# Small hexacronic icositetrahedron

Small hexacronic icositetrahedron Rank3
TypeUniform dual
SpaceSpherical
Notation
Coxeter diagramm4/3o3m4*a (      )
Elements
Faces24 darts
Edges24+24
Vertices8+6+6
Vertex figures8 triangles
6 squares
6 octagons
Measures (edge length 1)
Inradius$\frac{\sqrt{34(7+4\sqrt2)}}{17} \approx 1.22026$ Dihedral angle$\arccos\left(-\frac{7+4\sqrt2}{17}\right) \approx 138.11796^\circ$ Central density2
Number of external pieces48
Related polytopes
DualSmall cubicuboctahedron
Abstract & topological properties
Flag count192
Euler characteristic–4
OrientableYes
Genus3
Properties
SymmetryB3, order 48
ConvexNo
NatureTame

The small hexacronic icositetrahedron is a uniform dual polyhedron. It consists of 24 darts.

It appears the same as the small rhombihexacron.

If its dual, the small cubicuboctahedron, has an edge length of 1, then the short edges of the darts will measure $2\frac{\sqrt{2\left(26+17\sqrt2\right)}}{7} ≈ 2.85833$ , and the long edges will be $2\sqrt{2+\sqrt2} ≈ 3.69552$ . ​The dart faces will have length $2\frac{\sqrt{31-8\sqrt2}}{7} ≈ 1.26769$ , and width $2\left(1+\sqrt2\right) ≈ 4.82843$ . ​The darts have two interior angles of $\arccos\left(\frac14+\frac{\sqrt2}{2}\right) ≈ 16.84212^\circ$ , one of $\arccos\left(\frac12-\frac{\sqrt2}{4}\right) ≈ 81.57894^\circ$ , and one of $360^\circ-\arccos\left(-\frac14-\frac{\sqrt2}{8}\right) ≈ 244.73683^\circ$ .

## Vertex coordinates

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

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