Origami and geometry

A Platonic solid is a polyhedron with the same regular polygon on every side, and also the same number of sides at every vertex. The cube is the most well-known example, and the tetrahedron (pyramid with a triangular base) is another one. In fact, there are only five such figures possible! The other three are called the octahedron (8 sides), dodecahedron (12 sides), and icosahedron (20 sides).

How could these more obscure figures have been discovered in the first place? And how can we know for sure that there aren’t any more examples? We used origami paper to ponder these questions hands-on in this session, which was delivered by our special guest Paul Ellis of Manhattanville College and the Westchester Area Math Circle.

After a brief introduction, Paul showed us how to fold origami paper into a module (called a Sonobe unit) which can be used as the face of a polyhedron. It didn’t take us long to figure out how to put them together to make a cube! We eventually learned how to make larger polyhedra as well, though only in stellated versions: this means the faces are pointed pyramids instead of flat polygons:

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By trying in turn to put three, four, and five Sonobe units together at a corner, we successfully built a icosahedron, octahedron, and dodecahedron! So this is how these figures could have been discovered!

Paul then walked us through a short geometric proof that there cannot be any more examples. The main point is that one can’t use six Sonobe units together at a corner because the resulting figure would never “close up” (it would extend forever like a plane).

There is one last wrinkle to tell about. Our constructions of the stellated versions of the dodecahedron (twelve pentagonal faces) and the icosahedron (twenty triangular faces) actually produced the same result! Somehow, if you add pointed pyramids to the faces of either one you end up with exactly the same figure. It turns out that this is because the two shapes are dual to each other: if you start with the dodecahedron and replace each corner by a face and each face by a corner, you end up with the icosahedron (and vice versa). We discovered this hands-on with this activity!

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