# Triangular-gyroprismatic hexacosichoron

Triangular-gyroprismatic hexacosichoron
File:Triangular-gyroprismatic hexacosichoron.png
Rank4
TypeNoble
SpaceSpherical
Elements
Cells600 triangular gyroprisms
Faces1800 isosceles triangles, 600 triangles
Edges360+1800
Vertices360
Vertex figurePentagonal gyrotegum
Measures (edge length 1)
Central density1
Related polytopes
DualPentagonal-antiprismatic triacosihexecontachoron
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryH3●I2(30), order 3600
ConvexYes
NatureTame

The triangular-gyroprismatic hexacosichoron, also known as the ikeswirl 600, is a noble polychoron with 600 triangular gyroprisms for cells and 360 vertices. 10 cells join at each vertex.

It is the third in an infinite family of isogonal icosahedral swirlchora (the triacontafold icosaswirlchoron) and also the first in an infinite family of isochoric icosahedral swirlchora (the icosaswirlic hexacosichoron).

Each cell of this polychoron is a chiral variant of the triangular antiprism. If the edges of the base triangles are of length 1, half the side edges are also of length 1, while the other half are of length $\sqrt{\frac{15-3\sqrt5-\sqrt{30+6\sqrt5}}{12}} ≈ 0.37668$ .

The ratio between the longest and shortest edges is 1:$\frac{\sqrt{225+75\sqrt5+5\sqrt{1950+870\sqrt5}}}{10}$ ≈ 1:2.65475.

## Vertex coordinates

Coordinates for the vertices of a triangular-antiprismatic hexacosichoron of circumradius 1, centered at the origin, are given by:

• ±(0, 0, sin(kπ/15), cos(kπ/15)),
• ±(cos(kπ/15), sin(kπ/15), 0, 0),

along with 72°, 144°, 216° and 288° rotations in the xy axis of:

• ±(2sin(kπ/15)/10+25, 2cos(kπ/15)/10+25, 2cos(kπ/15)/10-25, 2sin(kπ/15)/10-25),
• ±(2sin(kπ/15)/10-25, 2cos(kπ/15)/10-25, -2cos(kπ/15)/10+25, -2sin(kπ/15)/10+25),

where k is an integer from 0 to 14.

## Isogonal derivatives

Substitution by vertices of these following elements will produce these convex isogonal polychora: