Step prism
A step prism is a type of convex isogonal polytope, with infinite families in all even dimensions greater than two. In the 4D case, the n vertices of an n-d step prism are given by:
- (a*sin(2πk/n), a*cos(2πk/n), b*sin(2πdk/n), b*cos(2πdk/n)),
where k is an integer ranging from 0 to n – 1 and a and b are two nonzero real numbers. The polytope itself is simply the convex hull of these points. Similarly to how the n vertices of an {n/d} star take d turns around a circle, the n vertices of a 4-dimensional n -d step prism take d turns around a Clifford torus. The n -d step prism is a faceting of an n -n duoprism, but in general has less symmetry than the latter.
The duals of the step prisms are called the gyrotopes. These are notable, as they are isotopic. Particularly, there exist fair dice in any even dimension with any amount of facets greater than the simplex. This is in contrast to 3D, where there are no fair dice with 5 faces, for instance.
In four dimensions, an n -d step prism has double symmetry if n is a divisor of d 2 +1 (which gives ionic symmetry similar to the tetragonal disphenoid) or d 2 -1 (which gives reflective symmetry similar to the rhombic disphenoid). 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).
In six dimensions, an n -m -k step prism can have double or triple symmetry, but the processes involved are much more complicated.
The convex hull of two or more step prisms in extended symmetry can be used to construct form new isogonal polytopes. For example, in four dimensions, the small 15-4 double step prism can be constructed from two 13-5 step prisms, and in six dimensions, the rectified heptapeton can be constructed from three 7-2-3 step prisms.
The symmetry of a step prism generally has the same order as the polygonal base, and is unique with respect to other step prisms based on the same polygon. For example, the 14-2 step prism and 14-3 step prism have different symmetries despite both having order 28, the same as the tetradecagon. In fact, only the 14-3 step prism has central inversion symmetry.
Construction[edit | edit source]
In four dimensions, to construct the n -d step prism, one starts with a n × n square grid. One can roll the grid so that opposite edges meet. This puts the vertices on the surfaces of a flat torus, and creates in essence a n -gonal duocomb. One takes any arbitrary vertex as the starting vertex, and repeatedly moves one step to the right, and d steps up. 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.
Alternatively, the step prisms can also be constructed as the convex hull of a regular skew polygon. Specifically the n -d step prism in four dimensions is the convex hull of {n}#{n/d}, that is the blend of {n} with {n/d}.
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. In addition, this construction can be extended for any higher-dimensional step prism based on a polygon. For example, in six dimensions, the n -m -k step prism is formed as the convex hull of {n}#{n/m}#{n/k}, or by 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.
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. This can also lead to extended symmetry, as the properties can be interchanged.
It is also possible to construct step prisms based on the blends of higher rank polytopes (i.e. not polygons). The simplest example is the 6D icosahedral-great icosahedral step prism, which is the hull of the skew icosahedron. Since the dimension of the blend (6) is half the number of vertices in the components (12), the skew icosahedron is an orthoplex realization and its convex hull is the 6-orthoplex.
Vertex coordinates[edit | edit source]
Coordinates for the vertices of an n-d step prism with height ratios a and b are given by:
- (a*sin(2πk/n), a*cos(2πk/n), b*sin(2πdk/n), b*cos(2πdk/n)),
for k ranging from 0 to n–1.
This can be generalized into higher dimensions, each time adding one degree of freedom per two dimensions.
Facets[edit | edit source]
In four dimensions, the cells of the n -d step prism are usually 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 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.
In higher dimensions, the facets are usually bilaterally-symmetric simplexes if all variables are coprime with the number of vertices of the polygon, with other facets possible if they are not coprime.
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 |
External links[edit | edit source]
- Bowers, Jonathan. "Four Dimensional Dice Up To Twenty Sides".