By George Hart for the Museum of Mathematics

If you’ve never made a set of the Platonic solids from paper, perhaps it’s time to try it. These shapes are the foundation for many aspects of three-dimensional design. Here is a set made with open faces, but the openings are strictly optional. You can just cut out regular polygons and tape them together so every vertex is identical, e.g., putting five triangles at each vertex leads to the icosahedron.

After mastering the five Platonic solids, there is a world of more complex models to explore. The polyhedron below consists of twelve regular pentagons and twenty (very slightly irregular) hexagons. It is made by cutting out paper polygons and taping them together on the inside. This design is often confused with the truncated icosahedron shape that is well known because of its use as a soccer ball.  But this shape is the truncated rhombic triacontahedron. To see the difference, notice that there are some vertices here with three hexagons and no pentagon, but in a soccer ball there is one pentagon and two hexagons at each vertex.

And if you become engaged in discovering the world of polyhedra, you will encounter the many additional families, including the stellated icosahedron below. Their intricacies can be quite a challenge to make from paper, especially when some components meet just at points.  I made the model below over thirty years ago, starting from a template in the book Polyhedron Models by Magnus Wenninger. If you want your models to last this long, be sure to use acid-free paper.

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