19-2 step prism

19-2 step prism
(OFF file)
Rank	4
Type	Isogonal
Elements
Cells	19+19+19+19+19+19+19+19 phyllic disphenoids
Faces	38+38+38+38+38+38+38 scalene triangles, 19+19 isosceles triangles
Edges	19+19+19+19+19+19+19+19+19
Vertices	19
Vertex figure	Ridge-sextitriakis bi-apiculated tetrahedron
Measures (circumradius ${\sqrt {2}}$, based on a uniform duoprism)
Edge lengths	17-valence (19): $2\sin {\frac {\pi }{19}}{\sqrt {3+2\cos {\frac {2\pi }{19}}}}\approx 0.72807$
	3-valence (19): $2{\sqrt {\sin ^{2}{\frac {2\pi }{19}}+\sin ^{2}{\frac {4\pi }{19}}}}\approx 1.38951$
	4-valence (19): ${\sqrt {4+2\sin {\frac {5\pi }{38}}-2\sin {\frac {7\pi }{38}}}}\approx 1.92600$
	4-valence (19): $2{\sqrt {\sin ^{2}{\frac {\pi }{19}}+\cos ^{2}{\frac {\pi }{38}}}}\approx 2.02017$
	4-valence (19): ${\sqrt {4-2\sin {\frac {7\pi }{38}}+2\cos {\frac {3\pi }{19}}}}\approx 2.15987$
	4-valence (19): ${\sqrt {4+2\cos {\frac {3\pi }{19}}-2\sin {\frac {3\pi }{38}}}}\approx 2.29521$
	4-valence (19): ${\sqrt {4+2\sin {\frac {\pi }{38}}+2\sin {\frac {9\pi }{38}}}}\approx 2.34941$
	4-valence (19): ${\sqrt {4+2\sin {\frac {\pi }{38}}+2\cos {\frac {\pi }{19}}}}\approx 2.47747$
	4-valence (19): ${\sqrt {4+2\sin {\frac {5\pi }{38}}+2\sin {\frac {9\pi }{38}}}}\approx 2.48152$
Central density	1
Related polytopes
Dual	19-2 gyrochoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	S2(I2(19)-2), order 38
Convex	Yes
Nature	Tame

The 19-2 step prism is a convex isogonal polychoron and a member of the step prism family. It has 152 phyllic disphenoids of eight kinds as cells, with 32 joining at each vertex. It can also be constructed as the 19-9 step prism.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is approximately 1:3.10417.

Vertex coordinates

Coordinates for the vertices of a 19-2 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(4πk/19), b*cos(4πk/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:${\frac {1}{\sqrt {2\cos {\frac {\pi }{19}}-2\cos {\frac {2\pi }{19}}}}}$ ≈ 1:3.51173.

External links

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