pentagon
tile 2
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
x-ray photo of a quasi-crystal forming five-sided 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 x-ray 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 quasi-crystal photo were ten dots
forming
pentagons aligned in a perfect circle arranged as a ten-pointed 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."
pentagon
tile 2