Mathematical models

I got interested in making models of polyhedra while at school. The Maths teachers organized a year-long project, leading up to a summer exhibition of all the different models made by pupils. I got hooked by the 3D geometrical figures – polyhedra – which were a particular interest of my teacher. Several boys were making large card models, painted to show off the different faces, following the descriptions in an excellent book: “Mathematical Models” by Cundy and Rollett, then newly published but still available today in paperback. I had not got much space at home, so I thought I would make miniature models, using a fairly stiff writing paper, which I left plain. In the end, I built up quite a large collection, most of which I still have. It's strange to reflect that nearly all of them are now over 50 years old.

This photo includes five models which I painted to illustrate the connection the ancient Greek geometers imagined between the so-called Platonic solids and the four elements (earth, air, fire and water) and the heavenly "quintessence".

In the course of this activity, my maths teacher showed me a monograph called “The 59 Icosahedra” by Coxeter, Du Val, Flather and Petrie. These are stellations of the basic icosahedron; to stellate a figure, the planes of the faces are extended until they meet up along new edges and so define new vertices. The results typically look like stars, hence the name. The illustrations looked attractive, so I set about making making my own models, working to a uniform scale. The photo shows all the ones I made, most of which I later painted white to cover the specks of glue which tended to get onto the paper. The basic icosahedron itself is at the front of the group, sitting on a 2p coin, 26mm in diameter. The most elaborate stellation is centre back.

The final stage came when I started to think about dual figures. Any polyhedron has a dual, basically because an edge is common to two faces and connects two vertices. A face is defined by the edges which link up to form its boundary and a vertex is defined by the edges which terminate at it. Consequently, in describing the figure, the words face and vertex can be exchanged and you get a description of the dual. So a cube has 6 faces, each defined by 4 edges, and 8 vertices, each defined by 3 edges. The dual is an octahedron which has 6 vertices, each defined by 4 edges, and 8 faces, each defined by 3 edges. It occurred to me that all these stellated icosahedra should have duals, which should be called “faceted dodecahedra”. Faceted because the dual process to stellation involves connecting the vertices of a polyhedron by new edges and faces making it look as though the original figure has been decorated by cutting away the surface and dodecahedra because you start from the dual figure and the dodecahedron (12 faces, 20 vertices) is dual to the icosahedron (12 vertices, 20 faces). Some known polyhedra obviously fitted the recipe: the "5 cubes in a dodecahedron" and the Great Stellated Dodecahedron both clearly do. Several stellated icosahedra are also faceted dodecahedra. So I already had some models and I made a few more.

It seemed however there was no systematic description of the faceted dodecahedra, leading to an exhaustive list of all the possible structures. I decided to have a go at doing the job and eventually decided that Coxeter's icosahedra didn't all have duals because he defined them as solids enclosed by faces, rather than continuous surfaces made up of linked faces. Using the second approach I found 22 facetions of the dodecahedron (a to v in the figure below) and 22 corresponding stellations of the icosahedron, including the base figures themselves.

The pictures below are reproduced from pen and ink drawings I made to illustrate a paper eventually published (1974) in Acta Crystallographica A. Download a pdf version of the paper. (The shading in the original drawings was achieved with little bits of adhesive plastic cut from larger sheets preprinted with a stippled pattern at the appropriate density. This makes them difficult to reproduce effectively because the regular array of dots generates a moire fringe when scanned, except at very high resolution.)

Thirty years on, I was pleasantly surprised to hear from a fellow enthusiast, Guy Inchbald, who has also been pursuing research on polyhedra. His website gives full information.