16-4 step prism

16-4 step prism
(OFF file)
Rank	4
Type	Isogonal
Elements
Cells	16+16 phyllic disphenoids, 4 square gyroprisms
Faces	32+32 scalene triangles, 16 isosceles triangles, 4 squares
Edges	16+16+16+16
Vertices	16
Vertex figure	Base-triakis tetragonal antiwedge
Measures (circumradius ${\sqrt {2}}$, based on a uniform duoprism)
Edge lengths	3-valence (16): ${\sqrt {2}}\approx 1.41421$
	6-valence (16): ${\sqrt {4-{\sqrt {2+{\sqrt {2}}}}}}\approx 1.46705$
	4-valence (16): ${\sqrt {4-{\sqrt {2-{\sqrt {2}}}}}}\approx 1.79851$
	3-valence (16): ${\sqrt {6-{\sqrt {2}}}}\approx 2.14144$
Central density	1
Related polytopes
Dual	16-4 gyrochoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	S2(I2(16)-4), order 32
Convex	Yes
Nature	Tame

The 16-4 step prism is a convex isogonal polychoron and a member of the step prism family. It has 4 square gyroprisms and 32 phyllic disphenoids of two kinds as cells, with 8 phyllic disphenoids and 2 square gyroprisms joining at each vertex.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\frac {\sqrt {4-2{\sqrt {2}}+4{\sqrt {2+{\sqrt {2}}}}}}{2}}$ ≈ 1:1.46310.

Vertex coordinates

Coordinates for the vertices of a 16-4 step prism inscribed in a hexadecagonal duoprism with base lengths a and b are given by:

• $\left(a\sin {\frac {k\pi }{8}},\,a\cos {\frac {k\pi }{8}},\,b\sin {\frac {k\pi }{2}},\,b\cos {\frac {k\pi }{2}}\right)$,

where k is an integer from 0 to 15. If the edge length differences are to be minimized, the ratio of a:b must be equivalent to 1:${\frac {\sqrt[{4}]{8+4{\sqrt {2}}}}{2}}$ ≈ 1:0.96119.

External links

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