By George Hart for the Museum of Mathematics

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A sphericon is a shape that you get by: (1) rotating a symmetric polygon about a mirror axis to get a solid of revolution, (2) cutting the solid into two equal pieces, and (3) putting the pieces back together differently. With a lathe or a 3D printing machine, it is easy to make many kinds of sphericons, with different starting polygons. Here is one based on a star.

sphericon 1 Math Monday: Star Sphericon

Start by rotating an ordinary 5-pointed star about any of its five mirror lines to create a kind of funny hat. Now cut the hat in half to reveal the original star cross section. I designed these halves to include cylindrical recesses for small magnets. I glued five magnets with North outwards in one half and five with South outwards on the other half. Other than that, the halves are identical.

sphericon 2 Math Monday: Star Sphericon

Because the star is symmetrical, the two parts can be put back together in five different ways. One way is the “hat” shape above and the others are simple to describe yet surprisingly difficult to visualize.

sphericon 3 Math Monday: Star Sphericon

Two of the ways are basically different and the other two are just their mirror images. They each follow a meandering path when you roll them on a flat surface.

sphericon 4 Math Monday: Star Sphericon

If you have access to a 3D printer, the STL file to make your own copies of this star sphericon are here.

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See all of George Hart’s Math Monday columns

Gareth Branwyn

Gareth Branwyn is a freelancer writer and the former Editorial Director of Maker Media. He is the author or editor of a dozen books on technology, DIY, and geek culture, including the first book about the web (Mosaic Quick Tour) and the Absolute Beginner’s Guide to Building Robots. He is currently working on a best-of collection of his writing, called Borg Like Me.


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