The 53 Semi-Regular Non-Convex Polyhedra

Introduction to the 53 Semi-Regular Non-Convex Polyhedra

Main Polyhedra Page   W67 Tetrahemihexahedron

We’re ready at last to look at the most wonderful polyhedra of the lot…well, that’s my view anyway. There are 53 to consider here, which is such a large number that I don’t think I’ll ever run out of admiration for them: I find them just staggering, mind-blowing, amazing. I could spend the rest of my life here.

The basic ideas are all familiar, though we haven’t put them all together yet. From the Platonics  – every vertex should be ‘the same’; from the Archimedeans – there can, however, be a combination of different polygons round each vertex; then from the Kepler-Poinsots the two ideas of letting the polygons penetrate through each other, and also star polygons.

The polygons we’ll work with here are also our old friends the triangle, square, pentagon, hexagon, octagon and decagon; and also the 5-pointed star (pentagram). But for the first time we find we can also use the 8-pointed star (octagram) and the 10-pointed star (decagram).

With 53 shapes it’s quite hard to find a way in to understanding them, so I’ll start by simply showing them all…

…and then I’ll list them by ‘families’ below. If you read to the bottom of the page you’ll see each one, though I rather hope you’ll also want to work through them slowly one by one by following the links on the top right.

My categorisation into ‘families’ is based on the way some of them just look obviously like each other: for instance 68 and 78 just below. So I’ve adopted 3 main criteria: do they fit into the same ‘external shell’; do they follow on from each other by some obvious process like stellation or ‘hollowing-out’; or is there some other idea which connects them. There’s no real mathematical reasoning behind this, but it certainly helps me to remember them.

At the start there is one which stands on its own. Tetrahedral symmetry is very simple and produces only the Tetrahemihexahedron here, which obviously sits inside an Octahedron:

W67 Tetrahemihexahedron

Octahedron (W3)

The next two look like inverses of each other, and both fit inside a Cuboctahedron. Aside from the obvious squares and triangles on the surface, each has hexagons which slice through the centre of the shape:

W68 Octahemioctahedron

W78 Cubohemioctahedron

Cuboctahedron

Since anything we can do with the Cuboctahedron generally means we can do the same, and better, with the Icosidodecahedron, here’s a lovely pair, both with decagons which slice right through the centre of the shape:

W89 Small Icosihemidodecahedron

W91 Small dodecahemidodecahedon

Icosidodecahedron

The same idea is here in these two shapes based on the Small Rhombic Cuboctahedron. Squares and Triangles on the surface, and Octagons which cut through the main part of the shape – though not actually through its centre.

W69 Small Cubicubocatahedron

W86 Small Rhombihexahedron

Shall Rhombic Cuboctahedron

…and again we can do this with a Small Rhombic Icosidodecahedron:

W72 Small Dodecicosidodecahedron

W74 Small Rhombidodecahedron

Small Rhombic Icosidodecahedron

At this point we run out of options for simply taking slices through the Archimedean solids: the reason the obvious one that we’ve used up all the Archimedeans where you can clearly do this ‘slicing’: we’ll see more of this with the Johnsons later on. But new shapes come in as prospective ‘external shells’, and ‘hollowing-out’ produces some new options.

The first family here comes from an external shell which has 3 stars meeting at each vertex. If you wanted to, you could say the external shell is a dodecahedron, though that observation feels less important than before in defining the new shapes:

W70 Small Ditrigonal Icosidodecahedron

W80 Ditrigonal Dodecahedron

W87 Great Ditrigonal Icosidodecahedron

Dodecahedron (W4)

Here’s a similar progression, but we start with the stars exploded away from each other and separated by triangles:

W71 Small Icosicosidodecahedron

W82 Small Ditrigonal Dodecicosidodecahedron

W90 Small Dodecicosahedron

And here is the same trick, but we start with a form where the stars are rotated: so instead of 3 points meeting at each vertex, we have 3 stars whose arms meet to surround ‘the space where the vertex used to be’.

W73 Dodecadodecahedron

W100 Small Dodecahemicosahedron

W102 Great Dodecahemicosahedron

As before we can explode the stars away from each other, but this time they separate with squares filling the gaps rather than triangles:

W76 Rhombidodecadodecahedron

W83 Icosidodecadodecahedron

W96 Rhombicosahedron

Now we start some work with the ‘big stars’ – the Octagram and Decagram. We start with a shape derived from a cube, and do the usual hollowing-out:

W77 Great Cubicuboctahedron

W103 Great Rhombihexahedron

W85 Quasirhombicuboctahedron

Then, as before, we pull the stars apart – first separating them with squares and then with hexagons; then finally we push them rather violently in:

W93 Quasitruncated Cuboctahedron

W79 Cuboctatruncated Cuboctahedron

W92 Quasitruncated Hexahedron

And as usual, what we can do to the cuboctahedral form we can also do to the icosidodecahedral:

W81 Great Ditrigonal Dodecicosidodecahedron

W101 Great Dodecicosahedron

W88 Great Icosicosidodecahedron

W98 Quasitruncated Dodecadodecahedron

W84 Icosidodecatruncated Icosidodecahedron

W97 Quasitruncated Small Stellated Dodecahedron

Now we’ll go back to some simpler forms which are just very obviously Truncated, or Quasitruncated, forms of the Kepler-Poinsot solids. Is there a version of the missing one, the Small Stellated Dodecahedron? Well – yes – it’s W97. But it fits nicely as the last member of the family we’ve just looked at, so we’ll let it stay there.

W75 Truncated Great Dodecahedron…	W95 Truncated Great Icosahedron…	W104 Quasitruncated Great Stellated Dodecahedron…
…is one of these with the corners cut off: Great Dodecahedron	…is one of these with the corners cut off: Great Icosahedron	..is one of these with the corners chopped out: Great Stellated Dodecahedron

Then there are a further three which share the external form of the Great Icosidodecahedron, a nice mix of triangles and pentagrams. And then the two hollowed-out versions show more of the underlying structure, which is basically decagrams slicing through the middle. .

W94 Great Icosidodecahedron

W106 Great Icosihemidodecahedron

W107 Great Dodecahemidodecahedron

A different arrangement of the decagrams brings them more to the surface and shrinks the exposure of the triangles and pentagrams – and by the time we get to hollowing it out there seems very little left…

W99 Great Dodecicosidodecahedron

W105 Quasirhombicosidodecahedron

W109 Great Rhombidodecahedron

Before we get to the 10 ‘snubs’ – which are really in a world of their own – there’s one left which seems to stand on its own. You could relate it to the decagrammic ones a few rows up and regard it as a slight ‘implosion’ of one of them, but really I think it deserves to be seen as a one-off:

W108 Great Quasitruncated Icosidodecahedron

Lastly there are the 10 snubs and John Skilling’s Polyhedron (S1), which we’ll include here since it turns out to be a very close relative. Among these there’s only really one family, made up of two of the snubs and Skilling’s Polyhedron. They have very similar external forms, all of them with the distinctive feature of paired pentagrams in the same planes:

W115 Great Snub Dodecicosidodecahedron

W119 Great Dirhombicosidodecahedron

John Skilling’s Polyhedron (S1)

After these, though, the snubs are so weird that they don’t really seem to belong in families. I’ll make some comment on them below, but on the whole this feels more like a world of geeky teenagers where ‘families’ don’t really operate.

W110 is always appealing – not only because it’s my own model but because it has a basic simplicity of form – but I don’t think the external form is shared with any other. And W114 is just, well, stunning:

W110 Small Snub Icosicosidodecahedron

W114 Inverted Snub Dodecadodecahedron

The next two look as if they might have the same vertices – the points of the stars – but there are elements of scaling which make that untrue. Indeed to calculate the vertices of W112 we have to work with the ‘plastic constant’ (1.3247180…) which I must admit I’d never heard of until I came to write this up…

W112 Snub Icosidodecadodecahedron

W111 Snub Dodecadodecahedron

The next three do have the same vertices, but other than that it’s hard to spot any relationship between them. Perhaps I’ll discover one if I dream about them long enough:

W113 Great inverted Snub Icosidodecahedron

W117 Great Inverted Retrosnub Icosidodecahedron

W116 Great Snub Icosidodecahedron

117 and 118 both have sharp edges, but apart from that there’s little to connect them: so 118 gets the final place on its own, at the very end of the list:

W118 Small Inverted Retrosnub Icosicosidodecahedron

So that’s all…and what’s left? Simply to click on W67 Tetrahemihexahedron and work through these delightful shapes one by one…!