We said on the last page, with some trepidation, that we had yet to consider the Duals of the Non-Convex Semi-Regular Polyhedra. There are, as we noted earlier, 53 of these. Oh dear…another 30 years’ work, and a lot more cupboards!
In the event the workload turns out to be not so bad. An odd feature of these Duals is that for some of them, some of the vertices of the Dual lie completely on the inside and are obscured by other parts of the structure: and since these tend to occurs in pairs or even triads, we can get away with making less than 53 – a quick look suggests it’s about 40.
We can further divide the work up by saying there are some Dual models that are easy, others that that will be challenging, and some that I can’t see myself ever getting round to at all – so it’s not worth feeling guilty about them. The workload reduces to perhaps 20 years…and given that colour is not such an issue with Duals that perhaps saves another 10. (The point here is that though each Dual has many strangely-shaped faces, those faces are all exactly the same – so there’s no need, for instance, to make squares one colour and triangles another.)
Along the way, there’s a curious subset relating to the 9 Hemi-Polyhedra, which have some faces slicing right through their centres. This raises the question how, given that Dualling is about taking the reciprocals of distances to the centre, can the corresonding vertices appear in the dual when the calculation seems to involve 1/r with r=0?
The answer, once you ‘get it’ is obvious: these vertices actually shoot off to infinity – and the mathematics all falls out nicely. But although the maths is tidy, it’s less easy to build a shape that is infinite!
We handle this with a compromise, again obvious once you ‘get it’ but nevertheless a neat piece of thinking when Wenninger first made it. Since the infinite vertices force the shape to have essentially parallel ‘legs’, we simply chop these off at a suitable distance from the centre and then pretend they carry on for ever. I’m skating past lots of mathematical subtleties here, but a picture will help explain what’s going on:
 Tetrahemihexacron |
The important thing here is the green ‘legs’, which you have to think of as shooting off into the far distance to infinity – they never get any wider or thinner, but just travel out endlessly to the furthest parts of outer space. In fact it helps if you can imagine think of them as only 3 legs, not 6 – the idea being that each leg shoots off to infinity and then comes back from the opposite direction. Mathematically there are really only three infinite square prisms intersecting here.
The yellow squares have no function at all except to ‘cap’ the square prisms in our real finite space, though they do helpfully remind us that the ‘legs’ are square in cross-section. But the main point is that this Dual is derived from a Hemi-polyhedron which originally had faces which passed through its very centre. Since there are three prisms there must have been three faces – and since the prisms meet in threes, there must be some other three-ishness about the original. So what what that original Hemi-polyhedron? It was the Tetrahemihexahedron, the very first one we encountered among the the Non-Convex Semi-Regular polyhedra:
 Tetrahemihexahedron |
For such a small shape this one is surprisingly tightly connected and really quite weird: it has a very strange total of 7 faces, and as we’ve said, the green, yellow and red faces all pass through it’s very centre. In the dual these 3 “hemi” faces become 3 “vertices at infinity” representated by the 3 prisms. As for the blue triangles… well each of them becomes a 3-sided vertex in the central crossing of the Tetrahemihexacron at the conjunction of three light green faces. This seems a rather feeble end for a good solid blue triangle, but that’s what the Dualling produces.
A more advanced example is the polyhedron below, the Small Icosi/Dodecahemidodecacron:
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Here the important thing is again the pink ‘legs’, which again you have to think of as shooting off into the far distance to infinity – they never get any wider or thinner, but just travel out endlessly to the furthest parts of outer space. The green decagons are merely the “bleeding ends” where we’ve chopped these legs off, and they have no mathematical significance other than to remind us that the ‘legs’ are decagonal in cross-section. There happen to be 12 of them, and the whole thing obviously has the usual icosahedral/dodecahedral symmetry – though again we have to think of them in pairs: so this is really just a conjunction of 6 decagonal prisms derived from a Hemi-polyhedron which originally had some faces passing through its very centre.
And which was that original Hemi-polyhedron? It turns out there are two, both leading to this same Dual: they’re our old friends the Small Dodecahemidodecahedron and the Small Icosihemidodecahedron.
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If you try to figure out how the Dual relates to these, it’s easy to see that both of them have 6 ‘hemi’ planes with 10 sides (those big orange and/or blue decagons) so these ‘obviously’ become 10-fold ‘vertices at infinity’ in the Dual. There are 6 of them in each case, and if you make the same jump of considering the ‘legs’ in pairs (so 12 becomes 6) we can see how it all works.
Again, it’s less clear where the blue pentagons and the red triangles go to in the Dual. They ought, in turn, to go to 5-fold and 3-fold vertices somewhere, but in fact they disappear somewhere into the inside and become part of the complex internal structure once the pink legs start intersecting each other. In these days of software rendering there ought to be ways to show this, but I’m not aware anyone has bothered: so we’re rather stuck with Wenninger’s view copied into Mathworld and Wikipedia: that the lovely 10-legged Dual we have here should be called both the Icosihemidodecacron and the Dodecahemidodecacron.
That’s quite enough hard thinking for the moment: on the next few pages we’ll record the detailed job of building these two, and a further three, Hemi-polyhedron duals.