15-5 step prism

15-5 step prism
(OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Elements
Cells	15 phyllic disphenoids, 3 pentagonal gyroprisms
Faces	30 scalene triangles, 15 isosceles triangles, 3 pentagons
Edges	15+15+15
Vertices	15
Vertex figure	Tetragonal antiwedge
Measures (circumradius $\sqrt2$, based on a unit duoprism)
Edge lengths	3-valence (15): $\sqrt{\frac{5-\sqrt5}{2}} ≈ 1.17557$
	4-valence (15): $\frac{\sqrt{19-\sqrt5-\sqrt{30-6\sqrt5}}}{2} ≈ 1.78127$
	3-valence (15): $\frac{\sqrt{19+\sqrt5-\sqrt{30+6\sqrt5}}}{2} ≈ 1.91357$
Central density	1
Related polytopes
Dual	15-5 gyrochoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	S2(I2(15)-5), order 30
Convex	Yes
Nature	Tame

The 15-5 step prism is a convex isogonal polychoron and a member of the step prism family. It has 3 pentagonal gyroprisms and 15 phyllic disphenoids as cells, with 4 phyllic disphenoids and 2 pentagonal gyroprisms joining at each vertex.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$\frac{\sqrt{5+\sqrt5-\sqrt{6-\frac{6}{\sqrt5}}}}{2}$ ≈ 1:1.16349.

Vertex coordinates

Coordinates for the vertices of a 15-5 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/3), b*cos(2πk/3)),

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:$\sqrt{\frac{3-\sqrt5+\sqrt{30-6\sqrt5}}{12}}$ ≈ 1:0.63484.

Isogonal derivatives

Substitution by vertices of these following elements will produce these convex isogonal polychora:

• Phyllic disphenoid (15): 15-5 step prism
• Scalene triangle (15): 15-5 step prism
• Scalene triangle (30): 30-5 step prism
• Edge (15): 15-5 step prism

External links

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