# 45-19 step prism

45-19 step prism
File:45-19 step prism.png
Rank4
TypeIsogonal
SpaceSpherical
Elements
Cells90+90+90 phyllic disphenoids
Faces180+180 scalene triangles, 90+90 isosceles triangles
Edges45+90+90+90
Vertices45
Vertex figure14-vertex polyhedron with 24 triangular faces
Measures (edge length 1)
Central density1
Related polytopes
Dual45-19 gyrochoron
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryS2(I2(45)-19)×2R, order 180
ConvexYes
NatureTame

The 45-19 step prism is a convex isogonal polychoron, member of the step prism family. It has 270 phyllic disphenoids of three kinds as cells. 24 disphenoids join at each vertex. It is also the 45-26 step prism.

The ratio between the longest and shortest edges is 1:$\frac{\sqrt{2-\cos\frac{2\pi}{45}+\cos\frac{7\pi}{45}}}{2\sin\frac\pi9}$ ≈ 1:2.01121.

## Vertex coordinates

Coordinates for the vertices of a 45-19 step prism of circumradius 2 are given by:

• (sin(2πk/45), cos(2πk/45), sin(38πk/45), cos(38πk/45)),

where k is an integer from 0 to 44.