15-3 step prism

15-3 step prism
(OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Elements
Cells	15+15+15 phyllic disphenoids, 5 triangular gyroprisms
Faces	30+30+30 scalene triangles, 15 isosceles triangles, 5 triangles
Edges	15+15+15+15+15
Vertices	15
Vertex figure	Base-ditriakis tetragonal antiwedge
Measures (circumradi $\sqrt2$, based on a uniform duoprism)
Edge lengths	8-valence (15): $\frac{\sqrt{17-3\sqrt5-\sqrt{30-6\sqrt5}}}{2} ≈ 1.24695$
	3-valence (15): $\sqrt3 ≈ 1.73205$
	4-valence (15): $\frac{\sqrt{17-3\sqrt5+\sqrt{30-6\sqrt5}}}{2} ≈ 1.89500$
	3-valence (15): $\frac{\sqrt{17+3\sqrt5-\sqrt{30+6\sqrt5}}}{2} ≈ 2.06876$
	4-valence (15): $\sqrt5 ≈ 2.23607$
Central density	1
Related polytopes
Dual	15-3 gyrochoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	S2(I2(15)-3), order 30
Convex	Yes
Nature	Tame

The 15-3 step prism is an isogonal polychoron and a member of the step prism family. It has 5 triangular gyroprisms and 45 phyllic disphenoids of three kinds as cells, with 12 phyllic disphenoids and 2 triangular gyroprisms joining at each vertex.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$\frac{\sqrt{60+6\sqrt5+6\sqrt{15+6\sqrt5}}}{6}$ ≈ 1:1.71108.

Vertex coordinates

Coordinates for the vertices of a 15-3 step prism inscribed in a pentadecagonal duoprism with base lengths a and b are given by:

• (a*sin(2πk/15), a*cos(2πk/15), b*sin(2πk/5), b*cos(2πk/5)),

where k is an integer from 0 to 14. If the edge length differences are to be minimized, the ratio of a:b must be equivalent to 1:$\frac{\sqrt{75+25\sqrt5+5\sqrt{150+30\sqrt5}}}{10}$ ≈ 1:1.43028.

External links

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