Pentagons that tile the plane
The problem here is to find a single tile shape with five sides
which can tile an infinite floor without leaving holes. I tried
finding new solutions, but I think all the tiles I found had been
previously found by someone else. Someone claimed the recipe I was using in my search was new.
Here are the known solutions:
I got started on this because I saw an article in the paper about
Marjorie Rice, an amateur mathematician with no training beyond high
school, who had discovered the tile used in the background of this
page and three others in 1976 and 1977. She used the same search
strategy I'm using, but by hand. At any rate, I thought, "I can do
that", so I did. (Thanks to H. Nelson Miller for refreshing my memory
on the details. Martin Gardner's "Time Travel and Other Mathematical
Bewilderments" covers Marjorie Rice.)
- The official list of all pentagons that tile the plane is here. The
official definition of a pentagonal tiling is slightly different from
mine. Here are a few known solutions.
- Here are more solutions, boring ones that you get by subdividing
more common tiles. (I think these are old, I'm not sure.)
Angles 120-60-180-60-120, all sides equal. This is definitely old, but
it forms some patterns that no other tile forms. The first pattern shown
is a "reptile": 4 tiles form a larger tile of the same shape, and 4 of those
form yet a larger tile, and so on recursively. (It's a pentagon
because there are five points at which tiles meet. It doesn't matter
that the angle at one of the points is 180 degrees.)
Here are the solutions I found. (I have about 2000 more candidate
new solutions, but I haven't developed the tools to judge them.
Probably about 6 will be real solutions.)
- Angles 45-270-22.5-112.5-90, four sides equal. This forms only
- Angles 100-140-60-160-80, all sides equal. The tile and its mirror
image are required for this tiling. I just learned the circle is
called a "Hisrchhorn Medallion", and was first found by Mike
Hirschhorn in the 1970's. I tiled the
floor of my half bath
with this tiling. Mathematician Ed Pegg, Jr. manufactured a
game (sort of like
dominoes) out of this tile and another that he found. (I had to get
a set to show my support for mathematicians actually building things.)
- Angles 108-36-252-36-108, all sides equal. This is easy to derive
from a pentagram. It forms many interesting tilings. I've seen it in
some of David Eppstein's tilings, so I didn't find it first. I
painted it on the ceiling of my half bath.
- Angles 60-90-150-30-210, all sides equal. It would be easy to lay
floor tiles or linoleum made with this.
Here is another solution by Mike Korn:
Here's the recipe I'm using in my search.
David Eppstein's tiling
David Eppstein's generally geometric
Ed Pegg's Chaotic Tiling combines
the 160-60-140-100-80 tile with another to produce chaotic tilings.
I decorated my half bath with these
Here's a gallery of my mom's art.
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