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.

Most often, the cells of the n-d step prism are phyllic disphenoids. More precisely, if n and d are coprime, except for the following conditions listed below, it will only contain phyllic disphenoids. However, if n and d are coprime and d² is equivalent to -1 mod n, then it will contain tetragonal disphenoids. If n and d are both even, or if n is even and d² is equivalent to 1 mod n and is a divisor of 2n, it will contain rhombic disphenoids. Finally, if n and d are not coprime, it will contain gyroprismatic cells (including those with n and d both even), such as the 8-2 step prism with rhombic disphenoids (considered as digonal gyroprisms), the 9-3 step prism with triangular gyroprisms and the 15-5 step prism with pentagonal gyroprisms.

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[edit | edit source]

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[edit | edit source]

Coordinates for the vertices of an n-d step prism with height ratio x are given by:

(sin(2πk/n), cos(2πk/n), x*sin(2πdk/n), x*cos(2πdk/n)),

for k ranging from 0 to n–1.

Special cases[edit | edit source]

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 1:1. Known examples for other cases are (using the ratio method):

7-2: 1:√2cos(π/7) ≈ 1:1.34236
8-2: 1:√4+2√2/2 ≈ 1:1.30656
9-2: 1:√1+2cos(π/9) ≈ 1:1.69688
9-3: 1:√3+6cos(2π/9)/3 ≈ 1:0.91871
10-2: 1:(1+√5)/2 ≈ 1:1.61803
10-4: 1:1
11-2: 1:1/√2cos(π/11)-2cos(2π/11) ≈ 1:2.05638
11-3: 1:√(1+2sin(3π/22))/(2cos(π/11)+2sin(π/22)-1) ≈ 1:1.23333
12-2: 1:√2+√3 ≈ 1:1.1.93185
12-3: 1:√2+2√3/2 ≈ 1:1.16877
12-4: 1:⁴√27/3 ≈ 1:0.75984
12-5: 1:1
13-2: 1:1/√2cos(π/13)-2cos(2π/13) ≈ 1:2.41846
13-3: 1:√(cos(2π/13)+sin(3π/26))/(cos(2π/13)-sin(π/26)) ≈ 1:1.27325
14-2: 1:1+2cos(2π/7) ≈ 1:2.24698
14-3: 1:√(2cos(π/7)+2sin(π/14)-1)/(1-2sin(π/14)) ≈ 1:1.49899
14-4: 1:√(1+cos(2π/7))/(1+cos(π/7)) ≈ 1:0.92414
14-6: 1:1
15-2: 1:√7+3√5+√150+66√5/2 ≈ 1:2.78203
15-3: 1:√75+25√5+5√150+30√5/10 ≈ 1:1.43028
15-5: 1:√9-3√5+3√30-6√5/6 ≈ 1:0.63484
15-6: 1:√50+10√75-30√5/10 ≈ 1:0.88396
16-2: 1:√8+4√2+2√20+14√2/2 ≈ 1:2.56292
16-3: 1:√1+√2 ≈ 1:1.55377
16-4: 1:⁴√8+4√2/2 ≈ 1:0.96119
16-6: 1:√8-4√2+2√4-2√2/2 ≈ 1:1.06159
16-7: 1:1
17-2: 1:1/√2cos(π/17)-2cos(2π/17) ≈ 1:3.14656
17-3: 1:√(sin(7π/34)+cos(2π/17))/(cos(2π/17)-sin(5π/34)) ≈ 1:1.77592
17-5: 1:√(cos(3π/17)+cos(4π/17))/(cos(2π/17)+cos(3π/17)) ≈ 1:0.94418
18-2: 1:csc(π/18)/2 ≈ 1:2.87939
18-3: 1:√1+2cos(π/9) ≈ 1:1.69688
18-4: 1:2cos(π/18)/√3 ≈ 1:1.13716
18-5: 1:√cos(2π/9)sec(π/9) ≈ 0.90289
18-6: 1:√6cos(π/9)-3/3 ≈ 1:0.54141
18-8: 1:1
19-2: 1:1/√2cos(π/19)-2cos(2π/19) ≈ 1:3.51173
19-3: 1:√(sin(5π/38)+cos(2π/19))/(cos(2π/19)-sin(7π/38)) ≈ 1:1.83802
19-4: 1:√(cos(2π/19)+sin(π/38))/(cos(2π/19)-sin(3π/38)) ≈ 1:1.21179
19-7: 1:√(cos(2π/19)+cos(3π/19))/(cos(2π/19)+sin(9π/38)) ≈ 1:1.06046
20-2: 1:√12+4√5+2√50+22√5/2 ≈ 1:3.19623
20-3: 1:√2+√5 ≈ 1:2.05817
20-4: 1:⁴√625+250√5/5 ≈ 1:1.17319
20-5: 1:√1-√5+√10+2√5/2 ≈ 1:0.80127
20-6: 1:1
20-8: 1:⁴√5000-1000√5/10 ≈ 1:0.72507
20-9: 1:1
n-2: 1:√(cos(2π/n)-cos(2π*floor(n/2)/n))/(cos(4π*floor(n/2)/n)-cos(4π/n))
n-3: 1:√(cos(2π*floor((n+1)/3)/n)-cos(2π/n))/(cos(6π/n)-cos(6π*floor((n+1)/3)/n))
2n-(n-1): 1:1
a*n-n: 1:√(cos(2π/n)-cos(2π/(a*n)))/(cos(2π/a)-1)

Examples[edit | edit source]

n-d	5-2	6-2	7-2	8-2	8-3
Name	Pentachoron	Triangular duotegum	7-2 step prism	8-2 step prism	Hexadecachoron
Image

n-d	9-2	9-3	10-2	10-3	10-4
Name	9-2 step prism	9-3 step prism	10-2 step prism	Bidecachoron	Pentagonal duotegum
Image

Higher dimensional generalizations[edit | edit source]

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.

In addition, step prisms can be constructed from multiprisms of polytopes in more than 2 dimensions, by following the edges of one polytope in one direction and a member of the same army in the other. The simplest example is the 6D icosahedral-great icosahedral step prism, which happens to be the regular hexacontatetrapeton.

External links[edit | edit source]

Bowers, Jonathan. "Four Dimensional Dice Up To Twenty Sides".