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Sat, 08 Sep 2007
The missing deltahedron
The number of edges that meet at a vertex is its valence. Vertices in convex deltahedra have valences of 3, 4, or 5. The valence can't be larger than 5 because only six equilateral triangles will fit, and if you fit 6 then they lie flat and the polyhedron is not properly convex. Let V_{3}, V_{4}, and V_{5} be the number of vertices of valences 3, 4, and 5, respectively. Then:
Well, this is all oversubtle, I realized later, because you don't need to do the V_{3}–V_{4}–V_{5} analysis to see that something is missing. There are convex deltahedra with 4, 6, 8, 10, 12, 14, and 20 faces; what happened to 18? Still, I did a little work on a more careful analysis that might shed some light on the 18-hedron situation. I'm still in the middle of it, but I'm trying to continue my policy of posting more frequent, partial articles. Let V be the number of vertices in a convex deltahedron, E be the number of edges, and F be the number of faces. We then have V = V_{3} + V_{4} + V_{5}. We also have E = ½(3V_{3} + 4V_{4} + 5V_{5}). And since each face has exactly 3 edges, we have 3F = 2E. By Euler's formula, F + V = E + 2. Plugging in the stuff from the previous paragraph, we get 3V_{3} + 2V_{4} + V_{5} = 12. It is very easy to enumerate all possible solutions of this equation. There are 19:
(3,1,1) fails completely because to have V_{5} > 0 you need V ≥ 6. There isn't even a graph with (V_{3}, V_{4}, V_{5}) = (3,1,1), much less a polyhedron. There is a graph with (3,0,3), but it is decidedly nonplanar: it contains K_{3,3}, plus an additional triangle. But the graph of any polyhedron must be planar, because you can make a little hole in one of the faces of the polyhedron and flatten it out without the edges crossing. Another way to think about (3,0,3) is to consider it as a sort of triangular tripyramid. Each of the V_{5}s shares an edge with each of the other five vertices, so the three V_{5}s are all pairwise connected by edges and form a triangle. Each of the three V_{3}s must be connected to each of the three vertices of this triangle. You can add two of the required V_{3}s, by erecting a triangular pyramid on the top and the bottom of the triangle. But then you have nowhere to put the third pyramid. On Thursday I didn't know what went wrong with (2,2,2); it seemed fine. (I found it a little challenging to embed it in the plane, but I'm not sure if it would still be challenging if it hadn't been the middle of the night.) I decided that when I got into the office on Friday I would try making a model of it with my magnet toy and see what happened. It turned out that nothing goes wrong with (2,2,2). It makes a perfectly good non-convex deltahedron. It's what you get when you glue together three tetrahedra, face-to-face-to-face. The concavity is on the underside in the picture.
There is a planarity failure, which is also topological, but less severe, as with (3,0,3). (3,0,3) also fails because you can't embed it into R^{3}. (I mean that you cannot embed its 3-skeleton. Of course you can embed its 1-skeleton in R^{3}, but that is not sufficient for the thing to be a polyhedron.) I'm not sure if this is really different from the previous failure; I need to consider more examples. And (3,0,3) fails in yet another way: you can't even embed its 1-skeleton in R^{3} without violating the constraint that says that the edges must all have unit length. The V_{5}s must lie at the vertices of an equilateral triangle, and then the three unit spheres centered at the V_{5}s intersect at exactly two points of R^{3}. You can put two of the V_{3}s at these points, but this leaves nowhere for the third V_{3}. Again, I'm not sure that this is a fundamentally different failure mode than the other two. Another failure mode is that the graph might be embeddable into R^{3}, and might satisfy the unit-edge constraint, but in doing so it might determine a concave polyhedron, like (2,2,2) does, or a non-polyhedron, like (2,0,6) does. I still have six (V_{3},V_{4},V_{5}) triples to look into. I wonder if there are any other failure modes? I should probably think about (0,1,10) first, since the whole point of all this was to figure out what happened to D_{18}. But I'm trying to work up from the simple cases to the harder ones. I suppose the next step is to look up the proof that there are only eight convex deltahedra and see how it goes. I suspect that (2,1,4) turns out to be nonplanar, but I haven't looked at it carefully enough to actually find a forbidden minor. One thing that did occur to me today was that a triple (V_{3}, V_{4}, V_{5}) doesn't necessarily determine a unique graph, and I need to look into that in more detail. I'll be taking a plane trip on Sunday and I plan to take the magnet toy with me and continue my investigations on the plane. In other news, Katara and I went to my office this evening to drop off some books and pick up some stuff for the trip, including the magnet toy. Katara was very excited when she saw the collection of convex deltahedron models on my desk, each in a different color, and wanted to build models just like them. We got through all of them, except D_{10}, because we ran out of ball bearings. By the end Katara was getting pretty good at building the models, although I think she probably wouldn't be able to do it without directions yet. I thought it was good work, especially for someone who always skips from 14 to 16 when she counts. On the way home in the car, we were talking about how she was getting older and I rhapsodized about how she was learning to do more things, learning to do the old things better, learning to count higher, and so on. Katara then suggested that when she is older she might remember to include 15.
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