Here's a puzzle for which I found a nice but degenerate solution: Find a toroidal polyhedron of genus 1 whose corners are all "flat" in the sense that the angles of the faces meeting at each corner add up to 360 degrees. (I agree that there a number of inequivalent ways to interpret the problem, but I don't know how to clarify the statement of the problem without giving too big a hint!) I'm curious whether equally nice but less degenerate examples are known. Jim
Before Bill Thurston steps in and cleans up here, I think I'll just put in my two-penn'orth ... A rather elegant class of degenerate solutions --- almost certainly more disreputable than you had in mind --- are provided by "kaleidocycles", or rotating rings of tetrahedra. See Marcus Engel's elegant "AniKa" animations at http://www.kaleidocycles.de/anim.shtml I'd guess that you found an example with 12 vertices, 24 edges, 12 faces which are all self-intersecting quadrilaterals --- though I must admit to not having actually computed any coordinates! Fred Lunnon On 8/7/09, James Propp <jpropp@cs.uml.edu> wrote:
Here's a puzzle for which I found a nice but degenerate solution: Find a toroidal polyhedron of genus 1 whose corners are all "flat" in the sense that the angles of the faces meeting at each corner add up to 360 degrees.
(I agree that there a number of inequivalent ways to interpret the problem, but I don't know how to clarify the statement of the problem without giving too big a hint!)
I'm curious whether equally nice but less degenerate examples are known.
Jim
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On 8/7/09, James Propp <jpropp@cs.uml.edu> wrote:
... Find a toroidal polyhedron of genus 1 whose corners are all "flat" in the sense that the angles of the faces meeting at each corner add up to 360 degrees. ... I'm curious whether equally nice but less degenerate examples are known.
On 8/8/09, Fred lunnon <fred.lunnon@gmail.com> wrote:
... I'd guess that you found an example with 12 vertices, 24 edges, 12 faces which are all self-intersecting quadrilaterals --- though I must admit to not having actually computed any coordinates!
A construction similar to that suggested earlier for the kaleidocycle should successfully de-degenerate this --- always assuming an example with the correct --- i.e. equal --- angle-sums exists initially. The 4 false vertices are excised, and replaced by triangular antiprisms, with sufficient freedom to juggle the new angle-sums into equality. The result would have 48 faces, 96 edges, 48 vertices --- the same statistics as has a surgically enhanced 6-cell kaleidocycle. And I still don't fancy actually doing it! WFL
On 8/7/09, James Propp <jpropp@cs.uml.edu> wrote:
... Find a toroidal polyhedron of genus 1 whose corners are all "flat" in the sense that the angles of the faces meeting at each corner add up to 360 degrees. ... I'm curious whether equally nice but less degenerate examples are known.
Thinking further about the kaleidocycle --- one could of course freeze it in (say) the position with all hinge angles equal, then replace the neighbourhoods of the hinge edges by (congruent) trapezohedra having 2 opposite faces rectangular, and 2 opposite pairs of congruent trapezia. By adjusting the angles of these last, with a little spherical trig. it should be possible to persuade the polyhedral vertices to have the required sum; but just now I don't think I can summon the required enthusiasm ... WFL
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