# 16-6 step prism

16-6 step prism
Rank4
TypeIsogonal
Elements
Cells16+16+16 phyllic disphenoids, 8 rhombic disphenoids
Faces32+32+32 scalene triangles, 16 isosceles triangles
Edges8+16+16+16+16
Vertices16
Vertex figureEdge-expanded hexagonal tegum
Measures (circumradius ${\displaystyle {\sqrt {2}}}$, based on a uniform duoprism)
Edge lengths6-valence (16): ${\displaystyle {\sqrt {4-{\sqrt {2}}-{\sqrt {2-{\sqrt {2}}}}}}\approx 1.34923}$
5-valence (16): ${\displaystyle {\sqrt {4-{\sqrt {2}}}}\approx 1.60804}$
4-valence (16): ${\displaystyle {\sqrt {4-{\sqrt {2}}+{\sqrt {2-{\sqrt {2}}}}}}\approx 1.83061}$
4-valence (16): ${\displaystyle {\sqrt {4+{\sqrt {2}}-{\sqrt {2+{\sqrt {2}}}}}}\approx 1.88851}$
4-valence (8): 2
Central density1
Related polytopes
Dual16-6 gyrochoron
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryS2(I2(16)-6), order 32
ConvexYes
NatureTame

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:${\displaystyle {\frac {\sqrt {10088+4656{\sqrt {2}}+194{\sqrt {170+71{\sqrt {2}}}}}}{97}}}$ ≈ 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:

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

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