# Great rhombihexacron

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Great rhombihexacron
Rank3
TypeUniform dual
Elements
Faces24 butterflies
Edges24+24
Vertices12+6
Vertex figures12 squares
6 octagrams
Measures (edge length 1)
Inradius${\displaystyle {\frac {\sqrt {34(7-4{\sqrt {2}})}}{17}}\approx 0.39751}$
Dihedral angle${\displaystyle \arccos \left(-{\frac {7-4{\sqrt {2}}}{17}}\right)\approx 94.53158^{\circ }}$
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 butterflies will measure ${\displaystyle {\sqrt {2\left(2-{\sqrt {2}}\right)}}\approx 1.08239}$, and the long edges will be ${\displaystyle 2{\sqrt {2-{\sqrt {2}}}}\approx 1.53073}$. The butterflies have two interior angles of ${\displaystyle \arccos \left({\frac {2+{\sqrt {2}}}{4}}\right)\approx 31.39971^{\circ }}$, and two of ${\displaystyle \arccos \left({\frac {-1+2{\sqrt {2}}}{4}}\right)\approx 62.79943^{\circ }}$. The intersection has an angle of ${\displaystyle \arccos \left({\frac {2-{\sqrt {2}}}{8}}\right)\approx 85.80086^{\circ }}$.

## Vertex coordinates

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

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