Step prism

A step prism is an isogonal polychoron whose vertices swirl in correspondence to a star polygon. Similarly to how 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. The n-d step prism is a subsymmetrical faceting of an n-n duoprism. Some step prisms can be swirlchora based on dihedra.

Most often, the cells of the n-d step prism are phyllic disphenoids. More precisely, if either n and d are coprime and the step prism does not have doubled symmetry, or if n is odd and the step prism is a swirlchoron with doubled symmetry, it will only contain phyllic disphenoids. However, if n and d are coprime, if the step prism has double symmetry, and if isn't a swirlchoron, then it will contain tetragonal disphenoids. If n and d are both even, or if n is even and d is coprime with doubled symmetry and the step prism is a swirlchoron, it will contain rhombic disphenoids. Finally, if n and d have a greatest divisor larger than 2, it will contain antiprismatic cells, such as the 9-3 step prism with triangular antiprisms and 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. Particularly, since there's at least one non-degenerate n-d step prism for each n≥5, there exist fair four-dimensional dice with any amount of cells, starting from 5. This is in contrast to 3D, where there are no fair dice with 5 faces, for instance.

An n-d step prism has double symmetry if n is a divisor of d2+1 or d2–1. Examples include the 13-5 step prism (13 divides 52+1 = 26), 17-4 step prism (17 divides 42+1 = 17), and the 30-11 step prism (30 divides 112–1 = 120).

Step prisms with double symmetry can be compounded to form new isogonal polychora, such as the tetragonal-antiwedge 8-3 double step prism from two 8-3 step prisms.

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.

This construction only works for 2 ≤ d ≤ n–2. When d ∈ {0, 1, n–1, n}, all constructed points are coplanar, and the step prism degenerates into a regular polygon.

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, so that 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.

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

Special cases
In four dimensions, an n-d step prism can have the least possible edge length difference by varying the height ratio (the ratio of the edge lengths of the orthogonal n-gons of an n-gonal duoprism used to create a step prism). For double symmetry cases, the ratio is 1. Known examples for other cases are (using the absolute value method):
 * 7-2: 1:$\sqrt{1–sec(π/7)/2}$ ≈ 1:0.66711
 * 11-2: 1:$\sqrt{1–1/(1+2cos(2π/11))}$ ≈ 1:0.79197
 * 11-3: 1:$\sqrt{2sec(2π/11)}$/2 ≈ 1:0.77094
 * 12-2: 1:$\sqrt{27|4}$/3 ≈ 1:0.75984
 * 13-3: 1:$\sqrt{(cos(3π/13)+cos(4π/13))/(cos(π/13)+cos(4π/13))}$ ≈ 1:0.92492
 * 14-3: 1:1/$\sqrt{2+2cos(2π/7)|4}$ ≈ 1:0.74496
 * 14-4: 1:$\sqrt{(1+cos(2π/7))/(1+cos(π/7))}$ ≈ 1:0.92414
 * 15-2: 1:$\sqrt{2√15+6√5-2√5-4}$/2 ≈ 1:0.73981
 * 15-3: 1:$\sqrt{75-25√5+5√150-30√5}$/10 ≈ 1:0.80392
 * 15-6: 1:$\sqrt{50+10√75-30√5}$/10 ≈ 1:0.88396
 * 16-2: 1:1/$\sqrt{2+√2|4}$ ≈ 1:0.73566
 * 16-3: 1:$\sqrt{1-√2+√4-2√2}$ ≈ 1:0.81742
 * 16-6: 1:$\sqrt{8-4√2+2√4-2√2}$ ≈ 1:1.06159
 * 17-3: 1:$\sqrt{(cos(5π/17)+sin(π/34))/(cos(2π/17)+sin(3π/34))}$ ≈ 1:0.75904
 * 17-5: 1:$\sqrt{(cos(3π/17)+cos(4π/17))/(cos(2π/17)+cos(3π/17))}$ ≈ 1:0.94418
 * 18-3: 1:$\sqrt{2}$/2 ≈ 0.70711
 * 18-4: 1:cos(π/9)sec(π/18) ≈ 0.95419
 * 18-5: 1:$\sqrt{cos(2π/9)sec(π/9)}$ ≈ 0.90289
 * 20-4: 1:$\sqrt{125|4}$/5 ≈ 1:0.66874
 * 20-8: 1:$\sqrt{5000-1000√5|4}$/10 ≈ 1:0.72507
 * (4n+1)-2: 1:1/$\sqrt{2cos(π/(4n+1))-2sin(pi/(8n+2))}$
 * (4n+2)-2: 1:cot(nπ/(4n+2))/$\sqrt{2+2sin(π/(4n+2))}$
 * (2n+2)-n: 1:1
 * 3n-n: 1:2$\sqrt{sin(2π/3n)sin(4π/3n)}$/$\sqrt{3}$
 * 4n-n: 1:$\sqrt{cos(π/n)-cos(2π/n)}$/$\sqrt{2}$

Using the ratio method: specific: 5-2: 1:1 6-2: 1:1 7-2: 1:sqrt(2cos(pi/7)) = 1:1.34236 8-2: 1:sqrt(4+2sqrt(2))/2 = 1:1.30656 8-3: 1:1 9-2: 1:sqrt(1+2cos(pi/9)) = 1:1.69688 9-3: 1:sqrt(3+6cos(2pi/9))/3 = 1:0.91871 10-2: 1:(1+sqrt(5))/2 = 1:1.61803 10-3: 1:1 10-4: 1:1 11-2: 1:1/sqrt(2cos(pi/11)-2cos(2pi/11)) = 1:2.05638 11-3: 1:sqrt((1+2sin(3pi/22))/(2cos(pi/11)+2sin(pi/22)-1)) = 1:1.23333 12-2: 1:sqrt(2+sqrt(3)) = 1:1.1.93185 12-3: 1:sqrt(2+2sqrt(3))/2 = 1:1.16877 12-4: 1:sqrt(sqrt(27))/3 = 1:0.75984 12-5: 1:1 13-2: 1:1/sqrt(2cos(pi/13)-2cos(2pi/13)) = 1:2.41846 13-3: 1:sqrt((cos(2pi/13)+sin(3pi/26))/(cos(2pi/13)-sin(pi/26))) = 1:1.27325 13-5: 1:1 14-2: 1:1+2cos(2pi/7) = 1:2.24698 14-3: 1:sqrt((2cos(pi/7)+2sin(pi/14)-1)/(1-2sin(pi/14))) = 1:1.49899 14-6: 1:1 15-2: 1:sqrt(7+3sqrt(5)+sqrt(150+66sqrt(5)))/2 = 1:2.78203 15-3: 1:sqrt(75+25sqrt(5)+5sqrt(150+30sqrt(5)))/10 = 1:1.43028 15-4: 1:1 15-5: 1:sqrt(9-3sqrt(5)+3sqrt(30-6sqrt(5)))/6 = 1:0.63484 15-6: 1:sqrt(50+10sqrt(75-30sqrt(5)))/10 = 1:0.88396 16-2: 1:sqrt(8+4sqrt(2)+2sqrt(20+14sqrt(2)))/2 = 1:2.56292 16-3: 1:sqrt(1+sqrt(2)) = 1:1.55377 16-4: 1:sqrt(sqrt(8+4sqrt(2)))/2 = 1:0.96119 16-7: 1:1 17-2: 1:1/sqrt(2cos(pi/17)-2cos(2pi/17)) = 1:3.14656 17-3: 1:sqrt((sin(7pi/34)+cos(2pi/17))/(cos(2pi/17)-sin(5pi/34))) = 1:1.77592 17-4: 1:1 18-2: 1:csc(pi/18)/2 = 1:2.87939 18-3: 1:sqrt(1+2cos(pi/9)) = 1:1.69688 18-6: 1:sqrt(6cos(pi/9)-3)/3 = 1:0.54141 18-8: 1:1 19-2: 1:1/sqrt(2cos(pi/19)-2cos(2pi/19)) = 1:3.51173 19-3: 1:sqrt((sin(5pi/38)+cos(2pi/19))/(cos(2pi/19)-sin(7pi/38))) = 1:1.83802 20-2: 1:sqrt(12+4sqrt(5)+2sqrt(50+22sqrt(5)))/2 = 1:3.19623 20-3: 1:sqrt(2+sqrt(5)) = 1:2.05817 20-4: 1:sqrt(sqrt(625+250sqrt(5)))/5 = 1:1.17319 20-5: 1:sqrt(1-sqrt(5)+sqrt(10+2sqrt(5)))/2 = 1:0.80127 20-8: 1:sqrt(sqrt(5000-1000sqrt(5)))/10 = 1:0.72507 20-9: 1:1 general: n-2: 1:sqrt((cos(2pi/n)-cos(2pi*floor(n/2)/n))/(cos(4pi*floor(n/2)/n)-cos(4pi/n))) n-3: 1:sqrt((cos(2pi*floor((n+1)/3)/n)-cos(2pi/n))/(cos(6pi/n)-cos(6pi*floor((n+1)/3)/n))) 2n-(n-1): 1:1 a*n-n: 1:sqrt((cos(2pi/n)-cos(2pi/(a*n)))/(cos(2pi/a)-1))

Higher dimensional generalizations
Analogs of 4D step prisms exist in all even dimensions, formed in an analogous way from higher multiprisms. For example, a 6D n-m-k step prism would be formed from a faceting of the n-gonal trioprism, where we start with a 3D grid of cubes and move 1 step in one direction, m steps in another, and k steps in the third direction.