Great rhombihexacron

Great rhombihexacron
(3D model) (OFF file)
Rank	3
Type	Uniform dual
Elements
Faces	24 butterflies
Edges	24+24
Vertices	12+6
Vertex figures	12 squares
	6 octagrams
Measures (edge length 1)
Inradius	${\frac {\sqrt {34(7-4{\sqrt {2}})}}{17}}\approx 0.39751$
Dihedral angle	$\arccos \left(-{\frac {7-4{\sqrt {2}}}{17}}\right)\approx 94.53158^{\circ }$
Central density	odd
Number of external pieces	48
Related polytopes
Dual	Great rhombihexahedron
Conjugate	Small rhombihexacron
Convex core	Triakis octahedron
Abstract & topological properties
Flag count	192
Euler characteristic	–6
Orientable	No
Genus	8
Properties
Symmetry	B3, order 48
Convex	No
Nature	Tame

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 ${\sqrt {2\left(2-{\sqrt {2}}\right)}}\approx 1.08239$, and the long edges will be $2{\sqrt {2-{\sqrt {2}}}}\approx 1.53073$. The butterflies have two interior angles of $\arccos \left({\frac {2+{\sqrt {2}}}{4}}\right)\approx 31.39971^{\circ }$, and two of $\arccos \left({\frac {-1+2{\sqrt {2}}}{4}}\right)\approx 62.79943^{\circ }$. The intersection has an angle of $\arccos \left({\frac {2-{\sqrt {2}}}{8}}\right)\approx 85.80086^{\circ }$.

Vertex coordinates

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

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

External links

• Wikipedia contributors. "Great rhombihexacron".
• McCooey, David. "Great Rhombihexacron"