Soccer anyone?

Hyperbolic soccer ball

That there is what one might euphemistically call a hyperbolic soccer ball. It’s a model of the hyperbolic plane that I made using this template. For those that aren’t up on their non-euclidean geometry, the hyperbolic plane is a 2-dimensional space of constant curvature -1 (for comparison, the sphere has curvature +1 and a regular plane has zero curvature); it was the first example of a consistent geometry in which Euclid’s parallel postulate doesn’t hold.

The above model is based on the standard soccer ball pattern, which has black pentagons surrounded by white hexagons. That pattern works nicely on a sphere, but you can’t flatten it out; to flatten it, you have to exchange the pentagons for hexagons and then you get a tiling of the regular flat plane. Going one step further gives you the above picture: black heptagons surrounded by white hexagons, which, as with the regular soccer ball, can’t be flattened out without ripping the paper.

See the bigger version for a closer look, where you can more easily discern the obscene geoboard, platonic solids and other geometrical miscellanea cluttered in the background (all extensively documented in the notes to the Flickr version). There’s also another view, which gives a better sense of how the hyperbolic plane is sort of a bunch of saddle shapes all nested together. The best visual model might be the crochet model made by Daina Taimina. Of course, all of these models are incomplete: the actual hyperbolic plane extends out forever but gets totally curled up on itself when you try to embed it into regular 3-dimensional space.

And yes, before you ask, I do get paid for this.

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