# 19-3 step prism

(Redirected from 19-6 step prism)
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19-3 step prism
Rank4
TypeIsogonal
SpaceSpherical
Info
SymmetryI2(19)+×2×I, order 38
Elements
Vertex figure14-vertex polyhedron with 24 triangular faces
Cells19+19+19+19+19+19 phyllic disphenoids
Faces19+19+38+38+38+38+38 scalene triangles
Edges19+19+19+19+19+19+19
Vertices19
Measures (circumradius $\sqrt2$ , based on a uniform duoprism)
Edge lengths12-valence (19): $2\sqrt{\sin^2\frac{\pi}{19}+\sin^2\frac{3\pi}{19}} ≈ 1.00721$ 6-valence (19): $2\sqrt{\sin^2\frac{\pi}{19}+\cos^2\frac{7\pi}{38}} ≈ 1.70639$ 3-valence (19): $\sqrt{4+2\sin\frac{5\pi}{38}-2\cos\frac{4\pi}{19}} ≈ 1.79586$ 4-valence (19): $\sqrt{4+2\sin\frac{\pi}{38}-2\sin\frac{3\pi}{38}} ≈ 1.91682$ 4-valence (19): $\sqrt{4+2\cos\frac{5\pi}{19}-2\cos\frac{4\pi}{19}} ≈ 1.94327$ 4-valence (19): $\sqrt{4-2\sin\frac{3\pi}{38}+2\sin\frac{9\pi}{38}} ≈ 2.20535$ 4-valence (19): $\sqrt{4-2\sin\frac{7\pi}{38}+2\cos\frac{\pi}{19}} ≈ 2.20881$ Central density1
Euler characteristic0
Related polytopes
Dual19-3 gyrochoron
Properties
ConvexYes
OrientableYes
NatureTame

The 19-3 step prism is a convex isogonal polychoron and a member of the step prism family. It has 114 phyllic disphenoids of six kinds as cells, with 24 joining at each vertex. It can also be constructed as the 19-6 step prism.

## Vertex coordinates

Coordinates for the vertices of a 19-3 step prism inscribed in an enneadecagonal duoprism with base lengths a and b are given by:

• (a*sin(2πk/19), a*cos(2πk/19), b*sin(6πk/19), b*cos(6πk/19)),

where k is an integer from 0 to 18. If the edge length differences are to be minimized, the ratio of a:b must be equivalent to 1:$\sqrt{\frac{\cos\frac{7\pi}{19}+\cos\frac{2\pi}{19}}{\cos\frac{2\pi}{19}-\cos\frac{6\pi}{19}}}$ ≈ 1:1.83802.