Compound of six decagonal prisms

Compound of six decagonal prisms
Rank3
TypeUniform
Notation
Bowers style acronymRassid
Elements
Components6 decagonal prisms
Faces60 squares, 12 decagons
Edges60+60+60
Vertices120
Vertex figureIsosceles triangle, edge length (5+5)/2, 2, 2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {7+2{\sqrt {5}}}}{2}}\approx 1.69353}$
Volume${\displaystyle 15{\sqrt {5+2{\sqrt {5}}}}\approx 46.16525}$
Dihedral angles4–4: 144°
4–10: 90°
Central density6
Number of external pieces600
Level of complexity34
Related polytopes
ArmySemi-uniform Grid, edge lengths ${\displaystyle 1-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}}$ (dipentagon-ditrigon), ${\displaystyle {\sqrt {\frac {5-{\sqrt {5}}}{10}}}}$ (dipentagon-rectangle), ${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}}{10}}}}$ (ditrigon-rectangle)
RegimentRassid
DualCompound of six decagonal tegums
ConjugateCompound of six decagrammic prisms
Convex coreDodecahedron
Abstract & topological properties
Flag count720
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The rhombisnub dodecahedron, rassid, or compound of six decagonal prisms is a uniform polyhedron compound. It consists of 60 squares and 12 decagons, with one decagon and two squares joining at a vertex.

Its quotient prismatic equivalent is the dipentagonal trapezoprismatic hexateroorthowedge, which is eight-dimensional.

Vertex coordinates

Coordinates for the vertices of a rhombisnub dodecahedron of edge length 1 are given by all even permutations of:

• ${\displaystyle \left(\pm {\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{40}}}\right)}$,
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}-{\sqrt {\frac {10-2{\sqrt {5}}}{5}}}}{4}},\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}+{\sqrt {\frac {10+2{\sqrt {5}}}{5}}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}-{\sqrt {\frac {10-2{\sqrt {5}}}{5}}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {\frac {5+{\sqrt {5}}}{10}}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}+{\sqrt {\frac {10-2{\sqrt {5}}}{5}}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}+{\sqrt {\frac {10-2{\sqrt {5}}}{5}}}}{4}},\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}-{\sqrt {\frac {10+2{\sqrt {5}}}{5}}}}{4}}\right)}$.