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Pentagon Tile 2

by Alexander Braun

(60x60) 3/10 flurecent acrylic with gold leif on canvas. Painted in May 2005 .Toronto .Ontario .Canada

Inspired by quasi crystal's pentagon symmetry.

pentagon tile by sasha. home page

Released under the Creative Commons Share Alike License.

[edit] Licensing

This work is licensed under the Creative Commons Attribution-ShareAlike 1.0 License.

[edit] Press Release

New pentagon pattern discovered by a Toronto artist.

It is impossible to tile a pentagon with the same size pentagons in 2D plane without leaving unaccounted for space. By many many attempts were made to come up with a way to tile this basic geometric shape and some were successful, others managed to design nice images containing no tile pattern algorithm.

On December 28, 2004, a Toronto artist Alexander Braun aka sasha. was visiting long-time friend Yehudah Lionel Cullman who showed sasha. an x-ray photo of a quasi crystal forming five-sided symmetry in a science book and said, that as far as I am aware, there is still no known way to tile a pentagon. Inspired by an x-ray picture and challenged by the mystery cloud around the pentagon tile, sasha. started to chart his attempt at the impossible. I have realized that the only way to tile a pentagon with only one size pentagons would be in 3D forming the fifth Platonic volume - dodecahedron (i.e. "two plus ten faces" in Greek), where the twelve pentagons enclose 3D space. Still, currious, I decided to try to design a pentagon pattern.

"The first thing I have noticed", said sasha. in an interview with his PR, "is that at the centre of the quasi crystal photo were ten pentagons aligned in a perfect circle forming ten-pointed star at the centre of it. The fact that here I see ten pentagons in a circle gave me an idea that if I only tile 1/10th slice of it, which is 360/10=36 degrees, then it would be enough to create a tile pattern.

So, I drew one 36 degrees slice of infinity and placed one pentagon at the bottom corner of it where it perfectly matches the 108 degrees of the inner pentagon angles. The rest came in place naturally, as I just continued the pentagon lines to determine what other basic building shapes of the tile are there at the very bottom of the pentagon slice. Soon I have discovered other three shapes, and that all four of them create the entire pattern; those basic shapes are: two pentagons (which also can be seen as two triangles but I prefer to keep it as pentagons), and two triangles. The main dilemma for many who tried to organize pentagons is what to do with the unaccounted for space.

In my attempt at it I have soon realized that the unaccounted space between pentagons when tiled within the 36 degrees slices form perfect stars, pentagrams, which makes total sense since it is the shape contained within the pentagon boundaries. The next challenge was to figure out the actual tiling algorithm of the pentagon tile's perpetual expansion of its infinite outer rings. I realized that whatever happens at the bottom is what happens at the top of it, only on a different scale. Soon I noticed that the ring of pentagons is followed by a ring of stars and then by pentagons again in a perpetual rotation based on the power of six, i.e.,

* 1, 
* 6, 
* 36, 
* 1296, 
* 46656, 
* 1679616, 
* 60466176, 
* 2176782336, 
* 78364164096, 
* 2821109907456, 
* 101559956668416,

etc."

Alexander Braun

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