Step prism

A step prism is an isogonal polychoron whose vertices swirl in correspondence to a star polygon. The n-d step prism is a subsymmetrical faceting of an n-n duoprism.

Most often, the cells of step prisms are phyllic disphenoids. However, antiprisms can also appear as cells, as in the 9-3 step prism with triangular antiprisms, or as in the 15-5 step prism with pentagonal antiprisms.

The duals of the step prisms are called the gyrochora. These are notable, as they make fair dice in four dimensions.

Construction
To construct the n-d step prism, one starts with a grid of n × n squares. One takes any vertex as the starting vertex, and repeatedly moves one step to the right, and d steps up. One identifies opposite edges of the square sheet: in other words, when one reaches the rightmost or the uppermost edge, one wraps around to the leftmost or to the lowermost edge, respectively. When one reaches the starting vertex, the grid is folded into an n-n duoprism. Finally, one takes the convex hull of all the traversed vertices.

As a consequence of this construction, the n-d and n-(n–d) step prisms are congruent, as the latter can be constructed from the former by going d steps down instead of d steps up. Furthermore, when n and d are coprime and the modular inverse d–1 of d modulo n exists, the n-d–1 and n-(n–d–1) will also be congruent to the aforementioned step prisms, as these can be constructed by exchanging horizontal steps with vertical steps. If n-(n-d) is equal to d, then the n-d step prism will have double the symmetry. An example is the 13-5 step prism, where 13-(13-5) = 5.

This construction also reveals a correspondence between step prisms and star polygons: just like the n vertices of an {n/d} star take d turns around a circle, the n vertices of an n-d step prism take d turns around a duocylinder.

Vertex coordinates
Coordinates for the vertices of an n-d step prism with circumradius $\sqrt{2}$ are given by: for k ranging from 0 to n–1.
 * (cos(2πk/n), sin(2πk/n), cos(2πdk/n), sin(2πdk/n)),

Examples
11-2 and 11-5: 11-2 step prism

11-3 and 11-4: 11-3 step prism

12-2: 12-2 step prism

12-3: 12-3 step prism

12-4 12-4 step prism

12-5: 6-6 duopyramid

13-2 and 13-6: 13-2 step prism

13-3 and 13-4: 13-3 step prism

13-5: 13-5 step prism