# Small 12-5 double step prism

Small 12-5 double step prism
File:Small 12-5 double step prism.png
Rank4
TypeIsogonal
SpaceSpherical
Elements
Cells48 irregular tetrahedra, 24 phyllic disphenoids, 12+12 rhombic disphenoids, 12 tetragonal disphenoids
Faces24+48 isosceles triangles, 48+48+48 scalene triangles
Edges12+24+24+24+24+24
Vertices24
Vertex figure11-vertex polyhedron with 18 triangular faces
Measures (edge length 1)
Central density1
Related polytopes
DualSmall 12-5 bigyrochoron
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryS2(I2(12)-5)×2R, order 48
ConvexYes
NatureTame

The small 12-5 double step prism is a convex isogonal polychoron that consists of 12 tetragonal disphenoids, 24 rhombic disphenoids of two kinds, 24 phyllic disphenoids, and 48 irregular tetrahedra. 2 tetragonal disphenoid, 4 rhombic disphenoids, 4 phyllic disphenoids, and 8 irregular tetrahedra join at each vertex. It can be obtained as the convex hull of two orthogonal 12-5 step prisms.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$\frac{\sqrt{7+\sqrt{21}}}{2}$ ≈ 1:1.70166.

## Vertex coordinates

Coordinates for the vertices of a small 12-5 double step prism are given by:

• (a*sin(2πk/12), a*cos(2πk/12), b*sin(10πk/12), b*cos(10πk/12)),
• (b*sin(2πk/12), b*cos(2πk/12), a*sin(10πk/12), a*cos(10πk/12)),

where a = 7-27+33-127/4, b = 7+27+33+127/4 and k is an integer from 0 to 11.