# Tessellations: Tiling the Plane

Tessellations are patterns made up of repeating shapes that completely cover a surface without overlapping or leaving any gaps. The individual shapes in a tessellation are called tiles or tessellating shapes. They can be regular or irregular polygons.

In nature they are found in honeycombs or in the scales on a snake’s skin. They’re also commonly used in art and design, such as in Islamic geometric patterns or as a decorative element in tile work, fabrics, and wallpaper.

by Makoto Nakamura

M.C. Escher (Maurits Cornelis Escher, 1898-1972) is most famous for this design style. He was a Dutch graphic artist known for his mind-bending etches that incorporated elements of mathematics and geometry. His use of geometric patterns, symmetry, and perspective has inspired new ways of thinking about space, dimensionality, and the relationship between art and science.

Of the contemporary tessellators I’ve searched for, my favorite has to be Nicolas. Here’s what he had to say on the subject.

“Among the disciplines that come from a mixture of Mathematics and Art, figurative tiling is the most fascinating of all, as there is something magical in seeing complex identical forms assemble without leaving any gap in between them into infinity. This process requires imagination rather than artistic qualities or geometrical knowledge. However, the use of the usual 17 symmetry groups in the plane in crystallography is complex and, historically, has been poorly adapted.” – Alain Nicolas

## Alain Nicolas’ Tessellations

Here are some of my favorites that I’ve collected from his website
https://en.tessellations-nicolas.com

## Makoto Nakamura

Remember that animated gif from the intro? That was made by Makoto Nakamura, a Japanese man whose name translates to “good middle village” for what it’s worth. I don’t know much about him, but he’s very talented.

## Makoto Nakamura’s Animated Tessellation GIFS

more general GIFS

## Mr. Nakamura also has some 3-dimensional models

by Makoto Nakamura

https://tessella.sakura.ne.jp/animation/animation.html

https://web.archive.org/web/20170121050311/http://tessella.sakura.ne.jp/home.index.html (backup)

## More Tessellation Sources

http://www.tess-elation.co.uk : Reviews, methods, Escher, essays and a lot of articles. The very informative site of David Bailey.

http://tessellations.org : One of the two main sites dealing with tessellations. All aspects of tessellations and their history. Galleries and workshops.

http://ozbird.net : Periodic and non-periodic tilings, the inventive art of John A.L. Osborn.

http://www.josleys.com/galleries.php?catid=6 : Marvellous Mathematical Imagery by Joe Leys. His Escher flavoured pages. Where we find impossible geometry, tessellations, circular limits and even the Droste effect.

https://kickstarter.com/projects/mathspace/tessellation-masters-puzzle-series : Wooden puzzles with artwork from masters of mathematical art from around the world. Beautiful and particularly inventive puzzles.

https://mcescher.com/ : The Official M.C.Escher Website, published by the M.C. Escher Foundation and the M.C. Escher Company B.V.

http://mcescher.com/ : Stories, discuss M.C. Escher’s works, provide insight, and offer our high quality products promoting the intriguing work of Escher.

https://scottkim.com/ : Designer of visual thinking puzzles for web and print.

https://tessellation.info/ : Tessellation Database. Over 400 tessellations, 46 artists. The best site about the tessellation artists.

https://en.tessellations-nicolas.com/  (my personal favorite)

https://en.tessellations-nicolas.com/staging.php (my favorite page from his site)

http://www.cromp.com/tess/   :Andrew Crompton

http://www.tesselmaniac.com  Kevin Lee

http://www.kaleidomania.com/Chessboards.html Escheresque Chessboards

## Joe Dubs’ Rep-tiles!

These cute little rep-tiles could be yours for the low price of \$29.99! Don’t delay! Get yours today!!!

Choose your 3 colors: (Red, Orange, Yellow, Green, Blue, Purple, White, Black, Gold, Silver, Green-bleu-swirl, Khaki)

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Custom Orders: Email me at joe@joedubs.com

## Joe Dubs

I write about philosophy, geometry, health, politics and other stuff that interests me.