a circle and a square with equal perimeters instead approximations of the modern version of Pi must be used to create a circle and a square with equal perimeters. Approximations for the modern version of Pi can be achieved by reducing readings for the circumference of the circle and also reading s for the diameter of the circle to 4 or 5 decimal places. To create a circle with a circumference equal to the perimeter of a square the diameter of the circle divided by the width of the square must be the ratio of 1.27. To create a circle and a square with equal areas the diameter of the circle must measure 9 equal units of measure while the width of the square measures 8 equal units of measure. If the modern version of Pi is used to obtain the circumference of a circle then an infinite amount of decimal places will also be shown with the calculation and this is not good.

The modern version of Pi can be obtained if 1759.2918860102842 is divided by 560. Squaring the circle can be achieved exactly with the 13 ancient versions of Pi. The 13 ancient versions of Pi are all approximations of the modern version of Pi. The 13 ancient versions of Pi that the author of this book knows that can be used to square the circle with complete 100% accuracy include:

• 1. 22 divided by 7 = 3.142857142857143.

• 2. 43.2 divided by 13.75 = 3.141818181818182.

• 3. 35.5 divided by 11.3 = 3.141592920353982.

• 4. 36 divided by 11.46 = 3.141361256544503.

• 5. 28 divided by 8.9 = 3.146067415730337.

• 6. 16 divided by 5.09295 = 3.141597698779686.

• 7. 125 divided by 39.78 = 3.14228255404726.

• 8. 50 divided by 15.91 = 3.142677561282212.

• 9. 40 divided by 12.7325 = 3.141566856469664.

• 10. 666 divided by 212 = 3.141509433962264.

• 11. 2714336 divided by 864000 = 3.141592592592593.

• 12. 24881.41 divided by 7920 = 3.141592171717172.

• 13. 40075.16 divided by 12756.319 = 3.141592805887027.

The 13 known ancient versions of Pi allow the circumference of a circle to be read with out an infinite amount of decimal places. All the 13 ancient versions of Pi are approximations of modern Pi and have been discovered by reducing the decimal places to 4 or 5 from the results that were obtained from using modern Pi. Modern Pi allows other versions of Pi that are approximations to be discovered when the results of Modern Pi are reduced to 4 or 5 decimal places. The 13 known ancient versions of Pi allow the circumference of a circle to be known exactly 100%. The 13 ancient versions of Pi Square the circle numerically with 100% accuracy and visually share the same results. To Square the circle exactly with the 22 divided 7 version of Pi a square must be used that has an edge length that is capable of being divided into 11 equal units of measure while the diameter of the circle must have 14 equal units of measure when compared to the edge length of the square. Both the square and the circle must share the same centre when the desire is to create the circle with a circumference that is the same measure as the perimeter of the square. To prove that a circle can be created with the same circumference of a square again the edge of the square must be 11 equal units of measure or a multiple of 11 equal units of measure while the diameter of the circle must be 14 equal units of measure or a multiple of 14 equal units of measure.

• 1.Now the proof for Squaring the circle with 100% accuracy is demonstrated with dividing 22 by 7 that is 3.142857142857143 and multiplying this ratio the old version of Pi by the measure of the circle’s diameter for the purpose of discovering the measure of the circle’s circumference and then compare the measure of the circle’s circumference to the perimeter of the square and the measure for circumference of the circle will be the same as the perimeter of the square. Also if half of the Square’s perimeter is divided by the radius of the circle then the result is the ancient version of Pi that is 22 divided by 7 = 3.142857142857143. Also if the perimeter of the square that has a edge length of 11 equal units of measure is divided by the circle that has 14 units of equal measure the result again will be the ancient version of Pi = 3.142857142857143. The perimeter of a square is 4 times larger than the edge length of the square. 44 divided by 14 again is the ancient version of Pi = 3.142857142857143. Remember that the circumference of a circle with 14 equal units of measure has 44 equal units of measure according to the ancient version of Pi = 3.142857142857143. A circle that has a diameter that is capable of being divided by 14 equal units of measure has a circumference that is equal to the perimeter of a square with an edge length that is capable of being divided into 11 equal units of measure according to the first ancient version of Pi = 3.142857142857143.

14 divided by 11 is 1.272727272727273. 1.272727272727273 is the square root of the Golden ratio because if 1.272727272727273 is squared the result will be 1.619834710743802 and the Square root of 1.619834710743802 is 1.272727272727273. So the Square root of 1.618 the Golden ratio is 1.272 and 1.272 is the result of dividing 14 by 11. The ratio 1.272727272727273 in Trigonometry has an angle that measures 51.84277341263095 degrees.

• 2. The proof of Squaring the circle with the second ancient version of Pi 43.2 divide by 137.5 with 100% accuracy can be demonstrated if the circumference of a circle is 432 or 43.2 and the diameter of the circle is 137.5 or 13.75. 432 divide by 4 equals 108. A quarter of the circle’s circumference gives us the edge of the square’s edge length and that is 108. 432 divide by 137.5 = 3.141818181818182.

• 3. The proof of Squaring the circle with the third ancient version of Pi 35.5 divide by 11.3 with 100% accuracy can be demonstrated if the circumference of a circle is 355 or 35.5 and the diameter of the circle is 113 or 11.3. 355 divided by 4 equals 88.75. A quarter of the circle’s circumference gives us the edge of the square’s edge length and that is 88.75. 355 divided by 113 = 3.141592920353982. Doubling the ratios above allow other measurements to be found when squaring the circle for example 22 could be used for the edge of the square and 28 could be used for the diameter of the circle when squaring the circle with the first ancient version of Pi = 3.142857142857143. Multiples of 13.75 for the diameter of the circle and multiples of 10.8 can be used for the edge of the square when Squaring the circle is desired by using the second ancient version of Pi = 3.141818181818182. Multiples of 11.3 for the diameter of the circle and multiples of 8.875 can e used when squaring the circle is desired by using the third ancient version of Pi = 3.141592920353982.

• 4. Proof of squaring the circle with the fourth ancient version of Pi is when the circumference of a circle is 360 and the diameter of the circle is 114.6 alternatively the circumference of the circle can be 36 and the diameter of the circle can be 11.46. 36 divided by 11.46 = 3.141361256544503. 360 divided by 114.6 = 3.141361256544503. The fourth version of Pi works with circles that have a circumference that is 18 or multiples of 18 or divisible by 18. 11.46 divided by 9 = 1.273333333333333. 5.73 divided by 4.5 is also 1.273333333333333. The ratio 1.273333333333333 in Trigonometry has an angle that measures 51.85602405520532 degrees.

“Attempting to square the circle with the fifth ancient version of Pi”:

• 5. The fifth ancient version of Pi is 3.146067415730337 and is the result of 28 divided by 8.9. The fifth ancient version of Pi is the least accurate because the closest approximation of Pi is 3.141 or 3.142. Both the circle and Square can share the same centre or the edge of the square can be the second shortest length of a Kepler scalene triangle while the diameter of the circle can be the longest part of the Kepler scalene triangle. The fifth ancient version of Pi works with a circle that has a circumference of 28 or a multiple of 28 or is divisible by 28, while the diameter of the circle is 8.9 or a multiple of 8.9 or is divisible by 8.9. Remember that the perimeter of the square is equal to the circumference of the circle.

• 6. Squaring the circle with the sixth ancient version of Pi:

The sixth ancient version of Pi is 3.141597698779686 and can be obtained if 16 is divided by 5.09295.Both the circle and square must share the same centre. The sixth ancient version of Pi works with a circle that has a circumference of 16 or a multiple of 16 or is divisible by 16, while the diameter of the circle is 5.09295 or a multiple of 5.09295 or is divisible by 5.09295. Remember that the perimeter of the square is equal to the circumference of the circle.

• 7. Squaring the circle with the seventh ancient version of Pi:

The seventh ancient version of Pi is 3.14228255404726 and can be obtained if 125 is divided by 39.78.Both the circle and square must share the same centre. The sixth ancient version of Pi works with a circle that has a circumference of 125 or a multiple of 125 or is divisible by 125, while the diameter of the circle is 39.78 or a multiple of 39.78 or is divisible by 39.78. Remember that the perimeter of the square is equal to the circumference of the circle.

• 8. Squaring the circle with the eighth ancient version of Pi:

The eighth ancient version of Pi is 3.142677561282212 and can be obtained if 50 is divided by 15.91.Both the circle and square must share the same centre. The eighth ancient version of Pi works with a circle that has a circumference of 50 or a multiple of 50 or is divisible by 50, while the diameter of the circle is 15.91 or a multiple of 15.91 or is divisible by 15.91. Remember that the perimeter of the square is equal to the circumference of the circle.

• 9. Squaring the circle with the ninth ancient version of Pi:

The ninth ancient version of Pi is 3.141566856469664 and can be obtained if 40 is divided by 12.7325.Both the circle and square must share the same centre. The ninth ancient version of Pi works with a circle that has a circumference of 40 or a multiple of 40 or is divisible by 40, while the diameter of the circle is 12.7325 or a multiple of 12.7325 or is divisible by 12.7325. Remember that the perimeter of the square is equal to the circumference of the circle.

• 10. Squaring the circle with the tenth ancient version of Pi:

The tenth ancient version of Pi is 3.141509433962264 and can be obtained if 666 is divided by 212.Both the circle and square must share the same centre. The tenth ancient version of Pi works with a circle that has a circumference of 666 or a multiple of 666 or is divisible by 666, while the diameter of the circle is 212 or a multiple of 212 or is divisible by 212. Remember that the perimeter of the square is equal to the circumference of the circle.

• 11. Squaring the circle with the eleventh ancient version of Pi:

The eleventh ancient version of Pi is 3.141592592592593 and can be obtained if 2714336 is divided by 864000.Both the circle and square must share the same centre. The eleventh ancient version of Pi works with a circle that has a circumference of 2714336 or a multiple of 2714336 or is divisible by 2714336, while the diameter of the circle is 864000 or a multiple of 864000 or is divisible by 864000. Remember that the perimeter of the square is equal to the circumference of the circle

• 12. Squaring the circle with the twelfth ancient version of Pi:

The twelfth ancient version of Pi is 3.141592171717172 and can be obtained if 24881.41 is divided by 7920.Both the circle and square must share the same centre. The twelfth ancient version of Pi works with a circle that has a circumference of 24881.41 or a multiple of 24881.41 or is divisible by 24881.41, while the diameter of the circle is 7920 or a multiple of 7920 or is divisible by 7920. Remember that the perimeter of the square is equal to the circumference of the circle.

• 13. Squaring the circle with the thirteenth ancient version of Pi:

The thirteenth ancient version of Pi is 3.141592805887027 and can be obtained if 40075.16 is divided by 12756.319. Both the circle and square must share the same centre. The thirteenth ancient version of Pi works with a circle that has a circumference of 40074.16 or a multiple of 40075.16 or is divisible by 40075.16, while the diameter of the circle is 12756.319 or a multiple of 12756.319or is divisible by 12756.319. Remember that the perimeter of the square is equal to the circumference of the circle.

• Extra proof that a circle with a circumference equal to the perimeter of a square is the result of the width of the square being divisible by 11 equal units of measure and the circle’s diameter being divisible by 14 equal units of measure is if the width of the same square is divided by 9 equal units of measure then the circumference of the circle can have 36 equal units of measure. 9 is one quarter of 36 and both the circumference of the circle and the perimeter of the square can have 36 equal units of measure and be equal. The diameter of the circle will be 11.46 and the fourth ancient version of Pi is 36 divided by 11.46 = 3.141361256544503 and will be the ratio when the circumference of the circle is compared to the circle’s diameter. When a circle that is divisible by 14 equal units of measure shares the same centre as a square width that is divisible by 11 equal units of measure the circumference of the circle can be divided into equal parts that are 10 and also above 10 with only compass and straight edge and stylus and the circumference of the circle will be equal to the perimeter of the square. The circumference of a circle is naturally 360 degrees so any division of the circle’s circumference that is desired must be the result of 360 being divided into rational numbers, meaning no infinite decimals.

Liddz says

Examples of approximating squaring the circle include the measurement of our Earth’s diameter in miles in comparison to the diameter of our Earth’s moon in miles. The diameter of our Earth is 7920 miles and the diameter of our Earth’s moon is 2160 miles. The Earth moon ratio is of 11 divided by 3 and produces an irrational number of 3.6666666666667. Both the radiuses and the diameters of the Earth and moon when compared to each other produce the ratio of: 3.6666666666667.The symbolic square that contains the earth is 7920 miles. 7920 multiplied by 4 equals: 31680 equal units. Because the diameter of the Earth is 7920 miles the perimeter of the Earth is 31680 miles. If the diameter of the earth is divided by 11 the result will be 720 miles and 720 multiplied by 44 equals the perimeter of the Earth in miles 31680. The circumference of a circle that has a radius equal to both the radius of our Earth and our moon added is equal to the perimeter of Earth because an approximation of Pi is 22 divided 7 and when Pi multiplied by the diameter of the circle that has a radius equal to our Earth with the radius of the moon added to the measure the result is also 31680, the perimeter of our Earth. 31680 divided by 10080 is an approximation of Pi 3.142857142857143 .The Earth’s diameter divided by 11 and multiplied 8 and a half of one is 6120 equal units or miles. If the edge of a square is divided by 11 and then multiplied by 8 and a half of one then the ratio of 11 divided by 8 and a half can be achieved and that is an irrational ratio that reads: 1.2941176470588. The radius of the Earth plus the diameter of the moon is equal to the measure of the Earth’s square that intersects the circumference of a circle that has a radius equal to the radius of the Earth and plus the radius of the moon when this largest circle shares the same centre as the Earth.

Kepler’s Triangle is involved with squaring the circle with the ratio of 11 equal units of measure for the width of the square and 7 equal units of measure for the radius of the circle. Remember that both the square and circle must share the same centre.

A scalene triangle that has its shortest edge equal to half the width of the largest of the 3 pyramids at Egypt’s Giza plateau can be found between the centre of a circle that’s represents the moon above a square that represents the earth in a diagram. This Scalene triangle has been called The Kepler triangle. The shortest length of this scalene triangle is the shorter part of the golden ratio when the shortest length of this scalene triangle is compared to its longest edge length.

The second shortest length of the Kepler triangle divided by the shortest length of the Kepler triangle is the square root of the Golden ratio of 1.618 or 1.619 and that is: 1.272727272727273.

4 divided by Pi approximated to 3.142857142857143 also equals 1.272727272727273. 1.272727272727273 multiplied by 1.272727272727273 or 1.272727272727273 squared also equals 1.619. 1.619 is an approximation of the Golden ratio. The ratio 1.272727272727273 in Trigonometry has an angle that measures 51.84277341263095 degrees.

If a circle is created with a diameter that is equal to the measure of the second shortest length of the Kepler scalene triangle then the circumference of that circle will be equal to the measure of the perimeter of a square with a width that is equal in measure to the shortest length of the Kepler scalene triangle.

“The human figure”:

Example 1:

Please remember that the distance between the human individual’s outstretched arms is equal to the height of the human individual from the top of the head to the bottoms of the feet. Because the distance between the outstretched arms of a human individual is equal to the total height of a human individual from top of the head to the bottoms of the feet a square can be drawn around the physical figure of the human individual.

A circle with a circumference equal to the perimeter of the square that has width equal to the height and outstretched arms from fingertip to fingertip of the human individual can be created if the width of the square is used as the second shortest length of a Kepler scalene triangle and the diameter of the circle with a circumference equal to the perimeter of the square that has a width equal to the height and outstretched arms of the human individual is used as the longest length of the Kepler scalene triangle. The longest length of Kepler scalene triangle divided by the shortest length of a Kepler scalene triangle results in the Golden ratio of: 1.618. Only 2 of the 4 corners of the square can touch the circumference of the circle that has a measure equal to the square’s perimeter with this version of squaring the circle with the human figure. The shortest length of the Kepler scalene triangle has 2 points that both touch the circumference of this circle that has a measure equal to the perimeter of the square with a width that is equal to both the breadth and height of the human figure. The shortest length of the Kepler scalene triangle also forms a rectangle that has the second longest length of the Kepler scalene triangle as the longer length of this rectangle. Remember that the longer length of this rectangle with a diagonal that is also the longest length of a Kepler scalene triangle is also equal to the height and breadth of the human figure. Both of the poles of the circle’s diameter are outside of the square. The square and the circle do not share the same centre.

The hands of the human individual can also be moved to touch the circumference of the circle with limited results including the hands being outside the square when the hands touch the circumference of the circle. Only the hands can touch he circumference of the circle and not the human individual’s feet.

Example 2:

Reminding you of the fact that the distance between the human individual’s outstretched arms is equal to the height of the human individual from the top of the head to the bottoms of the feet. Because the distance between the outstretched arms of a human individual is equal to the total height of a human individual from top of the head to the bottoms of the feet a square can be drawn around the physical figure of the human individual.

Another version of the creation of a circle with a circumference equal to the perimeter of a square can be made involving the height and breadth of the human individual being contained in the square with a perimeter equal to the circumference of a circle .The creation of a circle with a circumference equal to the perimeter of a square can be made involving the height and breadth of the human individual being contained in the square with a perimeter equal to the circumference of a circle when 1 of the poles of the circle’s diameter are located on the bases of the feet of the human individual. The height of the human individual divided by the radius of the circle with a circumference of equal measure to the perimeter of the square containing the height and breadth of the human individual is 1.57. 1.57 is half of Pi. The diameter of the circle divided by the height of the human individual is 1.27.

1 of the poles of the circle’s diameter touch the soles of the human individual’s feet in this version of squaring the circle with the human figure. Very important thing to remember with this version of squaring the circle with the human figure is that none of the 4 corners of the square can touch the circumference of the circle that is equal to the perimeter of the square that contains both the height and breadth of the human individual.

Both hands and feet can touch the circumference of the circle that is equal to the perimeter of the square that contains both the height and breadth of the human individual.

The height and breadth of the human individual is equal to the shortest length of a Kepler scalene triangle when the diameter of the circle that has a circumference equal to the perimeter of the square that contains both the height and breadth of the human individual is taken as the second longest length of a Kepler scalene triangle.

Example 3:

Remember again that the distance between the human individual’s outstretched arms is equal to the height of the human individual from the top of the head to the bottoms of the feet. Because the distance between the outstretched arms of a human individual is equal to the total height of a human individual from top of the head to the bottoms of the feet a square can be drawn around the physical figure of the human individual.

This method of Squaring the circle with the human figure requires a square to be constructed around the human figure that has its edge length being equal to both the total standing height of the human figure and also the outstretched arms of the human individual from longest fingertip to longest fingertip. Squaring the circle with the human figure also requires a circle to be created that has a diameter of 5 equal cubits while the standing height of the human figure that is contained in a square is made of 4 equal cubits. To obtain the length of the diameter of the circle that is required to square the circle for the human figure the total standing height of the human figure must be divided into equal quarters. So the square that contains both the total standing height of the human figure and the also the breadth of the outstretched arms of the human figure from longest fingertip to longest fingertip must be divided into equal fourths, so that a rectangle that has 1 of its longest edges being the same as the top of the square that touches the top of the human’s head can be created. The longest length of this rectangle that is also part of the square for the height of the human is 4 cubits while the shortest length of the rectangle is 3 cubits. The diameter of the circle must be 5 cubits and the south pole of the circle’s diameter must touch the bases of the standing human’s feet. If a circle or a Sphere is placed above the head of the standing human with a diameter of 1 cubit then when the distance from the centre of the sphere to the centre of the standing human’s height is compared to the total standing height and breadth of the outstretched arms the result will be 1.6 and 1.6 is an approximation of the Golden ratio of 1.618. So the square that contains the standing height of the human figure must be 4 cubits and the diameter of the circle that has its south pole located at the bases of the feet of the standing human figure must be 5 cubits. The outstretched arms with longest fingertips of the human figure can also touch the circumference of this circle that has a diameter of 5 cubits with limited results similar to a circle that has a diameter of 4.8 equal units with the South Pole also located at the bases of the human feet. Squaring the circle with the human figure is based upon the ratio 4 out of 5 or four fifths. Please note that if the measurement for the height of the human figure is compared to the radius of the circle that is 2 and half of 1 cubit then the ratio of 1.6 can be achieved and 1.6 is an approximation of the Golden ratio. Please remember that the distance from the centre of the square that contains the height of the human figure and the breadth of the human figure’s outstretched arms from fingertip to fingertip and the centre of the cubit that is above the head of the human figure is also equal to the radius of the circle that has its south pole on the bases of the human feet. Please note that the centre of the circle is above the centre of the square. The perimeter of the Square with 4 equal cubits is equal 16 cubits and the circumference of the circle above with a diameter of 5 equal cubits is 15.714285714286. 15.714285714286 divided by 5 is an approximation of Pi 3.1428571428572. So with out the infinite amount of decimal places the circumference of this circle is 15.7 while the perimeter of the square is 16. 15.7 is not so far from 16. The perimeter of a square with an edge length of 4 is 16 and a circle’s circumference is capable of being divided into 16 equal units of measure .If the circumference of a circle is divided into 16 equal units of measure then the diameter of that circle in comparison to the circumference of a circle with 16 equal units of measure is 5.093 according to Pi approximated to: 3.141566856469664. 5.093 is close to 5.

• Another example of squaring the circle with a circle having a circumference equal to the perimeter of a square is the ground plan of Stone Henge in England. Regarding Stone Henge the circumference of the largest circle that is made from the 30 large stones is equal to the perimeter of a square that contains the second largest circle. The diameter of the second largest circle in ground plan of Stone Henge is 79.2 feet. The perimeter of a square that can be drawn around the second largest circle that has a diameter of 79.2 feet is 316.8 feet. 316.8 feet is also the measure for the circumference of the largest circle that is part of the Stone Henge ground plan. The diameter of the largest circle that is part of the Stone Henge’s ground plan is 100.8 feet. 316.8 divided by 100.8 is the ancient version of Pi 22 divided by 7 = 3.142857142857143.

• Another example of Squaring the circle is when the width of a Pyramid’s square base is measured to be 440 cubits that is 756 feet and the height of the pyramid is measured to be is 280 cubits that is 481.09090909090909 feet. If a circle is created with a radius that is equal to the height of the Egyptian Great Pyramid of Giza the circumference of that circle will be equal to the perimeter of the Great Egyptian Giza Pyramid’s Square base. 1760 cubits is the measure for the perimeter of the Great Egyptian Pyramid of Giza’s Square base. 1760 divided by 560 is an approximation of Pi 3.142857142857143. 440 divided by 280 is equal to 11 divided by 7 and this is a ratio that has an irrational number of 1.571428571428571. 1.571428571428571 is half of 3.142857142857143. These measurements are claimed by some researchers to be of the Greatest of the 3 largest Pyramids in Egypt’s Giza Plateau, meanwhile other researchers and scholars claim that the width of the Great pyramid’s square base measures 759 feet instead of the more commonly accepted claimed measure of 756 feet and also the height of the Great pyramid measures 483 feet instead of the commonly more claimed measure of 481.09090909090909 feet. Other researchers instead have claimed that the width of the square for the base of the Great Egyptian pyramid of Giza has a measure of 755.92 feet meanwhile the height of the Great Egyptian Pyramid of Giza has 481.04 feet.

Again 1 of the 13 ancient versions for the approximation of Pi is 22 divided by 7 = 3.142857142857143. The width of the Great Pyramid’s square base divided by the height of the Great Pyramid is half of Pi that is 1.57 and 1.57 multiplied 2 times is Pi = 3.14.

• Also if half of the Pyramid’s perimeter is divided by the height of the Egyptian Great Pyramid then the result is the ancient version of Pi that is 880 cubits divided by 280 cubits = 3.142857142857143. Also if the perimeter of the Great Pyramid’s square base is divided by the diameter of a circle that can be created with a radius equal to the height of the Great Pyramid then the result again will be 1 of the 13 ancient versions of Pi = 3.142857142857143. The perimeter of the Great Pyramid’s square base again is 1760 cubits and if a circle is drawn with a radius equal to the height of the Great Pyramid the diameter of that circle will be 560 cubits. 1760 dived by 560 again is 1 of the 13 versions of Pi that can be used for Squaring the circle.

• The Greatest of the 3 Egyptian Giza Pyramids is also based upon the Earth and moon ratio because if half of the width of the Greatest of the Egyptian Giza Pyramid’s square base is subtracted from it’s height the remainder will represent the diameter of the moon meaning that half of the width of the largest of the 3 Great Egyptian Giza pyramid’s square base represents the diameter of the Earth. If a circle is created with a diameter that measures equal to the height of the Egyptian Great pyramid of Giza that is 280 cubits then the circumference of that circle is 880 cubits and is equal to the perimeter of a square that has a width equal to the measure of half the width of the Great Egyptian Pyramid’s square base according to 1 of the 13 ancient versions of Pi 22 divided by 7 = 3.142857142857143. If an individual wants to create a replica of the largest of the 3 Great Egyptian Giza Pyramids and wants to know what measurement the height will be 2 solutions can be used, the first solution is to divide half of the width of the square base of the replica of the Greatest of the 3 Egyptian Giza pyramids into 5 and half of 1 equal units of equal measure and then add 1 and a half of those units of equal measure that have been derived from half of the Great Pyramid’s replica square base that has been divided into 5 and a half of 1 units of equal measure the remainder will be the desired height of 5 and half of 1 divided by 1 and a half and that is a ratio of: 3.6666666666667. Alternatively to obtain the height for a replica of the largest of the 3 Egyptian Giza Pyramids half of the width of the Square base can be divided by: 3.6666666666667 and the remainder when added to half of the width of the square base will be the height for the replica of the largest of the 3 Great Egyptian Giza Pyramids

The second method for obtaining the height for a replica of the largest of the 3 Egyptian Giza Pyramids involves: Dividing the width of the square base of the replica of the largest of the 3 Great Egyptian Giza Pyramids into 11 units of equal measure and using 7 of those equal units of measure to obtain the height for the replica of the largest of the 3 Great Egyptian Giza Pyramids. Alternatively to obtain the height for a replica of the largest of the 3 Egyptian Giza Pyramids the width of the Square base can be divided by: 1.571428571428571 and the remainder will be the height for the replica of the largest of the 3 Great Egyptian Giza Pyramids.

This Pyramid of 11 units for square base and 7 equal units for height can have 2 Pyramidions culminating in the total height of this Pyramid. Obviously 1 Pyramidion will be larger than the other Pyramidion. A Pyramidion is a smaller Pyramid that completes the total height of a Pyramid and share’s the same ratios as the full Pyramid. Regarding the replica of a Pyramid that has it’s square base made of 11 units of equal measure while it’s height is 7 units of equal measure the base of the Pyramid’s larger Pyramidion can be found if the width of the Pyramid’s square base is subtracted from the total height of the Pyramid. The height of this Pyramid’s larger Pyramidion is 1 and a half of 1 units of equal measure mean while the distance between the base of the larger Pyramidion and the large square base is 5 and a half of 1 units of equal measure. 5 and a half of 1 divided by 1 and a half is: 3.6666666666667. So the width of this Pyramid’s square base compared to the height of the larger Pyramidion is based upon the Earth and moon ratio of: 3.6666666666667 and is connected to squaring the circle. This Pyramid’s square base represents the diameter of the Earth meanwhile the height of this Pyramid’s larger Pyramidion represents the radius of the moon.

The length of this Pyramid’s square edge compared to the slant height that is found in each of its four triangular faces has an irrational ratio of: 1.2359550561798. Remember that if the slant height of the Pyramid is divided by half of the width of Pyramid’s square base then the result will be the Golden ratio approximation of 1.618181818181818. After the height of this larger Pyramidion has been multiplied twice and the width of the longest square base has been divided by 2 multiples for the measure for the height of this Pyramid’s larger Pyramidion the ratio will be: 3.6666666666667. Again 11 divide by 3 is: 3.6666666666667. The smaller Pyramidion can be found by subtracting half of the width of the larger Pyramidion from its height. Remember that both the larger Pyramidion and the smaller Pyramidion are made from each other and can help to complete the total height of this Pyramid. The smallest Pyramidion is also known as the capstone of the Pyramid and should be made from a different material as the rest of the Pyramid. The capstone of the Pyramid should also be made from a reflective material and the outer material of the rest of the Pyramid should also be made from a reflective material.

Please remember that the division of the whole large Pyramid into smaller Pyramids (Pyramidions) or addition of the smaller Pyramidion to the whole Pyramid structure is only an option and is not compulsory. When representing the Earth and moon ratio it is better to just use one capstone, the capstone that leaves the distance between its base and the base of the whole pyramid equal to half of the Pyramid’s square edge length.

The measurements for the slopes that are located on the 4 corners of the Pyramid’s square base can be known if the diagonal length of the Pyramid’s square base is compared to the height of the Pyramid, because all the slopes of the Pyramid converge at the height of the Pyramid.

The total surface area of a Pyramid with 11 equal units of measure for the width of the square base and 7 equal units of measure for the height of the Pyramid is 316.8 equal square units of measure. The surface area of a Pyramid with a square base is derived by multiplying half the perimeter of the Pyramid’s square base by the slant height of the Pyramid and then adding the result to the area of square units from the square base of the Pyramid.

Liddz says

• Diagram for earth and moon ratio and squaring the circle:

A diagram for the earth and moon and Squaring the circle can be constructed when a square is constructed that represents the diameter of the moon that is divided into 3 equal units of measure so that two rectangles with a ratio that has its longest edge length being 4 equal units and its shortest edge length being 3 equal units and contains 3,4,5 triangles. 3.4,5 triangles can be constructed on either edges of the square that represents the diameter of the moon. The shortest lengths of both the 3 out of rectangles and 3,4,5 triangles must be equal to the edges of the square that represent the diameter of the moon. So if the square that represents the diameter of the moon is divided into 3 equal units then the diameter of the earth must be 11 equal units of measure. The result of this earth and moon ratio is a diagram with a larger square that has 11 equal units and a smaller square on top of the larger square that has 3 equal units of measure that are derived from the division of the larger square into 11 equal units. Seven equal units of measure is the radius for the circle that has a diameter of 14 equal units of measure. The measure for the radius of the circle with a diameter of 14 equal units of measure is equal to the measure between any of the top corners of the Earth Square with 11 equal units of measure and the furthest bottom corner of the moon’s square. Both the circle that has a circumference equal to the perimeter of the square that represents Earth and also the square that represents earth must share the same centre.

• Squaring the circle with Earth and moon and radius of the circle with a circumference equal to the perimeter of a square:

If a circle with a radius of 7 equal units of measure is created and combined with a square that shares the same centre as the circle and also the width of the square is made from 11 equal units of measure then the Earth and moon ratio can be illustrated when 5 and half of 1 units is taken form the radius of the circle with 7 equal to of measure for the purpose of representing the diameter of the Earth. The remainder of 5 and a half of 1 taken from the circle that has a radius of 7 equal units of measure is 1 and half of 1. The width of the square that has a perimeter equal to the circumference of the circle that has a diameter of 14 equal units of measure is 11. Half of 11 is 5 and a half of 1. 5 and a half of 1 divided by 1 and half of 1 is 3.666666666666667. So both the diameter of Earth and also the diameter of the moon can be illustrated with in the radius of a circle that is made from equal units of measure because the diameter of Earth again 7920 miles while the diameter of the moon is 2160 miles and 7920 dived by 2160 is 3.666666666666667. So if half the width of a square that has a perimeter equal to the circumference of a circle is subtracted from the radius of the circle then the remainder will represent the size of the moon while half the width of the square that has a perimeter equal to the circumference of the circle with a radius of 7 equal units of measure can represent the size of Earth. Again the Earth and moon ratio is 3.666666666666667. The diameter of the Earth divided by the diameter of the moon equals 3.666666666666667.

• Alternative placement of moon for squaring the circle with Earth and moon diagram:

Alternatively the circumference of the moon does not have to touch the circumference of the Earth, instead the circumference of the moon with 1 of the moon’s poles can touch the circumference of the circle that is equal to the Earth’s perimeter. Alternatively the circumference of the Earth can touch the centre of the moon. Remember that if a circle is created with a radius that has measure that is a combination of the Earth’s radius and also the moon’s radius the circumference of that circle will be equal to the Earth’s perimeter. If a circle is created that is made of 14 equal units of measure and 3 units from the circle’s radius are subtracted for the purpose of representing the moon’s diameter then the distance from the centre of the circle with a radius of 7 equal units of measure to the centre of the measure that represents the diameter of the moon can be used to represent the radius of the Earth. The square that has a perimeter equal to the circumference of the circle should share the same centre as the circle.

• Alternative method for creating a diagram that involves a circle with a circumference equal to the perimeter of a square:

The diagram for squaring the circle with a circle that has a circumference equal to the perimeter of a square can also be constructed if 2 circles that both have diameters equal to half the width of the square are constructed around the central width of the square and a line that passes through any of the centres of the circles that have diameters equal to half the width of the square is connected to the bottom or top centres of the square. The line that passes through the centres of the 2 smaller circles and originates at the bottom or top centre of the square must also touch the circumferences of the 2 smaller circles. This line that originates at the bottom or top centre of the square must also be used as an arc that touches the 2 horizontal poles of the circle that has a circumference equal to the perimeter of the square that contains the 2 smaller circles.

• Alternative placement of Earth square with Squaring the circle with circle’s circumference equal to perimeter of Square in Earth and moon diagram:

Alternatively the Circle that has a circumference equal to the perimeter of a square does not have to share the same centre as the square with a perimeter equal to the circle’s circumference. Alternatively 1 of the central edges of the Earth Square can align with 1 of the poles of the diameter of the circle that has a circumference equal to the perimeter of the Earth Square. Only 1 edge of the Earth Square can touch the circumference of the circle that has it’s circumference equal to the measure of the perimeter of the Earth square. Please note that none of the 4 corners of the Earth square and also none of 4 corners of the moon square can touch the circumference of the circle that is equal to the perimeter of the Earth square in this version of the Earth and moon squaring the circle diagram. If the pole of the circle’s diameter that touches 1 of the centre edges of the Earth square is used as south while the opposing pole of the diameter of the circle that has a circumference equal to the perimeter of the Earth square is used as north and touches the north pole of the moon’s diameter then the centre of the Earth square will be below the centre of the circle that has a circumference equal to the measure of the perimeter for the Earth square. The circle or square that represents the moon can then be placed on the edge of the Earth Square that aligns with the centre of the circle that has a circumference equal to the perimeter of the Earth Square. Remember that the measure for the diameter of the Earth plus the measure for the diameter of the moon is equal to the diameter of a circle that has a circumference equal to the perimeter of the Earth Square.

• Diameter of circle with circumference equal to the Earth’s perimeter divided by the radius of Earth plus the diameter of the moon:

If the diameter of the circle that has a circumference equal to the perimeter of the Earth’s square is divided by the radius of the Earth plus the diameter of the moon combined the result is a close approximation of the Golden ratio. The Golden ratio is 1.618 and the diameter of the circle that has a circumference equal to the perimeter of the Earth’s divided by the radius of the Earth plus the diameter of the moon combined results in the ratio of 1.647 and 1.6 is the second approximation of the Golden ratio. 1.6 is close to 1.647. 1.666666666666667 is the first approximation of the Golden ratio.

• Requirements for squaring the circle

Remember that squaring the circle is defined as creating a circle and a square and combining them so that the circumference of the circle can become equal to the perimeter of the square also squaring the circle can include the creation of a circle and a square with almost equal areas. Also Squaring the circle can involve harmonious relationships such as when the measure of the square that intersects the circle’s circumference can be the same measure to the radius of the circle or have a similar measure to the radius of the circle or equal to half of the square’s edge length. Remember that the Geometrical phenomena called squaring the circle with the circumference of the circle being equal to the perimeter of a square cannot be achieved exactly with numerical digits when using the modern version of Pi because the modern version of Pi is an irrational and a transcendental number 3.141592653589793. To read the circumference of a circle and also the diameter of a circle with out an infinite amount of decimal places an approximation of the modern version of Pi must be used. An approximation for the modern version of Pi can be achieved when the readings for both the circle’s circumference and also the circle’s diameter are reduced to 4 or 5 decimal places. Squaring the circle with equal perimeters can only be achieved exactly 100% with numerical digits if the 13 known ancient versions of Pi are used. All of the 13 ancient versions of Pi create the same visual results. All of the 13 ancient versions of Pi are approximations for the modern version of Pi. The 13 known ancient versions of Pi that the author of this book is promoting are:

• 1. 22 divided by 7 = 3.142857142857143.

• 2. 864 divided by 275 = 3.141818181818182.

• 3. 355 divided by 113 = 3.141592920353982.

• 4. 360 divided by 114.6 = 3.141361256544503.

• 5. 28 divided by 8.9 = 3.146067415730337. (The fifth ancient version of Pi produces the least accurate results because the fifth ancient version of Pi is based upon 3.146 and a more accurate version of Pi is 3.141 or 3.142).

• 6. 16 divided by 5.09295 = 3.141597698779686

• 7. 125 divided by 39.78 = 3.14228255404726.

• 8. 50 divided by 15.91 = 3.142677561282212.

• 9. 40 divided by 12.7325 = 3.141566856469664.

• 10. 666 divided by 212 = 3.141509433962264.

• 11. 2714336 divided by 864000 = 3.141592592592593.

• 12. 24881.41 divided by 7920 = 3.141592171717172.

• 13. 40075.16 divided by 12756.319 = 3.141592805887027.

“Width of triangle compared to height of triangle used to Square the circle. The following are versions of Isosceles triangles that are made from 2 combined Kepler triangles”:

• (1. Width of triangle is 11 divided by height of triangle 7 = 1.57142857142857).

• (2. Width of triangle is 108 divided by height of triangle 68.75 = 1.570909090909091)

• (3. Width of triangle is 88.75 divided by height of triangle 56.5 = 1.570796460176991).

• (4. Width of triangle is 90 divided by height of triangle 57.3 = 1.570680628272251).

• (5. Width of triangle is 7 divided by height of triangle is 4.45 = 1.573033707865169).

• (6. Width of triangle is 4 divided by height of triangle is 2.546475 = 1.570798849389843).

• (7. Width of triangle is 31.25 divided by the height of triangle 19.89 = 1.57114127702363).

• (8. Width of triangle is 12.5 divided by the height of the triangle 7.995 = 1.571338780641106).

• (9. Width of triangle is 10 divided by the height of the triangle 6.36625 = 1.570783428234832).

• (10. Width of triangle is 166.5 divided by the height of the triangle 106 = 1.570754716981132).

• (11. Width of triangle is 678584 divided by the height of the triangle 432000 = 1.570796296296296).

• (12. Width of the triangle is 6220.3525 divided by the height of the triangle 3960 = 1.570796085858586)

• (13. Width of the triangle is 10018.79 divided by the height of the triangle 6378.1595 = 1.570796402943514).

The above ratios of width compared to height for the isosceles triangles mentioned above can be used to create a circle that has a circumference equal to the perimeter of a square with the most accuracy.

Please remember that 1.57 is half of Pi. Half of Pi multiplied by the radius of the circle also results in 1 quarter of the circle’s circumference.

• (1. 1.57142857142857 multiplied by 2 = 1.57142857142857).

• (2. 1.570909090909091 multiplied by 2 = 3.141818181818182).

• (3. 1.570796460176991 multiplied by 2 = 3.141592920353982).

• (4. 1.570680628272251 multiplied by 2 = 3.1413612565445).

• (6. 1.570798849389843 multiplied by 2 = 3.141597698779686).

• (5. 1.573033707865169 multiplied by 2 = 3.146067415730338.)

• (7. 1.57114127702363 multiplied by 2 = 3.14228255404726).

• (8.1.571338780641106 multiplied by 2 = 3.142677561282212).

• (9. 1.570783428234832 multiplied by 2 = 3.141566856469664).

• (10. 1.570754716981132 multiplied y 2 = 3.141509433962264)

• (11. 1.570796296296296 multiplied by 2 = 3.141592592592593).

• (12. 1.570796085858586 multiplied by 2 = 3.141592171717172).

• (13. 1.570796402943514 multiplied by 2 = 1.570796402943514

Liddz says

• “ There are 9 methods for squaring the circle with harmonious comparisons.”

The first method for squaring the circle includes creating a circle with a diameter that is capable of being divided into14 equal units of measure while the square has an edge length that is capable of being divided into 11 equal units of measure. Both the circle and the square must share the same centre. When 3 units is placed above the centre top of the square the distance between the centre of both square and circle and the top of the 3 equal units that is placed above the centre top of the square is equal to the 4 areas of the square that intersect the circumference of the circle that has a circumference equal to the perimeter of the square of 11 equal units of measure. Squaring the circle with this method also involves the circle with a diameter of 14 equal units of measure containing two 11 long and 8.5 short edge length rectangles, one horizontal and the other vertical and 8.5 equal units is between the centre of the square and the top pole of the smallest circle that has a diameter of 3 equal units of measure. The diameter of this circle is 14 equal units of measure and 14 divided by the edge of the rectangle that is 8.5 equal units of measure is 1.647. The Golden ratio is 1.618 and 1.6 is the first approximation of the Golden ratio. 1.6 is close to 1.647. 1.666666666666667 is the first approximation of the Golden ratio of 1.618. The radius of the circle that has a diameter of 14 equal units of measure is equal to the measure between any of the 2 top corners of the square with 11 equal units of measure and the furthest bottom corner of the square that is made of 3 equal units of measure. Remember again that the square that has a width of 3 equal units of measure is on top of the square that has 11 equal units of measure for its width. The perimeter of this square with 11 equal units of measure is 44 equal units of measure. Again 22 divided by 7 is an approximation of Pi 3.142857142857143. 22 divided by 7 again is 3.142857142857143 and if this number 3.142857142857143 is multiplied by the diameter of the above circle that has a diameter of 14 equal units of measure the circumference for the above circle can be known and that is 44 equal units of measure and that is the same as the perimeter of the square for this circle that has it’s edge length equal to 11 equal units of measure. Also if half of the Square’s perimeter is divided by the radius of the circle then the result is 1 of the 13 ancient versions of Pi that is 22 dived by 7 = 3.142857142857143. Alternatively the 12 other ancient versions of Pi can be used to create a circle that has a circumference equal to the perimeter of a square and again those 12 other versions of Pi can be achieved by:

1. 864 divided by 275 = 3.141818181818182.

2. 355 divided by 113 = 3.141592920353982.

3. 360 divided by 114.6 = 3.141361256544503.

4. 28 divided by 8.9 = 3.141573033707865.

5. 16 divided by 5.09295 = 3.141597698779686

6. 125 divided by 39.78 = 3.14228255404726.

7. 50 divided by 15.91 = 3.142677561282212.

8. 40 divided by 12.7325 = 3.141566856469664.

9. 666 divided by 212 = 3.141509433962264.

10. 2714336 divided by 864000 = 3.141592592592593.

11. 24881.41 divided by 7920 = 3.141592171717172.

12. 40075.16 divided by 12756.319 = 3.141592805887027.

• 1. So if the circumference of a circle is 864 then the diameter of that circle will be 275 and the square edge length that has a perimeter equal to the circumference of that square can be 216.

• 2. If the circumference of a circle is 355 then the diameter of that circle will be 113 and the square edge length that has a perimeter equal to the circumference of that square can be 88.75.

• 3. So if the circumference of a circle is 360 the diameter of the circle will be 114.6 and the edge of the square that has a perimeter equal to this circle that has a circumference of 360 will be 90. 90 is 1 quarter of 360.

• 4. If the circumference of a circle is 28 then the diameter of that circle will be 8.9. The edge of the square will be 7 equal units of measure derived from the circumference of a circle that has 28 equal units of measure while the diameter of the circle has 8.9 equal units of measure. (This method is the least accurate because the result of the circle’s circumference being divided by the circle’s diameter produces 3.146 for a value of Pi and a better approximation of Pi is 3.141 or 3.142.

• 6. If the circumference of a circle is 125 then the diameter of that circle will be 39.78. The square’s perimeter is equal to the circle’s circumference. So the width of the square is 31.25.

• 5. If the circumference of a circle is 16 then the diameter of that circle will be 5.09295. The square’s perimeter is equal to the circle’s circumference so the width of the square is 4.

• 7. If the circumference of a circle is 50 then the diameter of that circle will be 15.91. The square’s perimeter is equal to the circle’s circumference. So the width of the square is 12.5

• 8. If the circumference of a circle is 40 then the diameter of that circle will be 12.732. The square’s perimeter is equal to the circle’s circumference. So the width of the square is 10.

• 9. If the circumference of a circle is 666 then the diameter of that circle will be 212. The square’s perimeter is equal to the circle’s circumference. So the width of the square is 166.5.

• 10. If the circumference of a circle is 2714336 then the diameter of that circle will be 864000. The square’s perimeter is equal to the circle’s circumference. So the width of the square is 678584.

• 11. If the circumference of a circle is 24881.41 then the diameter of that circle will be 7920. The square’s perimeter is equal to the circle’s circumference. So the width of the square is 6220.3525.

• 12. If the circumference of a circle is 40075.16 then the diameter of that circle will be 12756.319. The square’s perimeter is equal to the circle’s circumference. So the width of the square is 10018.79.

Remember that the 12 other ancient versions of Pi when used for squaring the circle create similar visual results as the 22 divided by 7 version of Pi when the circle’s circumference is equal to the square’s perimeter.

Also if the perimeter of the square that has a edge length of 11 equal units of measure is divided by the circle that has 14 units of equal measure the result again will be 1 of the 4 ancient versions of Pi = 3.142857142857143. The perimeter of a square is 4 times larger than the edge length of the square. 44 divided by 14 again is 1 of the 12 ancient versions of Pi = 3.142857142857143. Remember that the circumference of a circle with 14 equal units of measure has 44 equal units of measure according to the ancient version of Pi = 3.142857142857143, A circle that has a diameter of 14 equal units of measure has a circumference that is equal to the perimeter of a square with an edge length of 11 equal units of measure according to the ancient version of Pi = 3.142857142857143. A circle that has a circumference equal to the perimeter of a square must be capable of being divided into 14 equal units of measure and a square that has a perimeter equal to the circumference of a circle must be capable of being dived into 11 equal units of measure. 14 divided by 11 is 1.272727272727273. 1.272727272727273 is the square root of the Golden ratio because if 1.272727272727273 is squared the result will be 1.619834710743802 and the Square root of 1.619834710743802 is 1.272727272727273. So the Square root of 1.618 the Golden ratio is 1.272 and 1.272 is the result of dividing 14 by 11. Remember again that the same visual results can achieved as when the circle is capable of having 14 equal units of measure for its diameter and the square is capable of having 11 equal units of measure for its edge length when the other 12 ancient versions of Pi are used to create a circle that has a circumference equal to the perimeter of a square. Again those 12 other versions of Pi are:

• 1. Circumference of circle is 864 divided by the diameter of the circle 275 = 3.141818181818182. Edge length of square is 216. Edge length of square 216 divided by radius of circle 137.5 = 1.570909090909091.

• 2. Circumference of circle is 355 divided by the diameter of the circle 113 = 3.141592920353982. Edge length of square is 88.75. Edge length of square 88.75 divided by radius of circle 56.5 = 1.570796460176991.

• 3. Circumference of circle is 360 divided by the diameter of the circle 114.6 = 3.141361256544503. Edge length of square is 90. Edge length of square 90 divided by the radius of the circle 57.3 = 1.570680628272251.

• 4. Circumference of circle is 28 divided by the diameter of the circle 8.9 =3.146067415730337. Edge length of square is 7. Edge length of square 7 divided by radius of circle 4.45 = 1.573033707865169. (This method is the least accurate because the version of Pi that is used is 3.146 and a better approximation of Pi is 3.141 or 3.142).

• 5. Circumference of circle is 16 divided by the diameter of the circle 5.09295 = 3.141597698779686. Edge length of square is 4. Edge length of square 4 divided by radius of circle = 2.546475 = 1.570798849389843.

• 6. Circumference of circle is 125 divided by the diameter of the circle 39.78 = 3.14228255404726. Edge length of square is 31.25. Edge length of square 31.25 divided by radius of circle 19.89 = 1.57114127702363.

• 7. Circumference of circle is 50 divided by the diameter of the circle 15.91 = 3.142677561282212. Edge length of square is 12.5. Edge length of square 12.5 divided by radius of circle 7.955 = 1.571338780641106.

• 8. Circumference of circle is 40 divided by the diameter of the circle 12.7325 = 3.141566856469664. Edge length of square is 10. Edge length of square 10 divided by radius of circle 6.366 = 1.570845114671693.

• 9. Circumference of circle is 666 divided by the diameter of the circle 212 = 3.141509433962264. Edge length of square is 166.5. Edge length of square 166.5 divided by radius of circle 106 = 1.570754716981132.

• 10. Circumference of circle is 2714336 divided by the diameter of the circle 864000 = 3.141592592592593. Edge length of square is 678584. Edge length of square 678584 divided by radius of circle 432000 = 1.570796296296296.

• 11. Circumference of circle is 24881.41 divided by the diameter of the circle 7920 = 3.141592171717172. Edge length of square is 6220.3525. Edge length of square 6220.3525 divided by radius of circle 3960 = 1.570796085858586.

• 12. Circumference of circle is 40075.16 divided by the diameter of the circle 12756.319 = 1.570796402943514. Edge length of square is 10018.79. Edge length of square 10018.79 divided by radius of circle 6378.1595 = 1.570796402943514.

•

• Extra proof that a circle with a circumference equal to the perimeter of a square is the result of the width of the square being divisible by 11 equal units of measure and the circle’s diameter being divisible by 14 equal units of measure is if the width of the same square is divided by 9 equal units of measure then the circumference of the circle can have 36 equal units of measure. 9 is one quarter of 36 and both the circumference of the circle and the perimeter of the square can have 36 equal units of measure and be equal. The diameter of the circle will be 11.46 and the fourth ancient version of Pi is 36 divided by 11.46 = 3.141361256544503 and will be the ratio when the circumference of the circle is compared to the circle’s diameter. When a circle that is divisible by 14 equal units of measure shares the same centre as a square width that is divisible by 11 equal units of measure the circumference of the circle can be divided into equal parts that are 10 and also above 10 with only compass and straight edge and stylus and the circumference of the circle will be equal to the perimeter of the square. The circumference of a circle is naturally 360 degrees so any division of the circle’s circumference that is desired must be the result of 360 being divided into rational numbers, meaning no infinite decimals.

• The second method for squaring the circle includes again the circle’s circumference and the square’s perimeter having equal measure, but the square and circle do not share the same centre in this version. None of the 4 corners of either the large square or the smaller square touch the circumference of the circle that is equal to the perimeter of the large square. The harmonies included in this version of squaring the circle are:

1. 1 of the centre edges of the square touch 1 of the poles of the circle’s diameter.

2. If a square that is divisible of 3 equal units of measure is placed above the square that is divisible of 11 equal units of measure while the diameter of the circle is divisible of 14 equal units of measure then 1 of the centre edges of the square that is divisible of 3 equal units of measure will touch 1 of the poles of the circle’s diameter that is divisible of 14 equal units of measure while the opposing pole of the circle with a diameter that is divisible of 14 equal units of measure touches 1 of the centre edges of the square with a perimeter equal to the circumference of a square.

3. If the pole of the circle that has a circumference equal with the perimeter of a square touches the centre edge of the square is used as the South Pole then the centre of the square that has its perimeter equal to the circumference of the circle will below the centre of that circle. The centre of the circle that has a circumference equal to the perimeter of the square is parallel to the centre of that square.

4. If the measure of the square that is divisible of 11 equal units of measure is added to the measure of the square that is divisible of 3 equal units of measure then the result will be the measure for the diameter of the circle that has its circumference equal to the perimeter of the square that is divisible of 11 equal units of measure when the ancient version of Pi that is 22 divided by 7 = 3.142857142857143.

• The third method of squaring the circle includes creating a circle with a diameter of 8.9 units of measure and the diameter of the circle is also the longest length of a Kepler scalene triangle while the second longest length of the Kepler scalene triangle has 7 equal units of measure and is also the edge length of the width of the square. If the diameter of the circle is divided by the shortest edge length of the Kepler scalene triangle the result will be the Golden ratio of 1.618181818181818. The circle and square do not share the same centre in this version of squaring the circle. Only 2 of the 4 corners of the square touch the circumference of the circle that is equal to the perimeter of the square. None of the poles of the diameter for the circle with circumference equal to the perimeter of the square touch any of the edges of the square. The 4 centre edges for the square are parallel to the poles of the circle’s diameter. The centre of the square is parallel to the centre of the circle. If the pole of the circle’s diameter that is above the 2 corners of the square that touch the circumference of the circle then the centre of the square will be below the centre of the circle that has its circumference equal to measure for the square’s perimeter. Both the circumference of the circle and the perimeter of the square are equal. Circumference of circle is 28 and the perimeter of the square is 28. If the circumference of the circle 28 is divided by the diameter of the circle 8.9 then the approximation of Pi that is achieved is 3.146067415730337 but 3.141 0r 3.142 is a better approximation of Pi.

(Regarding the creation of a circle with a circumference equal to the perimeter of a square:

“Please remember that If a circle is created with a circumference equal to the perimeter of a square then the width of the square divided by the radius of the circle is half of Pi = 1.57. Pi is 3.14.” Remember that if a circle and a square are created with equal perimeters meaning that the circumference of the circle is equal in measure to the perimeter of the square then the width of the square will be 1 quarter of the circle’s circumference in measure. Also if half of Pi is multiplied by the radius of the circle then the result will be the width of the square because the width of the square is equal to 1 quarter of the circle’s circumference when a square and a circle are created with the circle’s circumference being in equal in measure to the perimeter of the square.

“Also if a circle is created that has a circumference equal to the perimeter of a square in addition to the circumference of the circle divided by the diameter of the circle the perimeter of the square divided by the diameter of the circle will also be Pi because both the circumference of the circle and the square’s perimeter are of equal measure.” Also if half the perimeter of the square is divided by the circle’s radius then the result is also Pi. Also if a circle and a square are created with perimeters of equal measure meaning that the circumference of the circle is equal in measure to the perimeter of the square then both the measure for the perimeter of the square and the measure for the circumference of the circle can be obtained also if Pi is multiplied 2 times and then multiplied by the measure for the radius of the circle. Please remember that if Pi is multiplied twice the ratio is known as Tau and is also the result of a circle’s circumference being divided by the radius of the circle. Also the measure for both a square’s perimeter that is equal in measure to a circle’s circumference can be known if Pi is multiplied by the measure for the diameter of the circle.

“Please remember that if a circle is created with a circumference equal to the perimeter of a square then the radius of the circle is also the height of a isosceles triangle that has the width of the square for the base of the isosceles triangle. Also if any of the 2 longest lengths of this isosceles triangle that produces half of Pi when the width of the base is divided by the height is then divided by half the width of the base of the isosceles triangle the result will be the Golden ratio.”)

• The fourth method for Squaring the circle includes again the circle and square sharing the same centre. The diameter of the circle must measure 5 equal units while the edge length for the square must measure 4 equal units. If half of 1 cubit is added to the radius of the circle that has a diameter of 5 equal cubits for measure then the new measure is 3 cubits and is equal to any of the 4 sections of the square that intersect the circumference of the circle. Squaring the circle with this method also involves the circle containing two 3 out of 4 edge length rectangles, one horizontal and the other vertical. The shorter measures for the 3 out of 4 rectangles intersect with the circumference of the circle. If the radius of the circle is divided by half of the shorter edge of any of the two 3 our of the 4 rectangles that is formed when the circumference of the circle intersects the edges of the square then the result is 2.5 half divided by 1.5 and equals 1.6666666666667 the smaller part of the first approximation for the Golden ratio of 1.618. Also half of the shorter edge of the 3 out of 4 rectangle is equal to one tenth of the circle’s circumference the edge of a decagon. Remember again that the perimeter of the Square with 4 equal cubits is equal to 16 cubits and the circumference of the circle above with a diameter of 5 equal cubits is 15.7 cubits when read rationally or 15.714285714286 when read irrationally. 15.7 divided by 5 is the shortest written value of Pi 3.14. 15.714285714286 divided by 5 is also an approximation of Pi 3.1428571428572.

The above method is a close approximation for squaring the circle and the above circle’s circumference is close to 1 measure of unit away from the Perimeter of the square. Remember again that a square with an edge length of 4 equal units of measure has a Perimeter of 16 equal units of measure. The ancient version of Pi 22 divided by 7 = 3.142857142857143 multiplied by a circle with a diameter of 5.0909090909096 equals 16.0000000000016, so this method is close when the 0s are added.

• The fifth method for squaring the circle includes creating a squaring with 4 equal units and a circle with 5 equal units for diameter and the south pole of the diameter of the circle must touch the base of the square. The edge of the square must also be attached to the longest length of a rectangle that measures 4 equal units long and 3 equal units short. The 3 fourths rectangle must share the same centre as the circle that has a diameter of 5 equal units. Please remember that 4 divide by 2 and a half of 1 equals 1.6 and 1.6 is the second approximation of the Golden ratio of 1.618. 1.6 is the ratio of the edge of the square and the long part of the rectangle that measures 4 cubits when compared to the radius of the circle that measures 2 and a half of 1 cubit. This method contains only one 3 out of 4 rectangle meanwhile the method that involves the square and circle both sharing the same centre has two 3 out of rectangles. The distance between the centre of the square and the north pole of the circle with a diameter of 5 cubits is equal to the shorter edge of the 3 out of 4 rectangle that is contained by the circumference of the circle with a diameter of 5 cubits. The shorter measure for the 3 out of rectangle is contained with in the circle while the circumference of the circle intersects one cubit added to the shorter measure for the 3 out of rectangle resulting in the square of 4 cubits being created. The 3 out of 4 rectangle is also part of the square that is made of 4 cubits of equal measure. If the radius of the circle is divided by half of the short edge length for the 3 out of the rectangle the result is 1.6666666666667, the smaller part of the first approximation for the Golden ratio of 1.618 and is also equal to one tenth of the circle’s circumference. Remember again that the perimeter of the Square with 4 equal cubits is equal 16 cubits and the circumference of the circle above with a diameter of 5 equal cubits is 15.7 when read rationally and 15.714285714286 when read irrationally. 15.7 divided by 5 is the shortest written value for Pi 3.14. 15.714285714286 divided by 5 is also an approximation of Pi 3.1428571428572. The above method is also a close approximation for squaring the circle and the above circle’s circumference is close to 1 unit of measure away from the above square’s Perimeter. Remember again that a square with an edge length of 4 equal units of measure has a Perimeter of 16 equal units of measure .The ancient version of Pi 22 divided by 7 = 3.142857142857143 multiplied by a circle with a diameter of 5.0909090909096 equals 16.0000000000016, so this method is close when the 0s are added.

Also if the diameter of this circle with 5 equal units is divided by the short edge of the 3 out of 4 rectangle contained inside this circle the resulting sum is 1.6666666666667 because 5 divide by 3 is 1.6666666666667. Both 5 and 3 are Fibonacci numbers.

• The sixth method for squaring the circle includes the radius of the circle being the shorter part of the square root of 3 and the edge of the square is the larger part of the square root of 3. The width of the square is equal to one third of the circle’s circumference. The square root of 3 is also the edge of an equilateral triangle compared to its radius and also the height of an equilateral triangle compared to half of the edge length of the equilateral triangle. Remember that both the circle with a radius that is equal to the shorter part of the square root of 3 and the square that is equal to the larger part of the square root of 3 for edge length must share the same centre. The parts of the circle’s circumference that intersect the square are all each equal to the radius of the circle. The diameter of the circle is twice as large as any of the parts of the square that intersect with the circle’s circumference .The parts of the square that intersect with the circumference of this circle is equal to the shorter part of the Square root of 3 rectangle and is also equal to one sixth of the circumference of the circle and is also equal to the edge of a Hexagon inscribed around the circumference of the circle that has a radius equal to the shorter part of the square root of 3. Remember again that both the circle and the square must share the same centre and the radius of the circle when compared to the edge of the square must be the shorter part of the square root of 3, so a square with a edge length that is equal to the length of 1 third of this circle’s circumference must share the same centre as this circle. Squaring the circle with this method also involves the circle containing two square root of 3 edge length rectangles, one horizontal and the other vertical. Remember again that 22 divided by 7 is an approximation of Pi 3.142857142857143. Numerically an example of the above method can be stated that if the edge of the square is 4 equal units of measure then the radius will be 2.3094010767585 equal units of measure while the diameter will be twice as much and can read 4.618802153517 equal units of measure. 22 divided by 7 is an approximation of Pi 3.142857142857143. 3.142857142857143 multiplied by the diameter of this circle of 4.618802153517 equal units of measure is 14.516235339625 equal units of measure. So the circumference of the above circle is 14.516235339625 and the perimeter of the square that has edge length of 4 is 16. This square has a Perimeter of 16 and the circumference of the above circle again is 14.516235339625. The circumference of this circle is close to 2 equal units of measure from the Perimeter of the above square.

• The seventh method for squaring the circle includes a square and a circle that shares the same centre and the diameter of the circle in comparison to half the width of the square is the larger length of the square root of 5 ratio. The Square root of 5 is 2.236. The harmony of the part of the circle’s circumference that intersects the square being equal to half the edge length of the square. Squaring the circle with this method also involves the circle containing 2 double square rectangles, one horizontal and the other vertical. The diameter of the circle can be 89.44 while the edge length of the square is 80 equal units of measure and half the width f the square is obviously 40 equal units of measure. The circumference for this method of squaring the circle has 28.09840469370711 equal units of measure according to the modern version of Pi while the square’s perimeter has 32 equal units of measure. According to any version of Pi the perimeter of the square is larger than the circle’s circumference and the circumference of the circle is 28 plus an infinite amount of decimal points.

• The Eighth method for squaring the circle includes a square and a circle that shares the same centre. The diameter of the circle has 4 equal units of measure and the width of the square has 3 equal units of measure. The harmonies of combining a square with a width of 3 equal units of measure with a circle with a diameter of 4 equal units of measure and both the square and the circle share the same centre involves the circle’s circumference being equal in measure to the area of the circle. If the diameter of a circle has 4 equal units of measure then the circumference of that circle is 12.566370614359172 and 12.566370614359172 reduced to 4 decimal places is 12.56. Pi = 3.141592653589793 and 3.141592653589793 multiplied by the radius of a circle with 2 equal units of measure is Tau = 6.283185307179586. If Tau – 6.283185307179586 is multiplied by 2 equal times then the area of the circle is 12.566370614359172 and 12.566370614359172 is the same value for the circumference of a circle with a radius of 2 equal units of measure. The width of this square has 3 equal units of measure and the perimeter of a square with a width of 3 equal units of measure is 12 equal units of measure. So the perimeter of this square with a width of 3 equal units of measure and the radius of the circle with 2 equal units of measure involves the perimeter of the square with a width of 3 equal units of measure being close in measure to the circumference and area of a circle that has a radius equal in measure to 2 equal units of measure. Circumference of circle is 12.566370614359172 and area of circle is also 12.566370614359172. 12.566370614359172 reduced to 4 decimal places is 12.56 and the perimeter of a square with a width of 3 equal units of measure is 12 equal units of measure.

If the radius of the circle with 2 equal units of measure is divided by the width of the square with 3 equal units of measure then the result is the inverse of the Golden ratio approximated to: 0.666666666666667.

If the hypotenuse that is located between the radius of a circle with 2 equal units of measure and half the central width of a square with 3 equal units of measure is divided by half the width if the square with 3 equal units of measure the result is the Golden ratio approximated to: 1.666666666666667.

A square with a width of 3 equal units of measure and a circle with a radius of 2 equal units of measure encodes the therom of Pythagoras when both the circle with a radius of 2 equal units of measure and the square with a width of 3 equal units of measure share the same centre because the scalene triangle that is made from the radius of the circle with 2 equal units of measure and half the central square width of 3 equal units of measure is based upon a 3-4-5 scalene triangle with its edge lengths being half of the 3 lengths of 3-4-5 scalene triangle.

The hypotenuse of a scalene triangle with its second longest length the adjacent edge being equal to 2 equal units of measure and the shortest length being the opposite edge of the scalene results in the measure of the hypotenuse of this scalene triangle being 53.13010235415598 degrees and 53.13010235415598 reduced to 4 decimal places is 53.13 degrees.

A square with a width of 3 equal units of measure and a circle with a radius of 2 equal units of measure produce the ratio of 0.75 or 3 quarters when half the width of the square is divided by the radius of the circle with 2 equal units of measure. A square with a width of 3 equal units of measure and a circle with a radius of 2 equal units of measure produces the ratio of 1.25 or 1equal unit of measure plus a quarter of 1 equal unit of measure when the hypotenuse of the scalene triangle that is formed between the radius of the circle with 2 equal units of measure and half the central width of the square with 3 equal units of measure is divided by the radius of the circle with 2 equal units of measure. The diameter of a circle with 4 equal units of measure divided by the width of a square with 3 equal units of measure produces the ratio: 1.333333333333333. The ratio 1.333333333333333 when applied to the inverse of Tangent in Trigonometry produces the measure angle of 53.13010235415598 degrees. 53.13010235415598 degrees reduced to 4 decimal places is 53.13 degrees.

• The Ninth method for squaring the circle includes a square and a circle that do not share the same centre. In this diagram the central base width of the square is located on 1 of the poles for 1 of the diameters of the circle. The diameter of the circle has 4 equal units of measure and the width of the square has 3 equal units of measure. The harmonies of combining a square with a width of 3 equal units of measure with a circle with a diameter of 4 equal units of measure and both the square and the circle share the same centre involves the circle’s circumference being equal in measure to the area of the circle. If the diameter of a circle has 4 equal units of measure then the circumference of that circle is 12.566370614359172 and 12.566370614359172 reduced to 4 decimal places is 12.56. Pi = 3.141592653589793 and 3.141592653589793 multiplied by the radius of a circle with 2 equal units of measure is Tau = 6.283185307179586. If Tau – 6.283185307179586 is multiplied by 2 equal times then the area of the circle is 12.566370614359172 and the circumference of the circle is also 12.566370614359172. The width of this square has 3 equal units of measure and the perimeter of a square with a width of 3 equal units of measure is 12 equal units of measure. So the perimeter of this square with a width of 3 equal units of measure and the radius of the circle with 2 equal units of measure involves the perimeter of the square with a width of 3 equal units of measure being close in measure to the circumference and area of a circle that has a radius equal in measure to 2 equal units of measure. Circumference of circle is 12.566370614359172 and area of circle is also 12.566370614359172. 12.566370614359172 reduced to 4 decimal places is 12.56 and the perimeter of a square with a width of 3 equal units of measure is 12 equal units of measure.

The diameter of a circle with 4 equal units of measure divided by the width of a square with 3 equal units of measure produces the ratio: 1.333333333333333. The ratio 1.333333333333333 when applied to the inverse of Tangent in Trigonometry produces the measure angle of 53.13010235415598 degrees. 53.13010235415598 degrees reduced to 4 decimal places is 53.13 degrees.

The diameter of the circle has 4 equal units of measure and if 3 quarters of the circle’s diameter are used to create the length of a line then the length of that line will be equal in measure to the width of this appropriate square. The width of the square is 3 equal units of measure and the diameter of the circle is 4 equal units of measure and 3 divided by 4 is the ratio 0.75 or 3 quarters of 1.

• The Golden ratio and also the square root of 3 and also other irrational ratios can help with approximating the squaring of the circle.

The double square and the square root of 5 and also the Fibonacci sequence is required to obtain the Golden ratio and the Golden ratio and also the ratio 1.333333333333333 and also the square root of 3 and also the ratio of 9 divide by 8 and the ratio of 1.128379167095513 is also required for approximating the Squaring of the circle. In the past numerous attempts have been made by Geometrical mathematicians to Square the circle with equal perimeters and those attempts all failed because the attempts that were made by those Geometrical mathematicians to Square the circle with equal perimeters included the use of the modern version of the irrational number Pi 3.141592653589793 with out realizing the modern version of Pi must be approximated and that means the results given to us by the modern version of Pi must be reduced to 4 decimal places resulting in other values of Pi that are approximations of modern Pi. It should now be evident that the modern version of the irrational and transcendental number Pi cannot be used to Square the circle without our current value of Pi being approximated resulting in many other values of Pi. Squaring the circle can never be achieved with complete accuracy when using the modern version of Pi = 3.141592653589793 without the modern version of Pi 3.141592653589793 being approximated. Pi must be approximated. Instead the 13 ancient approximate versions of Pi should be used to know the ratio of a circle’s circumference compared to the circle’s diameter. The modern version of Pi can be good also because the modern version of Pi = 3.141592653589793 can be used to discover the 13 ancient approximate versions of Pi and also the modern version of Pi can be used to discover more approximate versions of Pi when the result is reduced to 4 or 5 decimal places.

Squaring the circle can also be achieved with less accuracy than the 9 methods mentioned above by combining 2 rectangles of equal size and proportion inside a circle with 1 rectangle being horizontal and the other rectangle being vertical. Both rectangles must be extended to form a square that has its four vertices extended beyond the circle’s circumference similar to previous examples of how the circle and square share the same centre.

Liddz says

Pi encoded in the measures of Earth and moon and the Great Egyptian pyramid of Giza:

The equatorial perimeter of our planet Earth divided by the combined equatorial diameters of both our Earth and our Earth’s moon = Pi approximated to 22 divided by 7 = 3.142857142857143. If a circle is created with a diameter equal in measure to 10080 statute miles the circumference of that circle will be equal in measure to the equatorial perimeter of our planet Earth. The equatorial diameter of our planet Earth is 7920 statute miles and the equatorial perimeter of our planet Earth is 31680 statute miles. The equatorial diameter of our Earth’s moon is 2160 statute miles. 7920 statute miles plus 2160 statute miles = 10080 statute miles. If a Pyramid is created with a square base width that is equal in measure to the equatorial diameter of our planet Earth = 7920 statute miles and the height of the pyramid is equal in measure to the combined equatorial radiuses of both our planet Earth = 3960 statute miles and our Earth’s moon’s equatorial radius = 2160 statute miles, the slant height of that pyramid is equal to the slant height of the Great Egyptian pyramid of Giza. 5040 statute miles is the measure for the combined equatorial radiuses of both our planet Earth and our Earth’s moon.3960 + 2160 = 5040. 5040 divided by 3960 = 1.272727272727273. In Trigonometry the ratio 1.272727272727273 when applied to the inverse of Tangent is 51.84277341263095 degrees. 51.84277341263095 degrees is also the slant height of the Great Egyptian pyramid of Giza because the height of the Great Egyptian pyramid of Giza is made with 280 cubits and half the width of the Great Egyptian Pyramid’s square base has 220 cubits for measure. 280 divided by 220 = 1.272727272727273.: https://www.youtube.com/watch?v=gStpRbF5o4Y&t=224s

Liddz says

Ultra Pyramid.

@SGD 22 divided by 7 is Pi approximated to: 3.142857142857143.

4 divided by pi approximated to: 3.142857142857143 is 1.272727272727273.

If 1.272727272727273 is applied to the inverse of Tangent in Trigonometry the measuring angle in degrees is 51.84277341263095.

51.84277341263095 reduced to 4 decimal places is 51.84.

7.2 squared is 51.84

51.84 multiplied by 500 is 25920. 25920 is the amount of years for the Sun to travel through the 12 constellations of the Zodiac.

51.84277341263095 is the angle for the slant height of a pyramid with a edge width of 110 equal units of measure and a height of 70 equal units of measure.

(220 is half of this Pyramid’s perimeter of square base)

(89 is the measure for the slant height of this Pyramid)

(12100 is the number of square equal units in the area of this Pyramid’s square base). “220 x 89 = 19580 + 12100 = 31680.

The surface area of this Pyramid is 31680 equal square units of measure and the area of this Pyramid’s square base is 12100 equal square units of measure. If the surface area of this Pyramid 31680 equal square units of measure is divided by the area of the square base that’s 12100 equal square units of measure the result will is the Golden ratio squared: 2.618181818181818. Also if the surface area of this Pyramid 31680 equal units of measure is divided into the Golden ratio the larger segment of the division is 19580 the result of half of this Pyramid’s square perimeter multiplied by the slant height of this Pyramid 89, while the smaller segment of the division is the area of this Pyramid’s square base 12100 equal square units of measure. 19580 divided by 12100 = 1.618181818181818. 19580 + 12100 = 31680. Half of this Pyramid’s square base perimeter multiplied by the central slant of this pyramid divided by the area of the square base of this Pyramid = 1.618181818181818. The volume of this Pyramid is 282333 equal units of measure. The longest edge lengths for the 4 isosceles triangles that make up this Pyramid is 104.64 equal units of measure.

31680 feet is also equal to 6 miles. 31680 miles is also equal to the equatorial perimeter of our Earth. If the circumference of a circle has 31680 equal units of measure then the diameter of that circle can have 10080 equal units of measure according to pi approximated to : 3.142857142857143.

A pyramid with a square base width of 110 equal units of measure and a height of 70 equal units of measure has 440 cubits for the width of the square base and 280 cubits for the square base width and this reminds us that a cubit is 1 quarter of a equal unit of measure because 110 divided by 440 and 70 divided by 280 is 0.25 and .025 is 1 quarter of 1.

Liddz says

Golden ratio Lidium

From: Liddz.

72.81 divided by 45 = 1.618. Also 40.45 divided by 25 = 1.618. Also. 9.708 divided by 6 = 1.618. The Golden ratio can also be obtained in Trigonometry through the formula Cosine (36) = 0.809016994374948 multiplied by 2 = 1.618033988749895. The formula for the Golden ratio in Trigonometry can be written in short as Cosine (36) = 0.809016994374948 x 2 = 1.618033988749895.

The Golden ratio can also be obtained in Trigonometry through the formula Cosine (72) = 0.309016994374948 multiplied by 2 = 0.618033988749895 plus 1 = 1.618033988749895. The formula for the Golden ratio in Trigonometry can be written in short as Cosine (72) = 0.309016994374948 x 2 = 0.618033988749895 + 1 = 1.618033988749895. The Golden ratio can also be obtained in Geometry through Cosines of both multiples of 36 and 72 and remember to multiply the result twice to get the Golden ratio. The Golden ratio in Trigonometry can also be obtained through the formula 1+SIN (18) = 1.309016994374947 x 2 = 2.618033988749895 – 1 = 1.618033988749895. Please remember that 2.618033988749895 is The Golden section or Golden ratio squared and also has many applications. The Golden ratio squared can also be obtained in Trigonometry through the formula Cosine (36) x 2 + 1 = 2.618033988749895. The Golden ratio squared can also be obtained in Trigonometry through the formula 2+SIN (18) x 2 = 2.618033988749895.

Numerically with a digital calculator the Golden ratio is expressed as 1.618. The Golden ratio is connected to the square root of 5 and can also be obtained from the Square root of 5 by adding 1 to the Square root of 5 and then dividing the Square root of 5 plus 1 into 2. An example of obtaining the Golden ratio is 2.236 plus 1 equals 3.236. 3.236 divide by 2 equals 1.618. The Golden ratio can also be approximated from a mathematical infinite progression called the Fibonacci series with the remainder being the result of adding together the previous 2 numbers in the sequence. The Fibonacci progression is as follows: 1plus1 equals 2, 2 plus 1 equals 3, 3 plus 2 equals 5, 5 plus 3 equals 8, 8 add 5 equals 13, 13 add 8 equals 21, 21 add 13 equals 34, 34 add 21 equals 55, 55 add 34 equals 89. 89 divide by 55 is: 1.618181818181818818181818181818. The sequence can also work with the numbers above being doubled such as 2,4,6,10,16,26,42,68,110,178. 178 divide by 110 is 1.618181818181818818181818181818. The further the progression is extended the closer to view the Golden ratio becomes.

Also 72.81 divided 45 = 1.618. Also 40.45 divided by 25 = 1.618. 9.708 divided by 6 = 1.618.

The Golden ratio can also be approximated very closely if 121393 divided by 75025 = 1.618033988670443. If 121393 is added to 75025 the result is 196418. If 196418 is divided by 121393 the result is 1.618033988780243.

If 196418 is added to 121393 the result is 317811. 317811 divided by 196418 = 1.618033988738303. If 317811 is added to 196418 = 514229. 514229 divided by 317811 = 1.618033988754323. If 514229 is added to 317811 the result is 832040. 832040 divided by 514229 = 1.618033988748204.

Please note the repeating numbers 1.6180339887….. after 196418 has been divided by 121393. So the Golden ratio can be approximated to 1.6180339887 plus an infinite amount of decimal point numbers that never repeat. So the Golden ratio can be reduced to 1.6180339887 because after 1.6180339887 there is an infinite amount of decimal point numbers that are never repeated but 1.6180339887 is repeated after 196418 is divided by 121393.

The Golden ratio can also be obtained in Trigonometry through the formula Cosine (36) = 0.809016994374948 multiplied by 2 = 1.618033988749895. The formula for the Golden ratio in Trigonometry can be written in short as Cosine (36) = 0.809016994374948 x 2 = 1.618033988749895.

The Golden ratio can also be obtained in Trigonometry through the formula Cosine (72) = 0.309016994374948 multiplied by 2 = 0.618033988749895 plus 1 = 1.618033988749895. The formula for the Golden ratio in Trigonometry can be written in short as Cosine (72) = 0.309016994374948 x 2 = 0.618033988749895 + 1 = 1.618033988749895. The Golden ratio can also be obtained in Geometry through Cosines of both multiples of 36 and 72 and remember to multiply the result twice to get the Golden ratio. The Golden ratio in Trigonometry can also be obtained through the formula 1+SIN (18) = 1.309016994374947 x 2 = 2.618033988749895 – 1 = 1.618033988749895. Please remember that 2.618033988749895 is The Golden section or Golden ratio squared and also has many applications. The Golden ratio squared can also be obtained in Trigonometry through the formula Cosine (36) x 2 + 1 = 2.618033988749895. The Golden ratio squared can also be obtained in Trigonometry through the formula 2+SIN (18) x 2 = 2.618033988749895.

Many mathematicians have called the Golden ratio the most pleasing ratio in existence.

Constructing Golden logarithmic spirals that all share the same centre in a circle:

A circle can be constructed with opposing logarithmic spirals originating from the centre of the circle. To divide a circle into spirals construct a rectangle that is based upon the Golden ratio with the longer length of the Golden rectangle with equal length to the radius of the circle. Now construct a second Golden rectangle that has its longest edge the same as the shorter edge of the larger Golden rectangle. Both of the large and small Golden rectangles must have an edge that emanates from the centre of the circle. The diagonals of the 2 Golden rectangles must then be divided into the golden ratio the same way that 3 divisions of the longest width of a Pentagram is divided. One rectangle must have a diagonal that is 31.71 degrees and the other rectangle must have a diagonal of 58.28 degrees or if the choice is to draw the rectangles tilted then another combination of 2 angles that add up to 90 degrees can be used just like both 31.71 and 58.28 add up to 90. 31.71 degrees and 58.28 degrees must come from the centre of the circle and can expand to infinity by the Golden ratio. Any 1 of the 2 longer divisions of the diagonals that are located on the sides of the smaller middle section must then be used to extend the diagonals through the centre of the circle, allowing 2 new larger golden rectangles to be constructed. Please remember for the purpose of accuracy that when a smaller Golden rectangle shares it’s longest edge length with the shorter edge length of a larger Golden rectangle the full length of the whole rectangle will have a ratio of the square root of 5. The longer length of the shorter Golden rectangle is the shorter length of the Square root of 5 rectangle. Remember again that the longest length of the larger Golden rectangle is equal to the radius of the circle so that the spirals that are drawn from the squares that make the Golden rectangle can all originate from the centre of the circle. Now Golden spirals can be constructed from the largest Golden rectangles with the spirals attempting to emanate from the centre of the circle. The circumference of the circle can be divided into equal sections with the process of constructing Golden spirals that attempt to emanate from the centre of the circle being repeated to infinity if desired by expanding the radius of the circle to form new circles that share the same centre with the method of the Fibonacci progression.

The radius of the circles that all share the same centre must expand by the Golden ratio if the sizes of the spirals are to be increased. If opposing spirals are drawn from the centre of the circle and are on opposite sides and above each other then they can be interpreted as the spirals of a Galaxy. Please note that an unfortunate fact exists and that is the spiral or spirals that can be created from the squares that form a Golden rectangle cannot completely touch the centre of the circle even though the spirals do come close to touching the centre of the circle. The reason why using squares that form the Golden rectangle to create spirals that are desired to touch the centre of the circle cannot be achieved 100% is because the Squares that form the Golden rectangle are created in ratio to each other based upon the Fibonacci sequence and the Fibonacci sequence begins with 1 plus 1 then 2 plus 1 equaling 3, so to obtain the size of the next square the number that proceeded the current number must be added to the current number.

If the Geometer is determined to make the Golden spiral proceed to the centre of the circle that must have both 31.71 degrees and 58.28 degrees emanating from the centre of the circle to indicate the point for the spiral to emanate then the only solution is to divide 1 of the 2 smallest squares into a Golden rectangle. The Golden rectangle that allows the logarithmic spiral to reach the centre of the circle must have its longest edge length equal to 1 of the 2 smallest squares that make up the whole largest Golden rectangle. The Golden rectangle that has its longest edge length equal to 1 of the 2 smallest squares that make up the largest Golden rectangle must also be divided into squares if possible.

Constructing a Golden rectangle inside of any random circle:

Please remember that a Golden rectangle can be constructed inside any random circle when a double square-rectangle is constructed inside of that circle that has the diameter of the circle being equal to the diagonal of the rectangle that is made of 2 squares and is constructed inside of the circle. Please remember that if a Golden rectangle is placed on the 2 parallel edges of a square then the total height of the new rectangle is Square root of 5. This knowledge is also used to construct an Icosahedron inside of a Sphere.

• The Golden ratio:

The Golden section is a ratio that has a line with the small section being the same as the larger section is to the whole length of the line. The Golden ratio first appears when a circle is divided into 5 equal parts producing a Pentagon with the Pentagon’s diagonals being connected to the 5 vertices of the Pentagon inside the circle touching the circle’s circumference. Applications for the Golden ratio are many and the author of this book does NOT have enough space to list all of them here. Some of the applications for the Golden ratio include, constructing a Pentagon inside a circle.

“Using the Golden ratio to construct a Pentagon and also a Decagon”:

The Golden ratio must be used to construct a Pentagon because the edge of a Pentagon in relation to it’s diagonal is the Golden ratio. Here is another method for constructing a Pentagon inside of a circle .The Golden ratio can also be used to divide the circumference of a circle into 10 equal parts because one tenth of a circle’s circumference is the smaller section of the Golden ratio in relation to the radius of a circle. Remember that a Pentagon is half of a Decagon. The radius of a circle can be divided into the Golden ratio Geometrically with just the use of a compass and a straight edge and obviously a stylus by first dividing the radius of a circle into half by constructing a Vesica Pisces with the base being equal to the radius of the circle, the base of the Vesica Pisces and the radius of the circle must share the same base. Next connect a line between the 2 opposing apexes of the Vesica Pisces so that the radius of the circle can be divided into 2. Next construct a semi-circle or a circle with the radius of the large circle that will contain the Pentagon being the diameter of the smaller circle, so that a rectangle made of double squares can be constructed on the radius of the large circle. The radius of the large circle must be the long edge of the rectangle that is made from 2 squares, the short length of the rectangle that is made from double squares is equal to half of the radius of the large circle that contains the Pentagon. Next swing an arc with a compass with a measurement of the short part of the rectangle that is made from double squares onto the diagonal of the rectangle that is made from double squares, next swing another arc from the larger division of the diagonal of the rectangle that is made from double squares onto the radius of the large circle that contains the Pentagon. The radius of the circle that is to contain the Pentagon has been divided into the Golden ratio, take the larger part of the circle’s radius that has been divided into the Golden ratio and multiply it 2 times from the point on the circumference of the circle that has the radius of the circle touching it so that one fifth of the circle’s circumference can be obtained. Now that one fifth of the circle’s circumference has been obtained by dividing the radius of the circle into the golden section and multiplying the larger part twice, next multiply one fifth of the circle’s circumference until the Pentagon is completed and connect the diagonals of the Pentagon up to the 5 vertices of the circle’s circumference so that a 5 pointed star that is based upon the Golden ratio can be observed.

“Inscribing a Pentagon in a circle when only the measure for the radius or diameter of the circle is known or finding the measure for the Pentagon’s radius”:

A method for determining the measure for the edge of a Pentagon when only the measure for the radius or diameter of the Pentagon’s circle is known is to add the result of the circle’s radius squared to the result of the circle’s decagon squared and then apply the result of adding together both the results of the circle’s radius squared and the circle’s decagon squared to square root. Remember the edge of a decagon is obtained by dividing the circle’s radius into the Golden ratio of 1.6180339887498. The edge of a decagon can easily be obtained by also dividing the circle’s circumference into 10 equal units of measure.

Another solution for determining the edge length of a Pentagon when only the radius and diameter of the Pentagon’s circle is known is to take the measure of the diameter of the Pentagon being added to the edge of a decagon that has a radius equal in measure to the Pentagon’s radius must be squared and added to the measure for the Pentagon’s diameter squared and then the result must be applied to square root and the result will be the hypotenuse of a scalene triangle with its shortest length being equal to the diameter of the Pentagon and the second longest length of this scalene triangle is equal in measure to the measure for the Pentagon’s diameter being added to and combined with the edge of a decagon with a radius equal in measure to the Pentagon’s radius. The hypotenuse of the scalene triangle with its shortest length being equal in measure to the Pentagon’s diameter while the second longest length of the scalene triangle is equal in measure to the Pentagons diameter plus the edge of a decagon with a radius equal in measure to the Pentagon’s radius must be divided into the square root of 3 = 1.732050807568877 and the result of dividing the hypotenuse of this scalene triangle into the square root of 3 = 1.732050807568877 and the larger part of the division of the hypotenuse into the square root of 3 must be divided into the Golden ratio and the larger part of that new division from the hypotenuse being divided into the Golden ratio is the measure for the edge length of the Pentagon. The equation above can be simply written as:

1. Diameter of circle added to the length of the edge of the circle’s decagon.

2. Diameter of circle added to the length of the edge of a decagon squared and the results remembered.

3. Diameter of circle squared and the results remembered.

4. The result of the circles diameter squared added to the length of the circle’s diameter combined with the circle’s decagon edge length squared

5. The sum of both the circle’s diameter squared and the circle’s diameter plus the edge length of the circle’s decagon squared and the result of both added together now being applied to Square root.

6. The result of both the circle’s diameter squared and the circle’s diameter plus the edge length of the circle’s decagon squared and the result of both added together now being applied to Square root results in a measure that must be divided by the square root of 3 = 1.732050807568877.

7. The result of both the circle’s diameter squared and the circle’s diameter plus the edge length of the circle’s decagon squared and the result of both added together now being applied to Square root results in a measure that must be divided by the square root of 3 = 1.732050807568877 and then divided by the Golden ratio of 1.6180339887498 and the result is the measure for the edge of the circle’s Pentagon.

Another method for determining the measure for the edge of a Pentagon when only the measure for the radius or diameter of the Pentagon’s circle is known is to multiply the measure for the radius of the Pentagon by: Cosine (54) multiplied by 2 So if the radius of the Pentagon is 55 equal units of measure then multiply 55 by Cosine (54) multiplied by 2 and the result for the edge of the Pentagon with a radius of 55 equal units of measure is 64.656377752172052. Remember to reduce the results to 6 or 5 or 4 or 2 decimal places then the measure for the edge of the Pentagon can also be known in addition to the radius and diameter for the Pentagon’s circle already being known. The ratio 1.175570504584946 is the result of dividing the edge length of a Pentagon by the radius of a Pentagon and can be obtained in Trigonometry through the formula Cosine (54) = 0.587785252292473 multiplied by 2 = 1.175570504584946. The ratio 1.175570504584946 can be achieved through the short written formula in Trigonometry as Cosine (54) = 0.587785252292473 x 2 = 1.175570504584946. The ratio 1.175570504584946 reduced to 4 decimal places reads as 1.175.

If only the measure for the edge of the Pentagon is known and the desire is to also know the measure for the Pentagon’s radius and diameter a solution is to divide the measure for the edge of the Pentagon by Cosine (54) and the result can be the diameter for the Pentagon and the radius of the Pentagon is obtained by dividing the diameter of the Pentagon by 2 equal halves. So if the edge length of the Pentagon has 34 equal units of the measure and the desire is to know the diameter and radius of a Pentagon with a edge length of 34 equal units of measure the solution can be written as 34 divided by Cosine (54) = 57.844254967938708 equal units is the measure of a diameter for a Pentagon with 34 equal units of measure and 28.922127483969354 equal units is the measure for the radius of a Pentagon with a edge length of 34 equal units of measure. Written shorter the formula for obtaining the radius of a Pentagon with an edge length of 34 equal units of measure is 34/Cosine (54) = diameter of Pentagon = 57.844254967938708. Diameter of Pentagon = 57.844254967938708 divided by 2 = radius of Pentagon: 28.922127483969354.

If only the measure for the edge of the Pentagon is known and the desire is to also know the measure for the Pentagon’s radius and diameter a solution is to divide the measure for the edge of the Pentagon by 0.6180339887498 or multiply the measure for the edge of the Pentagon by 1.6180339887498 and the result is the length of the Pentagon’s diagonal. Multiply the Pentagon’s diagonal by the Square root of 3 = 1.732050807568877. Remember the result of the Pentagon’s diagonal being multiplied by the Square root of 3. Divide the result of the Pentagon’s diagonal being multiplied by the Square root of 3 = 1.732050807568877 by the Square root of 5 = 2.23606797749979. Remember the result of the Pentagon’s diagonal being multiplied by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = 2.23606797749979. The result of the Pentagon’s diagonal being multiplied by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = must now be squared and the result must also be remembered. The result of the Pentagon’s diagonal being multiplied by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = 2.23606797749979 must now be divided into the Golden ratio of 1.6180339887498 and the result must also be squared and remembered. The results of the Pentagon’s diagonal being multiplied by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = 2.23606797749979 and then squared must then be added to the result of the Pentagon’s diagonal being multiplied by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = 2.23606797749979 and then being divided by the Golden ratio of 1.6180339887498 and then squared and then applied to Square root. The results of the Pentagon’s diagonal being multiplied by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = 2.23606797749979 and then squared must then be added to the result of the Pentagon’s diagonal being multiplied by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = 2.23606797749979 and then being divided by the Golden ratio of 1.6180339887498 and then squared and then applied to Square root and then divided by the Square root of 3 = 1.732050807568877 and the new result can now be the radius of the Pentagon. So a simple formula for the equation above can be written as:

1. Edge of Pentagon divided by 0.6180339887498 or Edge of Pentagon multiplied by 1.6180339887498 resulting in the measure for the Pentagon’s diagonal.

2. Pentagon’s diagonal multiplied by the Square root of 3 = 1.732050807568877.The result of the length of the Pentagon’s diagonal being divided by the Square root of 3 = 1.732050807568877 must then be divided by the Square root of 5 = 2.23606797749979.

3. The result of the length of the Pentagon’s diagonal being divided by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = 2.23606797749979 must now be squared and remembered.

4. The result of the length of the Pentagon’s diagonal being divided by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = 2.23606797749979 and then divided into the Golden ratio of 1.6180339887498.

5. The result of the length of the Pentagon’s diagonal being divided by the Square root of 3 = 1.732050807568877 and then be divided by the Square root of 5 = 2.23606797749979 and then divided into the Golden ratio of 1.6180339887498 and then squared and remembered.

6. The result of the length of the Pentagon’s diagonal being divided by the Square root of 3 = 1.732050807568877 and then be divided by the Square root of 5 = 2.23606797749979 and then squared and remembered must now be added to the length of the Pentagon’s diagonal being multiplied by the Square root of 3 = 1.732050807568877 and then divided by the Square root of 5 = 2.23606797749979 and then being divided into the Golden ratio of 1.6180339887498 and squared.

Now that the length of the Pentagon’s diagonal has been multiplied by the Square root of 3 = 1.732050807568877 and the result of such has been divided by the Square root of 5 = 2.23606797749979 and then squared and remembered. The result of the length of the Pentagon’s diagonal multiplied by the Square root of 3 = 1.732050807568877 has been divided by the Square root of 5 = 2.23606797749979 and has be divided by the Golden ratio of 1.6180339887498 and squared and remembered. The sum of the result of the length of the Pentagon’s diagonal being divided by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = 2.23606797749979 and then divided by the Golden ratio of 1.6180339887498 and then squared and remembered must now be added to the result of the length of the Pentagon’s diagonal being divided by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = 2.23606797749979 squared and remembered and the result is the measure for the length of the Pentagon’s radius.

If only the measure for the edge of the Pentagon is known and the desire is to also know the measure for the Pentagon’s radius and diameter of the Pentagon another solution is to:

1. Have half of the edge of the Pentagon squared and the result remembered.

2. Half of the edge of the Pentagon multiplied by the ratio 1.376381920471173 and the result also squared and remembered. The ratio 1.376381920471173 can be obtained on a digital calculator through the Trigonometric formula TAN (54). .

3. Add the result of half of the edge of the Pentagon squared to the result of half of the edge of the Pentagon multiplied by 1.376381920471173 and then apply the result of both combined to Square root and the final result is the measure for the radius of the Pentagon.

If only the measure for the edge of the Pentagon is known and the desire is to also know the measure for the Pentagon’s radius and diameter of the Pentagon another solution is to:

1. Multiply half the edge length of the Pentagon by TAN (54) = 1.376381920471173

2. Divide the result of multiplying the edge length of the Pentagon by TAN (54) = 1.376381920471173 by the Golden ratio of 1.618033988749895.

3. The result of the edge length for the Pentagon being multiplied by TAN (54) = 1.376381920471173 and then divided into the Golden ratio of 1.618033988749895 must now be multiplied by 2 resulting in the measure for the radius of the Pentagon.

“How to obtain the height of a Pentagon”:

The height of a Pentagon can be obtained by:

1. Multiplying half the edge of the Pentagon by the ratio 3.077683537175253 and the result will be the height of the Pentagon. The ratio 3.077683537175253 can be obtained digitally on a calculator through the Trigonometric formula TAN (72).

2. The height of a Pentagon can also be obtained if half of the edge of the Pentagon is multiplied by the ratio 1.376381920471173 and the result then added to the radius of the Pentagon from the centre of the circle. The ratio 1.376381920471173 can be obtained from a calculator through the Trigonometric formula TAN (54).

3. If the radius of a circle is divided in half and then divided by 0.6180339887498 or multiplied by the Golden ratio of 1.6180339887498 and then the result then added to the radius of the circle from the centre of the circle the total length is equal to the height of a Pentagon that can be created to fit in the circle.

“How to obtain the radius of a Pentagon with Trigonometry if the height of the Pentagon is already known”:

So the height of the Pentagon has already been determined by multiplying half the edge of the Pentagon by TAN (72) = 3.077683537175253 and the desire is to know the radius of the Pentagon and a simple solution is to multiply half the edge of the Pentagon by TAN (54) = 1.376381920471173 and subtract the result of multiplying the edge of the Pentagon by TAN (54) = 1.376381920471173 from the height of the Pentagon and the result is the radius of the Pentagon.

If only the measure for the edge of the Pentagon is known and the desire is to also know the measure for the Pentagon’s radius and diameter another solution is to divide the measure for the edge of the Pentagon by 1.170 or 1.175 or 1.175570504584997 or 1.175851692873989 or 1.175656984785615 or 1.1756124220922 or 1.175454545454545 or 1.175636363636364 or 1.175505023709698 or 1.175570504584965 and the result will be an approximation for the radius of the Pentagon and the result must be reduced to 6 or 5 or 4 or 2 decimal places for better accuracy. Remember that if the radius of a circle is divided into the Golden ratio the result is the larger part of the division being equal to the measure for the edge of a decagon that has a radius equal in measure to the radius of the circle.

Geometrically the radius of a Pentagon can become visible if all the 5 edges of the Pentagon are divided in half with lines that pass through the centre of the Pentagon’s circle that also touch the centre edges of the Pentagon.

It should become evident from reading above that any line can be divided into the Golden ratio when a double square is constructed over that line and then swing an arc with a compass measurement equal to half of the double square on to the diagonal of the double square and then swinging an arc with compass measurement of the larger division of the diagonal of the rectangle that is made from double the squares. If the desire is to find the shorter part of the Golden ratio when you have a line and your desire is to make the line longer the use of a square that is divided into half so that the double square can be used by swinging an arc from the centre edge of the large square with a compass measurement equal to the diagonal of the 2 rectangles that are formed of double squares that also make up the largest square onto the extension of the line. Remember that a square is also made up from 4 smaller squares. The height of a Pentagon can also be found if a Golden rectangle is constructed from the centre of the circle that contains the Pentagon with the shorter edge of the Golden rectangle being equal to half the radius of the circle that contains the Pentagon.

• Constructing the shape of the Great pyramid.

Constructing the Great pyramid isosceles triangle from 2 Golden rectangles”:

The slope that is located in the centre of the 4 triangular faces of the largest of the 3 of the Egyptian Great pyramids of Giza can be constructed from 2 vertical golden rectangles. When constructing the central slope of the largest of the 3 Great Egyptian Pyramids an arc that has equal measure to the longest length of any of the 2 golden rectangles can be used and swung unto the centre of the rectangle that is made from the 2 Golden rectangles. The result will be a triangle that has a base length that is the longest edge and equal to 11 units of equal measure while the height of the triangle will be 7 units of equal measure relating the edge length that is the base of the triangle.

“Constructing the Great pyramid triangle from 2 over lapping circles called the Vesica Pisces”:

The triangle that forms the 11 and 7 ratio triangle also forms the central slope of the largest of the 3 Great Egyptian Giza Pyramids and can also be obtained from 2 over lapping circles of equal size that have circumferences that touch each other’s centers. The height of the Pyramid’s triangle is 7 equal units while the base is 11 equal units and the height of the Pyramid’s triangle shares the area called the Vesica Pisces. When the Vesica Pisces is used to construct the shape of a Pyramid that has a square base width of 11 equal units of measure and 7 equal units of measure for the height of this Pyramid triangle then the height of the this pyramid triangle will be equal to the edge of an equilateral triangle that is contained with in the circumference of any of the 2 circles that are used to make the Vesica Pisces.

“Constructing the shape of the Great pyramid triangle upon the diameters of a circle:”

To construct the Great pyramid isosceles triangle upon any of the diameters of a circle the radius of the circle must be multiplied around the circumference of the circle 2 times from any of the circles poles that are on the end of the circles diameters. The result of multiplying the radius of the circle 2 times around the circumference of the circle from any of the circle’s pole diameters is the creation of a measure that is equal to one third of the circle’s circumference. From each end of the measure that is equal to one third of the circle’s circumference swing arcs down on the opposing diameter resulting in the creation of the Great pyramid triangle with the radius of the circle being the height of this Great pyramid isosceles triangle. The apex of this created Great pyramid isosceles triangle is also located on the pole of the appropriate diameter of the circle being also located on the circumference of the circle.

• Golden ratio derived from constructing a scalene triangle based upon the ratio of the moon and Earth touching each other ‘s circumferences:

The ratio of the radius and diameter of the Earth compared to the radius and diameter of the moon is 11 equal units of measure compared to 3 equal units of measure for the diameter of the moon. So the radius and Diameter of the Earth is 11 equal units of measure and from this measure of 11 equal units of measure 3 units of equal measure can be used to obtain the diameter of the moon. 11 divide by 3 produces the ratio: 3.6666666666667. Now if a Scalene triangle is constructed from the circumference of the moon touching the circumference of the earth the apex of the Scalene triangle will be located on the centre of the moon with the second shortest length of the Scalene triangle being equal to the radius of the Earth plus the radius of the moon. The smallest edge of the scalene triangle will only be equal to the radius of the Earth. The largest part of the scalene triangle will be the larger part of the Golden ratio when compared to the radius of the Earth that is the smaller part of the Golden ratio. If the radius of the Earth and the shortest length of the Scalene triangle is 5 and a half of 1 equal units of measure then the second longest edge of the Scalene triangle that is derived from the Earth and moon ratio will be 7 units of equal measure. Since the shortest edge of the Scalene triangle and the radius of the Earth is 5 and a half of 1 equal units of equal measure the longest edge of the Scalene triangle will be 8 point 9 units of equal measure. 8 point 9 divided by 5 and a half of 1 is: 1.618181818181818818181818181818. (An approximation of the Golden ratio).

The second longest length of the Kepler scalene triangle divided by the shortest length of the Kepler scalene triangle is the square root of the Golden ratio: = 1.272727272727273. 1.272727272727273 squared is also an approximation of the Golden ratio of 1.619834710743802.

The Scalene triangle that is used above in the Earth and moon ratio to find the Golden ratio is called a Kepler triangle. The Kepler triangle is also found in a Pyramid that has a square base width of 11 equal units of measure and 7 equal units of measure for the height of the Pyramid and the Great Egyptian pyramid of Giza is such as Pyramid.

• Constructing a Great pyramid Isosceles triangle inside the circumference of a circle, How to construct a isosceles triangle that is made from 2 Kepler scalene triangles inside the circumference of a circle:

If the diameter of a circle is divided into the Golden ratio then the larger part of the measure of the circle’s diameter that has been divided into the Golden ratio can be equal to the height of an isosceles triangle that is made from 2 Kepler Scalene triangles. Also the height of the Isosceles triangle that is made from 2 kepler Scalene triangles is equal to the distance between the pole of the circle’s diameter that is opposite to the apex of this Isosceles triangle and any of the 2 base points of this Isosceles triangle that touch the circumference of the circle. If the diameter of the circle that contains this isosceles triangle that is made from 2 Kepler scalene triangles is 140 equal units of measure then the height of this isosceles triangle will be 86.52 equal units of measure, meanwhile the base of this isosceles triangle will be 135.96 equal units of measure. Half of the base width of this isosceles triangle can be 67.98 equal units of measure while the length of the 2 longest edges can be 110.004 equal units of measure. If any of the longest edges of this isosceles triangle that is made from 2 Kepler scalene triangles is divided by half of the base width of this Isosceles triangle then again the result will be the approximation for the Golden ratio known as 1.618181818181818. Also if the diameter of the circle is divided by the height of this Isosceles triangle that is made from 2 Kepler Scalene triangles and is also contained with in the circle then the result will be the approximation for the Golden ratio known as 1.618122977346278. Please remember that this isosceles triangle that is made from 2 Kepler scalene triangles can be used to create a circle that has a circumference equal to the perimeter of a square if both the circle and square share the same centre and the base width of this isosceles triangle that is made from 2 Kepler scalene triangles is equal to the width of the square and also the height of this isosceles triangle that is made from 2 Kepler scalene triangles is equal to the radius of the circle.

• Combined diameters of Earth and moon forming the second shortest length of a Kepler Scalene triangle that also includes the Golden ratio:

If the measure for the diameter of the moon is added to the measure for the diameter of the Earth then this combined new measure can be used as the second shortest length of a Kepler Scalene triangle that includes the Golden ratio. While the diameter of the Earth alone can be used as the shortest length of the Kepler Scalene triangle. The diameter of Earth again is 7920 statute miles and the diameter of Earth’s moon is 2160 statute miles. 7920 plus 2160 = 10080. If 10080 statute miles is the second shortest length of a Kepler Scalene triangle then 7920 is the shortest length of this Kepler Scalene triangle, while the longest length of this Kepler scalene triangle is 12816 statute miles. 12816 divided by 7920 = 1.618181818181818. 1.618181818181818 is an approximation of the Golden ratio of 1.618 and 1.618181818181818 can also be obtained if 8.9 is divided by 5.5.

Examples for applications of the Golden ratio are abundant through out nature including the formation of galaxies and the spiral curve of the human ear. : https://en.wikipedia.org/wiki/Golden_ratio