# Great rhombihexacron

Great rhombihexacron
Rank3
TypeUniform dual
SpaceSpherical
Elements
Faces24 butterflies
Edges24+24
Vertices12+6
Vertex figures12 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 densityodd
Number of external pieces48
Related polytopes
DualGreat rhombihexahedron
ConjugateSmall rhombihexacron
Convex coreTriakis octahedron
Abstract & topological properties
Flag count192
Euler characteristic–6
OrientableNo
Genus8
Properties
SymmetryB3, order 48
ConvexNo
NatureTame

The great rhombihexacron is a uniform dual polyhedron. It consists of 24 butterflies.

If its dual, the great rhombihexahedron, has an edge length of 1, then the short edges of the bowties will measure ${\displaystyle \sqrt{2\left(2-\sqrt2\right)} ≈ 1.08239}$, and the long edges will be ${\displaystyle 2\sqrt{2-\sqrt2} ≈ 1.53073}$. The butterflies have two interior angles of ${\displaystyle \arccos\left(\frac12+\frac{\sqrt2}{4}\right) ≈ 31.39971^\circ}$, and two of ${\displaystyle \arccos\left(-\frac14+\frac{\sqrt2}{2}\right) ≈ 62.79943^\circ}$. The intersection has an angle of ${\displaystyle \arccos\left(\frac14-\frac{\sqrt2}{8}\right) ≈ 85.80086^\circ}$.

## Vertex coordinates

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

• ${\displaystyle \left(±\left(2-\sqrt2\right),\,0,\,0\right),}$
• ${\displaystyle \left(±1,\,±1,\,0\right).}$