16-6 step prism

16-6 step prism
(OFF file)
Rank	4
Type	Isogonal
Elements
Cells	16+16+16 phyllic disphenoids, 8 rhombic disphenoids
Faces	32+32+32 scalene triangles, 16 isosceles triangles
Edges	8+16+16+16+16
Vertices	16
Vertex figure	Edge-expanded hexagonal tegum
Measures (circumradius ${\sqrt {2}}$, based on a uniform duoprism)
Edge lengths	6-valence (16): ${\sqrt {4-{\sqrt {2}}-{\sqrt {2-{\sqrt {2}}}}}}\approx 1.34923$
	5-valence (16): ${\sqrt {4-{\sqrt {2}}}}\approx 1.60804$
	4-valence (16): ${\sqrt {4-{\sqrt {2}}+{\sqrt {2-{\sqrt {2}}}}}}\approx 1.83061$
	4-valence (16): ${\sqrt {4+{\sqrt {2}}-{\sqrt {2+{\sqrt {2}}}}}}\approx 1.88851$
	4-valence (8): 2
Central density	1
Related polytopes
Dual	16-6 gyrochoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	S2(I2(16)-6), order 32
Convex	Yes
Nature	Tame

The 16-6 step prism is a convex isogonal polychoron and a member of the step prism family. It has 8 rhombic disphenoids and 48 phyllic disphenoids of three kinds as cells, with 14 (2 rhombic and 12 phyllic disphenoids) joining at each vertex. It is also the hexagonal funk tegum.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\frac {\sqrt {10088+4656{\sqrt {2}}+194{\sqrt {170+71{\sqrt {2}}}}}}{97}}$ ≈ 1:1.45294.

Vertex coordinates

Coordinates for the vertices of a 16-6 step prism inscribed in a hexadecagonal duoprism with base lengths a and b are given by:

• $\left(a\sin {\frac {k\pi }{8}},\,a\cos {\frac {k\pi }{8}},\,b\sin {\frac {3k\pi }{4}},\,b\cos {\frac {3k\pi }{4}}\right)$,

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:${\frac {\sqrt {8-4{\sqrt {2}}+2{\sqrt {4-2{\sqrt {2}}}}}}{2}}$ ≈ 1:1.06159.

External links

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