# Transitional 10-3 double gyrostep prism

Transitional 10-3 double gyrostep prism
File:Transitional 10-3 double gyrostep prism.png
Rank4
TypeIsogonal
SpaceSpherical
Elements
Cells10 tetragonal disphenoids, 10 chiral gyrobifastigia
Faces40 isosceles triangles, 20 kites
Edges20+40
Vertices20
Vertex figureTetragonal antiwedge
Measures (based on unit decachoron)
Edge lengths4-valence (20): 1
3-valence (40): ${\displaystyle \sqrt3 ≈ 1.73205}$
Circumradius${\displaystyle \sqrt2 ≈ 1.41421}$
Central density1
Related polytopes
DualTransitional 10-3 antibigyrochoron
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryS2(I2(10)-3)×2I, order 40
ConvexYes
NatureTame

The transitional 10-3 double gyrostep prism is a convex isogonal polychoron that consists of 10 chiral gyrobifastigia and 10 tetragonal disphenoids. 2 chiral gyrobifastigia and 4 tetragonal disphenoids join at each vertex. It can be obtained as the convex hull of two orthogonal stretched 10-3 step prisms.

It can also be constructed as a subsymmetrical faceting of the decachoron, formed by removing a gyrochoron-symmetric subset of 10 vertices. The disphenoids are the decachoron's vertex figures, while the gyrobifastigium cells are subsymmetric facetings of the decachoron's truncated tetrahedron cells.

The ratio between the longest and shortest edges is 1:${\displaystyle \sqrt3}$ ≈ 1:1.73205.

## Vertex coordinates

Coordinates for the vertices of a transitional 10-3 double gyrostep prism are given by:

• (a*sin(2πk/10), a*cos(2πk/10), b*sin(6πk/10), b*cos(6πk/10)),
• (b*sin(2πk/10), b*cos(2πk/10), -a*sin(6πk/10), -a*cos(6πk/10)),

where a = ${\displaystyle \sqrt{\frac{5-2\sqrt5}{5}}}$, b = ${\displaystyle \sqrt{\frac{5+2\sqrt5}{5}}}$, and k is an integer from 0 to 9.