We’ve looked at the five ‘regular’ solids we get by taking just one flat polygon as the basis for each model. But what if we allow some variation? We begin by allowing more than one type of polygon in each model, but we keep the rule that round each vertex there must be consistent numbers of each. This leads us to the ‘semi-regular’ or ‘Archimedean’ Polyhedra. (The great man apparently wrote a book about them around the year 280BC, but sadly it got lost somewhere between then and now.)
The first five semi-regular solids look as if we’ve cut the corners off the existing Platonic solids – so we use the term Truncated. But the important point is that the faces themselves are all still regular polygons – hexagons, octagons etc – and that the combination round each vertex is always the same.
 Truncated Tetrahedron – 1 Triangle and 2 Hexagons at each vertex |
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 Truncated Cube – 1 triangle and 2 octagons at each vertex |
 Truncated Dodecahedron – 1 triangle and 2 decagons at each vertex |
 Truncated Octahedron – 1 square and 2 hexagons at each vertex |
 Truncated Icosahedron – 1 pentagon and 2 hexagons at each vertex |
If you imagine making these truncations by cutting thin slices off potato shapes, you might well find you’d taken too much off. (Try it next time you’re peeling a potato…) The Cube and Octahedron, and the Dodecahedron and Icosahedron, display a neat relationship here: it turns out that if you cut too much off the corners of any one of them it starts to look like its partner. Get thing just right in the middle and you end up with two lovely half-way shapes, named so as to highlight the partnerships:
 Cuboctahedron – half-way between a Cube and an Octahedron |
 Icosidodecahedron – half-way between an Icosahedron and a Dodecahedron |
We now have seven shapes derived by chopping the corners off the Platonic solids, so it’s tempting to think we could try chopping the corners off some of these Archimedeans too and discover some new ones. That doesn’t quite do the trick: instead of getting square corners we get rectangles. And since rectangles aren’t ‘regular’ they’re not allowed, though these shapes do have a certain charm of their own.
The Cuboctahedron and the Icosidodecahedron, though, have so much ‘good’ geometry around that we can make it work if we also pull the existing faces apart a little to square things up. To reflect this contortion we use the word Rhombic, which in this context means ‘we’ve stretched those rectangles a bit to make them squares’. And because we’ve chopped the corners off, triangles also become hexagons, squares become octagons, and pentagons become decagons. (This is the first time we’ve encountered octagons and decagons, but you’d do well to get used to them…)
 Large Rhombic Cuboctahedron |
 Large Rhombic Icosidodecahedron |
Curiously we can now collapse the more complex polygons down again: think of this as letting those squares expand a bit to soak up some of those extra edges. We still call these Rhombic, but the words Large and Small help us distinguish where we’ve got to.
 Small Rhombic Cuboctahedron (hexagons become triangles; octagons become squares) |
 Small Rhombic Icosidodecahedron (hexagons become triangles; decagons become pentagons) |
Stand back from these for a moment and you can see we’ve still got the basic forms of the cuboctahedron and icosidodecahedron: all we’ve done is pull them apart and rough them up a bit. There’s one more contortion that we can do, which involves inserting not squares but pairs of triangles in parallelogram formation. Making the thing fit together requires a bit of twisting, which distorts their perfect symmetry – but they’re still valid according to our definition of Archimedean, or semi-regular, polyhedra as having exactly the same configuration round each vertex. To reflect their twisted nature we introduce the word ‘Snub’:
 Snub Cube |
 Snub Dodecahedron |
This accounts for all 13 Archimedean solids – and by a similar argument to the Platonics, there can’t be any more because we’ve tried every possible combinations of polygons around the vertices. They make a beautiful collection, yet they really come from just two underlying structures – Cuboctahedral and Icosidodecahedral – with the Truncated Tetrahedron as an extra little baby one. If you can hold these two structures in your head, all the rest is just details!
Our next step is to go non-convex …but we’ll leave that for the next page.
You may be asking by now “has Colin invented all these shapes, or did he get them out of a book?” The truth is actually a mixture of the two: there are some wonderful classic books on Polyhedra (though even now there are only four worth owning), but actually I did work out some of this stuff in my own head and on my own desk before I even became aware of the books – let alone consulted them. I claim that I’ve done at least some of the thinking here myself!
The four books in question are:
Cundy & Rollett: Mathematical Models (First discovered in the 6th form library at Newcastle RGS, and loved ever since.)
Wenninger: Polyhedron Models (A marvellous updating and expansion of the topic in the 1970s.)
Wenninger: Dual Models (his Spherical Models has never particularly inspired me, but Dual Models is a triumphant follow-on from Polyhedron Models.)
Coxeter: The 59 Icosahedra (The daddy of them all, and still the classic text on the Stellations of the Icosahedron.)Wenninger’s books are published by Cambridge University Press: the others are nowadays available as reprints from the excellent Tarquin Publications of Norwich.
These four books are the “bibles” of this subject: a “must-have” if you want to build the more complex models, and an absolutely solid reference point for discussion. They are never far from my bedside, and in fact there’s rarely a day when I don’t look into one or more of them.
Beyond this I’d commend Coxeter’s Regular Polytopes if you want to get into higher-dimensional space, but actually it’s very hard-going for little visual reward and I don’t usually get very far with it. There are virtually no other books worth reading on good standard 3-dimensional polyhedron models. In a way this lack of publication seems sad, but perhaps once it’s been said it doesn’t need saying again; and to write a credible book on these topics you really ought to have built the models. I suspect very few are doing that these days: and the one or two books that do appear are basically just coffee-table fodder full of computer-generated images and meaningless writing around them.
From time to time booklets also appear with “nets” to cut out so that you can make the models without going too deeply into the fundamental geometry. I haven’t bothered discussing nets yet; but these booklets are a great resource, excellent for the general reader and certainly for children, and I’ve even made some of them myself – particularly the 6th, 9th and final Stellations of the Icosahedron, and a huge collection of Oddities. Tarquin have made a particular point of keeping these booklets in print, and they’re well worth a look.
It is, of course, all available on the web – Wolfram Mathworld – Polyhedra is a wonderful resource, and there are a few other model builders who’ve put up photos of their own work (I’ll write about them later.) But none of this comes close to the thrill of assembling one of these models from scratch with cardboard and glue, and then holding it in your own hands. It’s a sad fact that most professional mathematicians have never made any of these shapes: even if they make some of the Platonic solids in primary school, most never come near making even a Dodecahedron with an adult mathematician’s understanding of what they’re doing.
Colinspics 