15-4 step prism

15-4 step prism
File:15-4 step prism.png (OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Elements
Cells	30+30 phyllic disphenoids
Faces	60 scalene triangles, 30+30 isosceles triangles
Edges	15+30+30
Vertices	15
Vertex figure	Paratetraaugmented digonal scalenohedron
Measures (circumradius $\sqrt2$, based on a uniform duoprism)
Edge lengths	6-valence (30): $\sqrt{\frac{7-\sqrt5}{2}} ≈ 1.54336$
	6-valence (15): $\sqrt{5-\sqrt5} ≈ 1.66251$
	3-valence (30): $\sqrt{\frac{7+\sqrt5}{2}} ≈ 2.14896$
Central density	1
Related polytopes
Dual	15-4 gyrochoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	S2(I2(15)-4)×2R, order 60
Convex	Yes
Nature	Tame

The 15-4 step prism is a convex isogonal polychoron and a member of the step prism family. It has 60 phyllic disphenoids of two kinds as cells, with 16 joining at each vertex.

Compared to other 15-vertex step prisms, this polychoron has doubled symmetry, because 15 is a factor of 4²-1 = 15.

The ratio between the longest and shortest edges is 1:$\frac{\sqrt{594+154\sqrt5}}{22}$ ≈ 1:1.39239.

Vertex coordinates

Coordinates for the vertices of a 15-4 step prism of circumradius √2 are given by:

• (sin(2πk/15), cos(2πk/15), sin(8πk/15), cos(8πk/15)),

where k is an integer from 0 to 14.

External links

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