The Math and the History Behind the Archimedean Solids

Amherst College Science Center

In his colloquium “Polyhedra: Plato, Archimedes, Euler,” Professor Robert Benedetto explains the mathematical history of the Archimedean solids – which include geometric forms like the truncated icosahedron, very reminiscent of a soccer ball but with flat faces instead of imposed on a spherical surface – and the proof that defines this set of 13 polyhedra. Professor Benedetto explained the ingenuity of the math behind the Archimedean solids – how, out of infinite semi-regular polyhedra, it was discovered that there are 13 and only 13 special cases – and considers the amazing realization that Archimedes may have produced this proof over 2000 years ago.

First, Professor Benedetto introduced the regular polyhedra, spatial regions bounded only by congruent regular polygons and with all vertices congruent to one another. Of these regular polyhedra, only 5 exist. Professor Benedetto showed how if we define the faces of a regular polyhedra as regular n-gons, and if we set the number of faces that meet at each vertex as k, then (n, k) can only be (3, 3), (4,3), (3, 4), (5, 3), or (3, 5). Otherwise, with any other parameters, we would start making polyhedra with angles greater than 360 degrees, a condition that is inconsistent with regular polyhedra.

However, if the parameters are changed to allow for polyhedra that have congruent vertices and faces that are all regular polygons (but, unlike with regular polyhedra, not necessarily each the same regular polygon), then we can make infinite semi-regular polyhedra. These semi-regular polyhedra can be further separated into three distinct types. Prisms (of which there are infinitely many), antiprisms (of which there are also infinitely many), and the 13 Archimedean solids. Prisms are two regular n-gons bounded by a belt of n squares, antiprisms are two regular n-gons bounded by a belt of 2n equilateral triangles, and the Archimedean solids are all the rest of the semiregular polyhedra that are not regular polyhedra, prisms, or antiprisms.

The proof that establishes that there are 13 Archimedean solids is similar to the proof of the 5 regular polyhedra. We know that there is the same arrangement at each vertex, and that no vertex can have an angle exceeding 360 degrees. Additionally, there are a few lemmas which specify limitations on vertex configuration in a semiregular polyhedron. There are only 13 possible configurations that are consistent with the lemmas, that don’t make prisms, antiprisms, or regular polyhedra, and that do not have angles exceeding 360 degrees.

Professor Benedetto mentioned that while Johannes Kepler wrote the first surviving proof for the 13 Archimedean solids in the seventeenth century, we also know that Archimedes was aware of them. Archimedes’ work has not survived, but the writings of Pappus of Alexandria discuss the 13 Archimedean solids. While Pappus of Alexandria didn’t give a proof, could Archimedes have identified these 13 solids without at least approaching one?

Professor Benedetto concluded his lecture with two fascinating cases, the pseudorhombicuboctahedron, and Euler’s Polyhedron formula. Each introduces a further layer of nuance in the definition of what constitutes an Archimedean solid or a regular polyhedron, respectively. Overall, Professor Benedetto’s presentation exemplified the brilliance and also the complexity in elegantly describing these sorts of groups. That Archimedes was able to separate 13 semiregular polyhedra from the infinite that exist, and that this grouping has basically stood the test of mathematical rigor over the centuries, is quite the incredible feat.