# Small rhombihexacron

Small rhombihexacron
Rank3
TypeUniform dual
Elements
Faces24 butterflies
Edges24+24
Vertices6+12
Vertex figures12 squares
6 octagons
Measures (edge length 1)
Inradius${\displaystyle {\frac {\sqrt {34(7+4{\sqrt {2}})}}{17}}\approx 1.22026}$
Dihedral angle${\displaystyle \arccos \left(-{\frac {7+4{\sqrt {2}}}{17}}\right)\approx 138.11796^{\circ }}$
Central densityodd
Number of external pieces48
Related polytopes
DualSmall rhombihexahedron
ConjugateGreat rhombihexacron
Convex coreDeltoidal icositetrahedron
Abstract & topological properties
Flag count192
Euler characteristic–6
OrientableNo
Properties
SymmetryB3, order 48
ConvexNo
NatureTame

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

It appears the same as the small hexacronic icositetrahedron.

If its dual, the small rhombihexahedron, has an edge length of 1, then the short edges of the butterflies will measure ${\displaystyle {\sqrt {2\left(2+{\sqrt {2}}\right)}}\approx 2.61313}$, and the long edges will be ${\displaystyle 2{\sqrt {2+{\sqrt {2}}}}\approx 3.69552}$. ​The butterflies have two interior angles of ${\displaystyle \arccos \left({\frac {1+2{\sqrt {2}}}{4}}\right)\approx 16.84212^{\circ }}$, and two of ${\displaystyle \arccos \left({\frac {-2+{\sqrt {2}}}{4}}\right)\approx 98.42106^{\circ }}$. The intersection has an angle of ${\displaystyle \arccos \left({\frac {2+{\sqrt {2}}}{8}}\right)\approx 64.73683^{\circ }}$.

## Vertex coordinates

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

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