POLYHEDRON MODELS (Part Two)

Dominic Rowland (C, 2001-06; CoRo, 2016- ) and Mathmā Don, adds a sequel to the article he published in TeS 2407 last November.

Loyal readers of the Trust e-Servant will remember the introduction to Michael Longuet-Higgin’s polyhedron models in Div Room E2*.   The most loyal have, of course, been waiting anxiously for a few words on the contents of the right hand half of the cabinet.

Recall (a word often used by mathematicians to introduce new concepts) that a Platonic solid is one where all vertices (corners) are identical, every face is regular and all the faces are identical. These shapes are discussed by Plato is his dialogue the Timaeus, where they play a key role in his understanding of what the universe is made of. For example, on the second shelf we find an icosahedron and an octahedron. Plato considered these to be the shapes of particles of water and air respectively. A modern reader might encode the fact that five triangles meet at each corner of the icosahedron by saying it has vertex configuration 3.3.3.3.3 while the octahedron would be labelled 3.3.3.3. Directly beneath each model on the second shelf, is its dual that is the polyhedron formed by placing a vertex in the exact centre of each face of the polyhedron above it. Remarkably the dual of the octahedron is a cube (4.4.4 for us, ‘Earth’ for Plato) and the dual of the icosahedron is the dodecahedron (5.5.5 or ‘The Universe’).

The rest of the second shelf comprises polyhedra almost as symmetrical as the Platonic solids, namely the Archimedean solids. These also have identical vertices where regular polygons meet but now we allow ourselves to use a variety of shapes at each vertex and are rewarded with a wider array of shapes. For example, the cuboctahedron has vertex configuration (3.4.3.4). Like the cube it has six square faces and like the octahedron it has eight triangular ones. Its dual is the beautiful rhombic dodecahedron which can be found, bisected, at the ends of the hexagonal tubes making up a honey comb. Perhaps the most familiar Archimedean solid is the truncated icosahedron (6.6.5) which is commonly seen being kicked.

Compiling a complete list of the 13 Archimedean solids is a salutary exercise. The figure with vertex configuration (4.6.10) [image 5] rejoices in the name rhombitruncatedicosidocehedron and the reader may wish to ponder why some configurations give sensible shapes, while others, like (6.8.10) and (3.4.5), are impossible.

The top and bottom shelf of the cabinet contain a somewhat haphazard collection of compounds of multiple platonic solids. The most magnificent is the compound of five cubes [image 6], all of which fit exactly inside a regular dodecahedron.