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"Another non-flat geometry, hyperbolic geometry, even allows infinitely many ways to tile the plane by regular pentagons. ... The flatness of Euclidean space se

"Another non-flat geometry, hyperbolic geometry, even allows infinitely many ways to tile the plane by regular pentagons. ... The flatness of Euclidean space seems to interfere with the ability of five-fold shapes to tile. Somehow, fiveness and flatness have a difficult time getting along together." Could this be related to 5 being a prime number? Are there any examples or counter-examples of prime-number sided shapes tiling the plane?

drdeca 5y ago
regular triangles can tile the plane.
- alexvoda 4y ago
  Actually, I think any triangle can tile a plane.
basicoperation 5y ago
The first image shows a tiling with triangles. 3 is prime.
deleted 5y ago
[deleted]
rogual 5y ago
As a non-mathematician, one thing that helped me understand this stuff -- or at least, helped it seem not totally arbitrary which tilings "exist" -- was this table: https://en.wikipedia.org/wiki/Template:Regular_hyperbolic_tiling_table https://en.wikipedia.org/wiki/Template:Regular_hyperbolic_ti... (You gotta click [show] for some reason) These are all the possible tilings, organized by the number of sides of each polygon (Y axis) and the number of polygons meeting at each point (X axis). The table shows that all of the infinitely-many tilings are possible, but most of them (infinitely many) only work on the hyperbolic plane (cells with blue backgrounds). The cells with red backgrounds are tilings that work on the sphere, like {5, 3} (three pentagons around each point). And the cells with green backgrounds, the rarest of all, are the tilings that work on that knife-edge between the sphere and the hyperbolic plane: the flat, Euclidean plane. The green cells are at {6,3} (hexagontal tiling), {4,4} (square tiling) and {3,6} (triangular tiling). Anyway, just wanted to share this table because it's quite hard to find on Wikipedia and presents the subject in a way that I found enlightening and satisfying.

drdeca 5y ago

regular triangles can tile the plane.

alexvoda 4y ago
Actually, I think any triangle can tile a plane.

basicoperation 5y ago

The first image shows a tiling with triangles. 3 is prime.

deleted 5y ago

[deleted]

rogual 5y ago

As a non-mathematician, one thing that helped me understand this stuff -- or at least, helped it seem not totally arbitrary which tilings "exist" -- was this table: https://en.wikipedia.org/wiki/Template:Regular_hyperbolic_tiling_table https://en.wikipedia.org/wiki/Template:Regular_hyperbolic_ti... (You gotta click [show] for some reason) These are all the possible tilings, organized by the number of sides of each polygon (Y axis) and the number of polygons meeting at each point (X axis). The table shows that all of the infinitely-many tilings are possible, but most of them (infinitely many) only work on the hyperbolic plane (cells with blue backgrounds). The cells with red backgrounds are tilings that work on the sphere, like {5, 3} (three pentagons around each point). And the cells with green backgrounds, the rarest of all, are the tilings that work on that knife-edge between the sphere and the hyperbolic plane: the flat, Euclidean plane. The green cells are at {6,3} (hexagontal tiling), {4,4} (square tiling) and {3,6} (triangular tiling). Anyway, just wanted to share this table because it's quite hard to find on Wikipedia and presents the subject in a way that I found enlightening and satisfying.