how to mismake a soccer ball

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• Subject: how to mismake a soccer ball
• From: [email protected] (Anton Sherwood)
• Date: Tue, 3 Sep 1996 23:24:53 -0700

I wrote a little C program to enumerate the ways to make a closed ball
from 12 pentagons and up to 20 hexagons.  (I don't care whether the
faces are regular, because I got on this track by wondering about
the varieties of buckyballs.)

I thought I'd have to build (and debug) a representation of the polygons
and their connections.  Easier to do the search by hand... But eventually
it hit me that all I need is a bit string representing the convex and
concave vertices of the boundary of the incomplete ball.

        Solutions      = 526363
        Dead ends      = 21982613
        Face overflows = 19186580
        Edge overflows = 410

The output is also a bit string: 1 for a pentagon, 0 for a hexagon; and
they're added in a strict spiral sequence, so even though the program
knows nothing about the shape, it's easy for a human to build it up
from its code.  Each solution appears in the tree 120/G times, where
G is the size of its symmetry group.

And what's my point in doing this?  Well, if you should ever want to
build a dome house with a square floor, let me know!

Anton Sherwood   *\\*   +1 415 267 0685   *\\*   [email protected]

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