# Great hexacronic icositetrahedron

Great hexacronic icositetrahedron
Rank3
TypeUniform dual
SpaceSpherical
Notation
Coxeter diagramm4/3m3o4*a
Elements
Faces24 kites
Edges24+24
Vertices8+6+6
Vertex figures8 triangles
6 squares
6 octagrams
Measures (edge length 1)
Inradius${\displaystyle \frac{\sqrt{34(7−4\sqrt2)}}{17} ≈ 0.39751}$
Dihedral angle${\displaystyle \arccos\left(-\frac{7−4\sqrt2}{17}\right) ≈ 94.53158^\circ}$
Central density4
Number of external pieces48
Related polytopes
DualGreat cubicuboctahedron
Abstract & topological properties
Flag count192
Euler characteristic–4
OrientableYes
Genus3
Properties
SymmetryB3, order 48
ConvexNo
NatureTame

The great hexacronic icositetrahedron is a uniform dual polyhedron. It consists of 24 kites.

If its dual, the great cubicuboctahedron, has an edge length of 1, then the short edges of the kites will measure ${\displaystyle 2\frac{\sqrt{2\left(26-17\sqrt2\right)}}{7} ≈ 0.56545}$, and the long edges will be ${\displaystyle 2\sqrt{2-\sqrt2} ≈ 1.53073}$. The kite faces will have length ${\displaystyle 2\frac{\sqrt{31+8\sqrt2}}{7} ≈ 1.85854}$, and width ${\displaystyle 2\left(\sqrt2-1\right) ≈ 0.82843}$. The kites have two interior angles of ${\displaystyle \arccos\left(\frac14-\frac{\sqrt2}{2}\right) ≈ 117.20057^\circ}$, one of ${\displaystyle \arccos\left(-\frac14+\frac{\sqrt2}{8}\right) ≈ 94.19914^\circ}$, and one of ${\displaystyle \arccos\left(\frac12+\frac{\sqrt2}{4}\right) ≈ 31.39971^\circ}$.

## Vertex coordinates

A great hexacronic icositetrahedron with dual edge length 1 has vertex coordinates given by all permutations of:

• ${\displaystyle \left(±\left(2-\sqrt2\right),\,0,\,0\right),}$
• ${\displaystyle \left(±\sqrt2,\,0,\,0\right),}$
• ${\displaystyle \left(±\frac{4-\sqrt2}{7},\,±\frac{4-\sqrt2}{7},\,±\frac{4-\sqrt2}{7}\right).}$