The Trouble with Five (2007)

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.<p>(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.)

<i>&quot;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.&quot;</i><p>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?

Here&#x27;s a fun rabbit hole of substitution tilings: <a href="https:&#x2F;&#x2F;tilings.math.uni-bielefeld.de&#x2F;substitution&#x2F;" rel="nofollow">https:&#x2F;&#x2F;tilings.math.uni-bielefeld.de&#x2F;substitution&#x2F;</a><p>Run substitutionList.show(1, 1000) in the browser console to show them all on one page.