On Wed, Jun 11, 2003 at 06:34:03PM -0400, George W. Hart wrote:
asimovd@aol.com wrote:
...PUZZLE: Given a rhombus tiling of a 2n-gon (so that each polygon edge is a rhombus edge), show that the number of rhombi is determined by n.
The question of counting how many solutions there are, especially in the 3D case of filling a zonohedron with parallelepipeds, is an interesting open question. For a large model and some references, see: G. Hart, "A Color-Matching Dissection of the Rhombic Enneacontahedron", Symmetry: Culture and Science, vol. 11, 2000, pp. 183-199. online at: http://www.georgehart.com/private/hart-v1.doc
Any chance of a version that is not in .doc format (e.g., PostScript)? I don't have Microsoft Word, and for some reason the pictures of the sculptures didn't come out in my version. On trying to count efficiently: I wonder if it's possible to apply your technique of picking an equatorial band repeatedly, picking, say, two different directions and looking at all the possibilities for the two cases. Of course the two will interact, but maybe it's possible to control that. On the other hand, isn't it possible to think of a decomposition of a 3-d zonotope into parallelipipeds as a sequence of decompositions of a 2-d zonotope into parallelograms? I guess that's what this paper: http://www.liafa.jussieu.fr/~latapy/Publis/Pav/abstract http://www.liafa.jussieu.fr/~latapy/Publis/Pav/Pav.ps.gz is about. I'm surprised that it's not possible to extend the sequence at least one more step. Peace, Dylan Thurston