# Small rhombidodecacron

Small rhombidodecacron
Rank3
TypeUniform dual
Notation
Coxeter diagramm5/2m5m ()
Elements
Faces60 butterflies
Edges60+60
Vertices12+30
Vertex figure30 squares, 12 decagons
Measures (edge length 1)
Inradius${\displaystyle {\frac {\sqrt {205\left(19+8{\sqrt {5}}\right)}}{41}}\approx 2.12099}$
Dihedral angle${\displaystyle \arccos \left(-{\frac {19+8{\sqrt {5}}}{41}}\right)\approx 154.12136^{\circ }}$
Central densityodd
Number of external pieces120
Related polytopes
DualSmall rhombidodecahedron
ConjugateGreat rhombidodecacron
Convex coreDeltoidal hexecontahedron
Abstract & topological properties
Flag count480
Euler characteristic–18
OrientableNo
Genus20
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

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

It appears the same as the small dodecacronic hexecontahedron.

If its dual, the small 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 ${\displaystyle \arccos \left({\frac {5+{\sqrt {5}}}{8}}\right)\approx 25.24283^{\circ }}$, and two of ${\displaystyle \arccos \left({\frac {-5+2{\sqrt {5}}}{10}}\right)\approx 93.02584^{\circ }}$. The intersection has an angle of ${\displaystyle \arccos \left({\frac {5+2{\sqrt {5}}}{20}}\right)\approx 61.73132^{\circ }}$.

## Vertex coordinates

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

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