It is also possible to have convex (as well as non-convex) dodecahedra with all faces congruent non-regular pentagons. One can even make them with all vertices rational points in R^3. I think the faces can be equilateral as well, but it has been so long since I looked at this, I'd have to check that part. Jim Buddenhagen On Mon, Aug 8, 2011 at 5:37 AM, Bill Gosper <billgosper@gmail.com> wrote:

Julian flew back from math camp today (on one hr's sleep) and, generalizing a question of mine, found a dodecahedron with (planar) equilateral pentagonal faces, but not the usual ones. Before peeking <http://gosper.org/dodohedron.png>, you might try to find one. After peeking, please speak up if you've seen it before. Tnx, --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

Well, I guess I got it -- don't read the rest of this if you haven't tried to guess it yourself. I visualized a convex pentagon shaped like a letter M resting on a horizontal line, and figured that at least two different types of faces would be required to make the dodecahedron, and naturally it needs to a convex 3D shape. So I kinda got it although I didn't work out the whole thing (and no, I have not seen it before). How many internets did I win? (I waited most of the day before responding so as not to spoil it too quickly :-) On Mon, Aug 8, 2011 at 06:37, Bill Gosper <billgosper@gmail.com> wrote:

Julian flew back from math camp today (on one hr's sleep) and, generalizing a question of mine, found a dodecahedron with (planar) equilateral pentagonal faces, but not the usual ones. Before peeking < http://gosper.org/dodohedron.png>, you might try to find one. After peeking, please speak up if you've seen it before. Tnx, --rwg

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