# Dodecahedron atop small rhombicosidodecahedron

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Dodecahedron atop small rhombicosidodecahedron
Rank4
TypeSegmentotope
Notation
Bowers style acronymDoasrid
Coxeter diagramxx5oo3ox&#x
Elements
Cells20 tetrahedra, 30 triangular prisms, 12 pentagonal prisms, 1 dodecahedron, 1 small rhombicosidodecahedron
Faces20+60 triangles, 30+60 squares, 12+12 pentagons
Edges30+60+60+60
Vertices20+60
Vertex figures20 triangular antipodiums, edge lengths 1 (base 1), 2 (sides), and (1+5)/2 (base 2)
60 isosceles trapezoidal pyramids, base edge lengths 1, 2, (1+5)/2, 2, side edge lengths 1, 1, 2. 2
Measures (edge length 1)
Circumradius${\displaystyle 3+{\sqrt {5}}\approx 5.23607}$
Hypervolume${\displaystyle {\frac {335+143{\sqrt {5}}}{96}}\approx 6.82039}$
Dichoral anglesTet–3–trip: ${\displaystyle \arccos \left(-{\frac {{\sqrt {6}}+{\sqrt {30}}}{8}}\right)\approx 172.23876^{\circ }}$
Pip–4–trip: ${\displaystyle \arccos \left(-{\sqrt {\frac {10+2{\sqrt {5}}}{15}}}\right)\approx 169.18768^{\circ }}$
Doe–5–pip: 162°
Srid–3–tet: ${\displaystyle \arccos \left({\frac {\sqrt {7+3{\sqrt {5}}}}{4}}\right)\approx 22.23876^{\circ }}$
Srid–4–trip: ${\displaystyle \arccos \left({\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 20.90516^{\circ }}$
Srid–5–pip: 18°
Height${\displaystyle {\frac {{\sqrt {5}}-1}{4}}\approx 0.30902}$
Central density1
Related polytopes
ArmyDoasrid
RegimentDoasrid
DualIcosahedral-deltoidal hexecontahedral tegmoid
ConjugateGreat stellated dodecahedron atop quasirhombicosidodecahedron
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryH3×I, order 120
ConvexYes
NatureTame

Dodecahedron atop small rhombicosidodecahedron, or doasrid, is a CRF segmentochoron (designated K-4.152 on Richard Klitzing's list). As the name suggests, it consists of a dodecahedron and a small rhombicosidodecahedron as bases, connected by 20 tetrahedra, 30 triangular prisms, and 12 pentagonal prisms.

It is also sometimes referred to as a dodecahedral cupola, as one generalization of the definition of a cupola is to have a polytope atop an expanded version.

It can be obtained as a dodecahedron-first cap of the small disprismatohexacosihecatonicosachoron.

## Vertex coordinates

The vertices of a dodecahedron atop small rhombicosidodecahedron segmentochoron of edge length 1 are given by all permutations of the first three coordinates of:

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

Plus all even permutations of:

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