Pentagon Tile by Alexander Braun
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, I was visiting an old friend who showed me an xray photo of a quasicrystal forming fivesided symmetry in a science book and said, that as far as he is aware there is still no known way to tile a pentagon. Inspired by an xray picture and challenged by a mystery of the pentagon tile I started to chart my 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.
I noticed that at the centre of quasicrystal photo were ten dots forming pentagons aligned in a perfect circle arranged as a tenpointed star. 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. Then I have discovered that only two triangles with angles of 36'72'72' and 108'36'36' degrees can form a pentagon tile, both are in the gloden ratio to each other. Thus, these two triangles can form a pentagon, or a star, or a perpetual fractal pentagon pattern, as I have soon realised. 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. 6x6x6... etc.,
* 1,
* 6,
* 36,
* 216
* 1296,
* 7776,
* 46656,
* 1679616,
* 60466176,
* 2176782336,
* 78364164096,
* 2821109907456,
* 101559956668416,
etc."
