Pyrorectichoron images

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Pyrorectichoron images

Postby Keiji » Sun Mar 13, 2016 2:01 pm

I've been working on the Pyrorectichoron page. The stereographic projection was nabbed from Wikipedia, but the Petrie polygon was worked out and drawn by me. Can anyone check that it's correct?
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Re: Pyrorectichoron images

Postby wendy » Sat Apr 23, 2016 12:29 pm

Coxeter's definition of the Petrie Polygon is very messy, and the two definitions really work on regular cases only.

One definition is the 'petrie zigzag', where two edges make the hedron, three edges to each choron, and so forth, except for the point and content.

The second definition is the 'petrie path', where one reflects a point repeatedly through the mirrors in order, thus 1,2,3,4,1,2,3,4,&c. This means that the petrie polygon has xh verticies, where x is the marked nodes and h is the petrie polygon of the group. We also have nh = 2m, where n is the dimension, h the petrie polygon, and m is the number of mirrors, ie 4*5 = 2*10.

The figure here is o3x3o3o, has only one kind of edge. This means the petrie path is a pentagon, and the circumdiameter-square is less than 2.8944, but the figure itself has a diameter-square of 2.4000, which means that the thing is pretty flat. The vertex-figure of this is a triangle-prism, ie x(3.3)x3o, and i suggest that the polygon produced by the petrie path is a girthing polygon, which goes to two unconnected vertices of the vertex-figure (ie a diagonal of the square face).
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