User talk:Tomruen/regular polychora

Coxeter's appendix to Regular Polytopes lists them in pairs to make duals stand out. Is there a way to make some of the horizontal rules heavier than others? --Anton Sherwood 17:57, 10 January 2006 (UTC)[reply]

First attempt: Row colors added, gray=self-Dual, red/blue pairs for duals

I think I can make a blank row with tiny height black as well, but not tried yet

Tom Ruen 22:42, 10 January 2006 (UTC)[reply]

I'd like to add information about existence - WHY these 16 exist and why other {p,q,r} permutations don't.... Help is appreciated. Tom Ruen 20:53, 11 January 2006 (UTC)[reply]

Coxeter's Regular Polytopes §7.7–§7.8 explains it, but it's a bit over my head. --Anton Sherwood 23:27, 11 January 2006 (UTC)[reply]

Basically, {p,q} and {q,r} must be in the 3d polyhedra. Simple criteria like margin-angle dihedral angle rule out things like {3,5,3} etc. But beyond that, there is still permitted by this rule {3,5/2,3}, {5/2,3,5/2}, {5/2,3,4} and {4,3,5/2}. One needs subtler criteria here. {5/2,3,4} has as an equator {5/2,4}, which is not one of the nine. So this and its dual disappears. For {5/2,3,5/2}, one notes that the equator of x5/2o3o5/2x is x5/2o5/2x (ie {5/2,4}, and this also is ruled out. For the remaining figure {3,5/2,3}, one notes that it contains the vertices and edges of {5/2,4,3}, which also implies {5/2,4}. Wendy.krieger 09:56, 22 September 2007 (UTC)[reply]

Coxeter's ordering of these is not only the duals, but of increasing density, and order of stellation. {5/2,5,3} and {3,5,5/2} are density 4, the first derives from stellating the edges of the 533. {5,5/2,5} is d6, arises from replacing the pentagrams of {5/2,5,3} with pentagons: the vertex figure becomes stellated. {5/2,3,5} is then the stellation of the edges of {5,5/2,5}. Replacing the {5/2,3} with the normal dodecahedron, one gets the dual (also d=20), {5,3,5/2}. Stellating the dodecahedra of this leads to the d66 {5/2,5,5/2}. As before, one can replace pentagrams with pentagons, which stellates the vertex-figure. This leads to the d76 {5,5/2,3} and its dual {3,5/2,5}. Stellating {5,5/2,3} leads directly to {5/2,3,3} and its dual {3,3,5/2}. These are d191. This is listed in Coxeter's Regular polytopes. Wendy.krieger 07:11, 22 September 2007 (UTC)[reply]