# 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$\frac{\sqrt{34(7−4\sqrt2)}}{17} ≈ 0.39751$ Dihedral angle$\arccos\left(-\frac{7−4\sqrt2}{17}\right) ≈ 94.53158^\circ$ 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 $2\frac{\sqrt{2\left(26-17\sqrt2\right)}}{7} ≈ 0.56545$ , and the long edges will be $2\sqrt{2-\sqrt2} ≈ 1.53073$ . The kite faces will have length $2\frac{\sqrt{31+8\sqrt2}}{7} ≈ 1.85854$ , and width $2\left(\sqrt2-1\right) ≈ 0.82843$ . The kites have two interior angles of $\arccos\left(\frac14-\frac{\sqrt2}{2}\right) ≈ 117.20057^\circ$ , one of $\arccos\left(-\frac14+\frac{\sqrt2}{8}\right) ≈ 94.19914^\circ$ , and one of $\arccos\left(\frac12+\frac{\sqrt2}{4}\right) ≈ 31.39971^\circ$ .

## Vertex coordinates

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

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