# Great deltoidal hexecontahedron

Great deltoidal hexecontahedron
Rank3
TypeUniform dual
Notation
Coxeter diagramm5/3o3m
Elements
Faces60 darts
Edges60+60
Vertices12+20+30
Vertex figure20 triangles, 30 squares, 12 pentagrams
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 density13
Number of external pieces120
Related polytopes
DualQuasirhombicosidodecahedron
ConjugateDeltoidal hexecontahedron
Convex coreNon-Catalan pentakis dodecahedron
Abstract & topological properties
Flag count480
Euler characteristic2
OrientableYes
Genus0
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The great deltoidal hexecontahedron is a uniform dual polyhedron. It consists of 60 darts.

It appears the same as the great rhombidodecacron.

If its dual, the quasirhombicosidodecahedron, has an edge length of 1, then the short edges of the darts will measure ${\displaystyle {\frac {\sqrt {5\left(5+{\sqrt {5}}\right)}}{3}}\approx 2.00500}$, and the long edges will be ${\displaystyle {\frac {\sqrt {5\left(85+31{\sqrt {5}}\right)}}{11}}\approx 2.52523}$. ​The dart faces will have length ${\displaystyle {\frac {\sqrt {10\left(157-31{\sqrt {5}}\right)}}{33}}\approx 0.89731}$, and width ${\displaystyle {\frac {5+{\sqrt {5}}}{2}}\approx 3.61803}$. The darts have two interior angles of ${\displaystyle \arccos \left({\frac {5+2{\sqrt {5}}}{10}}\right)\approx 18.69941^{\circ }}$, one of ${\displaystyle \arccos \left({\frac {-5+2{\sqrt {5}}}{20}}\right)\approx 91.51239^{\circ }}$, and one of ${\displaystyle 360^{\circ }-\arccos \left(-{\frac {5+9{\sqrt {5}}}{40}}\right)\approx 231.08879^{\circ }}$.

## Vertex coordinates

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

• ${\displaystyle \left(\pm {\sqrt {5}},\,0,\,0\right),}$
• ${\displaystyle \left(\pm {\frac {25-9{\sqrt {5}}}{22}},\,\pm {\frac {15-{\sqrt {5}}}{22}},\,0\right),}$
• ${\displaystyle \left(\pm {\frac {5-{\sqrt {5}}}{6}},\,\pm {\frac {3{\sqrt {5}}-5}{6}},\,0\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {5}}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {5-{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {4{\sqrt {5}}-5}{11}},\,\pm {\frac {4{\sqrt {5}}-5}{11}},\,\pm {\frac {4{\sqrt {5}}-5}{11}}\right).}$