25-7 step prism

25-7 step prism
File:25-7 step prism.png (OFF file)
Rank	4
Type	Isogonal
Elements
Cells	50+50 phyllic disphenoids, 25 tetragonal disphenoids
Faces	100 scalene triangles, 50+100 isosceles triangles
Edges	50+50+50
Vertices	25
Vertex figure	Paratetraaugmented snub disphenoid
Measures (circumradius ${\sqrt {2}}$, based on a uniform duoprism)
Edge lengths	6-valence (50): $2{\sqrt {\sin ^{2}{\frac {3\pi }{25}}+\sin ^{2}{\frac {4\pi }{25}}}}\approx 1.21260$
	6-valence (50): $2{\sqrt {\sin ^{2}{\frac {\pi }{25}}+\sin ^{2}{\frac {7\pi }{25}}}}\approx 1.56128$
	3-valence (50): ${\sqrt {4+2\cos {\frac {3\pi }{25}}-2\cos {\frac {4\pi }{25}}}}\approx 2.02656$
Central density	1
Related polytopes
Dual	25-7 gyrochoron
Abstract & topological properties
Flag count	3000
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	S2(I2(25)-7)×2I, order 100
Convex	Yes
Nature	Tame

The 25-7 step prism is a convex isogonal polychoron, member of the step prism family. It has 25 tetragonal disphenoids and 100 phyllic disphenoids of two kinds as cells with 20 (4 tetragonal and 16 phyllic disphenoids) meeting at each vertex.

Compared to other 25-vertex step prisms, this polychoron has doubled symmetry, because 25 is a factor of 7²+1 = 50.

The ratio between the longest and shortest edges is 1:${\sqrt {\frac {4+2\cos {\frac {3\pi }{25}}-2\cos {\frac {4\pi }{25}}}{4-2\cos {\frac {6\pi }{25}}-2\sin {\frac {9\pi }{50}}}}}$ ≈ 1:1.67124.

Vertex coordinates

Coordinates for the vertices of a 25-7 step prism of circumradius √2 are given by:

• $\left(\sin {\frac {2k\pi }{25}},\,\cos {\frac {2k\pi }{25}},\,\sin {\frac {14k\pi }{25}},\,\cos {\frac {14k\pi }{25}}\right)$,

where k is an integer from 0 to 24.