Dictionary of Mathematical Eponymy: Johnson Solids

I am a big fan of polyhedra. I’ve raved elsewhere about the icosidodecahedron, and even something as dull as a cube is something I can get behind. And so, naturally, I wondered: is there a periodic table of polyhedra?

And the answer is “not exactly”. But there’s something pretty close to it, the list of Johnson solids.

What’s a Johnson solid?

Before we get to the Johnson solids, I have to tell you what they’re not. They are specifically not:

  • Platonic solids, convex polyhedra that have identical regular faces;
  • Archimedean solids, which have regular faces that aren’t necessarily identical (the truncated cube, for instance, has pentagons, squares and triangles as faces);
  • Prisms, which have parallel, identical upper and lower faces connected by congruent quadrilaterals on the sides; or
  • Antiprisms, which are like prisms but slighly twisted, so the connections between one layer and the other are triangles.

If a convex polyhedron is not in one of those families but consists of regular shapes with equal edge lengths, it is a Johnson solid.

Splendid solids they are, too: pyramids and cupolas, rotundas and fastigia… there are many, and they’re all gorgeous.

In fact, there are 92 of them; Johnson first enumerated them in 1966, and Zalgaller proved that he’d caught them all three years later.

Why are they important?

In honesty, I don’t think they’re that important in themselves. They’re lovely, but most of their interest to me is in knowing that all such shapes can be categorised and found.

One application I like is that some of the Johnson solids can be combined with other polyhedra to form honeycombs in three dimensions - which can (if you’re into that sort of thing) give a nice way to split space up into smaller bits without relying on boring old cubes.

Who was Norman Johnson?

Norman Johnson (1930-2017) studied under H.S.M. Coxeter at the University of Toronto, and taught at Wheaton College in Massachusetts until he retired. Apart from his work on convex solids and honeycombs, he’s also known for having named all of the uniform star polyhedra. He died in 2017.

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

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