Great rhombidodecacron

Great rhombidodecacron
(3D model) (OFF file)
Rank	3
Type	Uniform dual
Elements
Faces	60 butterflies
Edges	60+60
Vertices	12+30
Vertex figure	30 squares, 12 decagrams
Measures (edge length 1)
Inradius	${\frac {\sqrt {205\left(19-8{\sqrt {5}}\right)}}{41}}\approx 0.36816$
Dihedral angle	$\arccos \left(-{\frac {19-8{\sqrt {5}}}{41}}\right)\approx 91.55340^{\circ }$
Central density	odd
Number of external pieces	120
Related polytopes
Dual	Great rhombidodecahedron
Conjugate	Small rhombidodecacron
Convex core	Non-Catalan pentakis dodecahedron
Abstract & topological properties
Flag count	480
Euler characteristic	–18
Orientable	No
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

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 ${\sqrt {5-{\sqrt {5}}}}\approx 1.66251$, and the long edges will be ${\sqrt {5+{\sqrt {5}}}}\approx 2.68999$. The butterflies have two interior angles of $\arccos \left({\frac {5+2{\sqrt {5}}}{10}}\right)\approx 18.69941^{\circ }$, and two of $\arccos \left({\frac {-5+{\sqrt {5}}}{8}}\right)\approx 110.21180^{\circ }$. The intersection has an angle of $\arccos \left({\frac {5+9{\sqrt {5}}}{40}}\right)\approx 51.08879^{\circ }$.

Vertex coordinates

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

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

External links

• Wikipedia contributors. "Great rhombidodecacron".
• McCooey, David. "Great Rhombidodecacron"