# 25-7 step prism

25-7 step prism
File:25-7 step prism.png
Rank4
TypeIsogonal
SpaceSpherical
Elements
Cells50+50 phyllic disphenoids, 25 tetragonal disphenoids
Faces100 scalene triangles, 50+100 isosceles triangles
Edges50+50+50
Vertices25
Vertex figureParatetraaugmented snub disphenoid
Measures (circumradius $\sqrt2$ , based on a uniform duoprism)
Edge lengths6-valence (50): $2\sqrt{\sin^2\frac{3\pi}{25}+\sin^2\frac{4\pi}{25}} ≈ 1.21260$ 6-valence (50): $2\sqrt{\sin^2\frac{\pi}{25}+\sin^2\frac{7\pi}{25}} ≈ 1.56128$ 3-valence (50): $\sqrt{4+2\cos\frac{3\pi}{25}-2\cos\frac{4\pi}{25}} ≈ 2.02656$ Central density1
Related polytopes
Dual25-7 gyrochoron
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryS2(I2(25)-7)×2I, order 100
ConvexYes
NatureTame

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:

• (sin(2πk/25), cos(2πk/25), sin(14πk/25), cos(14πk/25)),

where k is an integer from 0 to 24.