What about stellated polyhedra? Jim Propp On Friday, July 17, 2015, Veit Elser <ve10@cornell.edu> wrote:
If the angle sums are 2pi/k1, 2pi/k2, … for integers k1, k2, … greater than 1, then the curvature sum-rule implies
4pi =2pi (1-1/k1)+2pi (1-1/k2)+ …
But the only solution is the case where there are just four terms, and k1=k2=k3=k4=2.
If you allowed for general rational multiples of 2pi there would be more solutions (cube, regular icosahedron, …).
-Veit
On Jul 16, 2015, at 11:35 PM, James Propp <jamespropp@gmail.com <javascript:;>> wrote:
I want to think about rolling polyhedra, and the polyhedra I want to think about rolling are polyhedra in which the sum of the angles at each vertex is a submultiple of 360 degrees.
For instance, given any triangle T, we can create a tetrahedron whose four faces are all congruent to T, with angle-sum 180 degrees at each vertex (there's a name for such tetrahedra but I forget what it is).
Are there other polyhedra in which the angles at each vertex sum to a submultiple of 360 degrees (not necessarily the same one at each vertex)?
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