I thought this was going to be about fifth degree polynomials, and now after seeing the sphere, it’s got me thinking about factorization on the toroidal number system.
(Note: my swipe keyboard got “toroidal” on a mangled swipe and turned “number” into “bumble”. Sometimes I wonder whether this thing is reading my mind or dictating thoughts to me.)
"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?
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:
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.
(Note: my swipe keyboard got “toroidal” on a mangled swipe and turned “number” into “bumble”. Sometimes I wonder whether this thing is reading my mind or dictating thoughts to me.)