# Great rhombidodecacron

Great rhombidodecacron
Rank3
TypeUniform dual
Elements
Faces60 butterflies
Edges60+60
Vertices12+30
Vertex figure30 squares, 12 decagrams
Measures (edge length 1)
Inradius${\displaystyle {\frac {\sqrt {205\left(19-8{\sqrt {5}}\right)}}{41}}\approx 0.36816}$
Dihedral angle${\displaystyle \arccos \left(-{\frac {19-8{\sqrt {5}}}{41}}\right)\approx 91.55340^{\circ }}$
Central densityodd
Number of external pieces120
Related polytopes
DualGreat rhombidodecahedron
ConjugateSmall rhombidodecacron
Convex coreNon-Catalan pentakis dodecahedron
Abstract & topological properties
Flag count480
Euler characteristic–18
OrientableNo
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The great rhombidodecacron is a uniform dual polyhedron. It consists of 60 butterflies.

It appears the same as the great deltoidal hexecontahedron.

If its dual, the great rhombidodecahedron, has an edge length of 1, then the short edges of the butterflies will measure ${\displaystyle {\sqrt {5-{\sqrt {5}}}}\approx 1.66251}$, and the long edges will be ${\displaystyle {\sqrt {5+{\sqrt {5}}}}\approx 2.68999}$. The butterflies have two interior angles of ${\displaystyle \arccos \left({\frac {5+2{\sqrt {5}}}{10}}\right)\approx 18.69941^{\circ }}$, and two of ${\displaystyle \arccos \left({\frac {-5+{\sqrt {5}}}{8}}\right)\approx 110.21180^{\circ }}$. The intersection has an angle of ${\displaystyle \arccos \left({\frac {5+9{\sqrt {5}}}{40}}\right)\approx 51.08879^{\circ }}$.

## Vertex coordinates

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

• ${\displaystyle \left(\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {3-{\sqrt {5}}}{2}},\,0\right),}$
• ${\displaystyle \left(\pm {\sqrt {5}},\,0,\,0\right),}$
• ${\displaystyle \left(\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm {\frac {\sqrt {5}}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right).}$