Step Prisms

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

Step Prisms

Postby Mercurial, the Spectre » Fri Aug 04, 2017 5:08 pm

Jonathan Bowers stated earlier the existence of step tegums, which have identical cells that have mirror-symmetric faces. However, we don't have much information of their duals, the step prisms, as Stella4D is not a free program.
Based on what I have researched over the past month, I have explored the world of these oddballs - a world no one has gone before.

Step prisms, as Bowers described, are formed from a square lattice. Starting from the first vertex, one goes a row down and moves a number of steps, marking each subsequent vertex, which is repeated until it reaches the first vertex again (the square lattice is a topological duocylinder, since it wraps over like a loop). Then if one folds the lattice into a duocylinder and selects the marked vertices, forming its convex hull, one gets a step prism.

These are all isogonal and are denoted by p-q, where p is the number of vertices (must be 5 or greater) selected and q is a step counter (determines how many steps should be moved in the lattice). q must be greater than 1 (or else the result is a flat polygon) and less than p/2 (if p is even and q = p/2-1, p-q is then a p/2-gonal duoprism).

Additionally, a p-q step prism will have double symmetry (its vertex figure has double the symmetry) if q^2 happens to be either congruent to 1 or p-1 (mod p). Thus, a 13-5 step prism (5^2 mod 13 = 12) has double symmetry, but a 13-2 step prism won't (since 2^2 mod 13 = 4).

Generalizing this, let's add another step: m. If q*m mod p is congruent to 1 or p-1 for some value of q or m and is within the constraints, then p-q is the same as p-m. For example, a 7-2 step prism is the same as the 7-3 step prism (since 2*3 mod 7 = 6), same goes for the 16-3 and 16-5 (3*5 mod 16 = 15).

In addition, p-q has central symmetry if and only if p is even and q is odd. Therefore, the 14-3 is even, but the 12-3 is not.

The total symmetry of these step prisms (based on the duoprism construction) is 2p, if it has double symmetry, then it is 4p.

The coordinates of a p-q step prism are:
(cos(2n*pi/p),sin(2n*pi/p),cos(2i*pi/p),sin(2i*pi/p)) where n is incrementally added by a multiple of q, and i ranges from 1 to p. I'm still not sure whether these set of coordinates would actually produce a step prism, so someone, please check the results.

Thank you,
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Mercurial, the Spectre
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