By George Hart for the Museum of Mathematics


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.

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.

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.

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.

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

See all of George Hart’s Math Monday columns

Gareth Branwyn

Gareth Branwyn

Gareth Branwyn is a freelance writer and the former Editorial Director of Maker Media. He is the author or editor of over a dozen books on technology, DIY, and geek culture. He is currently a contributor to Boing Boing, Wink Books, and Wink Fun. And he has a new best-of writing collection and “lazy person’s memoir,” called Borg Like Me.

  • James B

    Looks like one of those triple integral problems from Calc III.