Coxeter-Stott Variations

Notion

This is a systematic distortion of polytopes by contraction and dilation of the edges. The systen is very good for small dimensions, where there is not a lot of things to do, and lots of things to do it on.

Notation

The notation consists of adding a modifier prefix to the polytope being modified. For example, tt{3,3,5} is a truncate tetrahedron. It is often applied to the Schlafli Symbol.

(Unmodified): This is the unmodified polytope. The primitive polytope normally heads up the table column as the first row.

r

Rectified: This is a limiting case on truncations where the verticies meet again. The Nth rectified meets in the centres of spaces of N dimensions.

t

Truncates: The truncates are made by removing the verticies of a polytope.

bt

Bitruncate: This class has two kinds of facets, each of which are truncates themselves.

c

Cantellated:

rr

Runcinated: This us effected by radially moving the facets away from the figure, but not increasing the size. Margins are thence replaced by prisms, and assorted prisms appear in the higher margin. The vertex is replaced by the face of the dual.

ct

Cantetruncated:

rt

Runcitruncated:

ot

Omnitruncated:

s

Snub: This term properly means a figure deruived by taking alternate verticies of the omnitruncate.

In three dimensions, there are three degrees of freedom, and three edges, so the class regularly forms with equal edges.

In four dimensions, there are four degrees of freedom, and six edges. This rarely forms a figure with all edges equal.

ss

SSnub: This is not a singular operator itself. Its role is to accomidate the reaminig two uniform convex figures in 4D, which are read as ss{3,4,3} and ss{3,3,5}

The ss{3,4,3} is actually derived from alternate verticies of a truncated {3,4,3}, not an omnitruncated one. It may be derived by removing 24 verticies of an inscribed {3,4,3} from a {3,3,5}. This replaces clusters of 20 tetrahedra with an icosahedron.

The ss{3,3,5} is derived from the {3,3,5} by removing just two of its girthing decagons: two that are furtherest apart. Each vertex would had been replaced by an icosahedron, but each of these loose a polar region, leaving just a pentagonal antiprism.

Comment

The notation has a large number of parallel irregular terms. While this cuts the number of different figures down by paired constructions, the constructions themselves are not further analised.

None the less, this notation does not rely heavily on assorted assumptions about symmetry groups, and is fairly general. The notion of "define and extend" makes it useful for exploring and classifying new polygopes.

Copyright 2002 Wendy Krieger