More Compounds
We looked at a couple of 'compounds' earlier on, in the context of Duality. There's a little more to say on compounds, and a lot more on duality - so let's deal with the other compounds. We begin by noting you can actually fit a tetrahedron into a cube by cutting the corners off. The pictures aren't completely convincing, but here goes:
In fact you can do this two ways, by picking opposite sets of four corners: so there are two tetrahedra in any cube.
Does this sound familiar? If we show both tetrahedra we get the Stella Octangula, or Compound of Two Tetrahedra, that we saw earlier:
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So far, so good - but it turns out we can also fit a cube inside a dodecahedron; and we can do it five different ways.
Whew! Nice drawing, of course, but the next step is to build a model showing those five cubes. Aha! Here's one we made earlier:
 Compound of 5 Cubes in a Dodecahedron |
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As you look in detail at these models you can make out the coloured cubes quite easily - particularly the red one and the blue one. If you then focus on the detail you start to see all sorts of curious other shapes forming - stars, dimples, triangles and so on. You notice the skeletons of some of the non-convex semi-regular polyhedra...and then you can just stand back and enjoy the whole big thing again. If you sense it has an overall dodecahedral shape, well - yes it does - after all, this is the model of 5 cubes in a dodecahedron!
It doesn't stop there. If you consider each cube contains two tetrahedra, we can pick sets of either 5 or 10 tetrahedra and build the models of the compounds. Here is a model of 5 tetrahedra in a dodecahedron:
 Compound of 5 Tetrahedra |
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A stick model of the Compound of 5 Tetrahedra is a favorite machismo test among people who do Modular Origami. I've never yet made it without everything collapsing in a heap, but some people make a really nice job of it. It may be that the quality of the paper helps...or perhaps it's spit...
 Compound of 5 Tetrahedra in origami |
 ...and again |
Obviously there's a "twist" to this compound - i.e. left or right - and you could make a mirror image of it with the other "twist" (the technical name is "enantiomorph".) Putting the two together makes a Compound of 10 Tetrahedra. I don't have a photo of this since I've never managed to construct it in a way which makes the colours come right - but thankfully you can order a net for this one on the internet. When I've got it and made it up, pictures will appear here!
[Picture to come, showing the Compound of 10 Tetrahedra]
Lastly in this section, we note that the dual of a cube is an octahedron, and the dual of a dodecahedron is an icosahedron - so as well as the compound of 5 cubes inscribed inside a dodecahedron, we can make a compound of 5 octahedra circumscribed round the outside of an icosahedron.
 Compound of 5 Octahedra around an Icosahedron |
 ...and again |
At this point I think we've exhausted the main mathematical interest in compounds, but there are a number of other very appealing ones to look at. One is the Compound of Three Cubes, a shape which also appears in mediaeval etchings and carvings. Although it seems an easy model in concept, it's just complex enough to be tricky when you come to build it. In modern times, Escher featured it alongside the Stella Octangula in at least one of his works:
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There are other compounds beyond this (I once built a model of 5 dodecahedra intersecting each other, but it was huge and got lost somewhere along the way...) but it's time to move on again. Next we'll have a deeper look at Duality - that fascinating parallel between vertices and faces that leads to so many new and curious shapes.
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