Mathematical Papercraft

[picture of Niles]

Some of my 2105 holiday papercraft!

The Stellated Icosahedron

A stellated icosahedron made with 30 Sonobe modules in 6 colors. The colors are arranged in 6 interlocking bands, with 5 distinct colors meeting at each of 12 star-shaped faces. There are 20 triangular points in 10 antipodal pairs. These each have 3 colors, and the 10 pairs represent all possible choices of 3 colors.

Instructions for folding the modules are available at

The Klein Quartic

A Klein quartic made with 24 heptagons (7-sided shapes). In a 4-dimensional space (CP^2), this can be done with regular heptagons, and the shape has 168 symmetries. This is the largest possible for a 3-holed donught! To push the shape down to 3 dimensions destroys most of this symmetry, and distorts the regular heptagons severely. But by doing so we can construct it out of paper!

Instructions and an interactive model in python were made by Tim Hutton. There is now a more symmetric coloring scheme with 8-colors!

8-color base

For much much more, you could start with this article by John Baez. Among other things, he explains where the number 168 comes from, and how it's related to the fact that 1 - 1/2 - 1/3 - 1/7 = 1/42. "So, we see that far from being arbitrary flukes, the numbers 84 and 168 are burnt deep into the fabric of reality."

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