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 d^{2} 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 d^{2} 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 antiprismatic cells (including those with n and d both even), such as the 8-2 step prism with rhombic disphenoids (considered as digonal antiprisms), 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 d^{2}+1 or d^{2}–1. Examples include the 13-5 step prism (13 divides 5^{2}+1 = 26), 17-4 step prism (17 divides 4^{2}+1 = 17), and the 30-11 step prism (30 divides 11^{2}–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".