On Tuesday 18 November 2008, Mike Reid wrote:
[me, about making a polyhedron with many faces, all congruent:]
No. Take a skew prism -- lots of congruent triangular faces -- and put a cap on each end made out of triangles of the same dimensions.
[Mike:]
just a minor nit. this cannot always be done; it depends upon the antiprism. er, ... did i interpret this correctly "skew prism" = "(what i've always known as an) antiprism"?
Er, yes, "antiprism" was the word I was looking for. And I didn't mean to suggest that it can be done for an arbitrary antiprism -- of course it can't -- but only that there's a construction that works that way. Which there is :-).
anyway, it can be done if the isosceles triangles are sufficiently tall, and that suffices for the "no" answer above. this construction gives a polyhedron with 4n faces. however, the "belt" around the antiprism is not needed!
D'oh, of course. Silly me.
YOU feel silly? I implemented http://gosper.org/congru.htm < 4yrs ago! Of course, my original interest was "ball shaped" polyhedra, if we can make that precise. I think I can do 216. --rwg
are there examples of polyhedra with all faces congruent, that have an odd number of faces? (i did not see any parity restriction, although the faces must have an even number of sides.)
If you don't mind nonconvexity, it's pretty easy to make one out of squares (all meeting at right angles).
For a convex polyhedron, the number of sides per face must be less than 6 (think curvature)[1], so if it's even it must be 4. So, v-e+f=2; 4e=2f by counting (face,edge) pairs, so e=2f and v-f=2. Counting (face,vertex) pairs we have 4f = (avg vertex degree).v, so since v is bigger we have avg vertex degree < 4 but (for large face count) it's very close to 4. I suspect that this is impossible on the grounds that (handwaving again) many vertices must have at least 4 "average" angles at them, and therefore must be either flat (no!) or concave.
[1] Note: I'm handwaving here and haven't actually checked that what I say is true, but I'm pretty sure it's obvious :-).
Anyone fancy either filling in the details or refuting my handwaving?
-- g
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