Dodecahedron atop small rhombicosidodecahedron

Dodecahedron atop small rhombicosidodecahedron
(OFF file)
Rank	4
Type	Segmentotope
Notation
Bowers style acronym	Doasrid
Coxeter diagram	xx5oo3ox&#x
Elements
Cells	20 tetrahedra, 30 triangular prisms, 12 pentagonal prisms, 1 dodecahedron, 1 small rhombicosidodecahedron
Faces	20+60 triangles, 30+60 squares, 12+12 pentagons
Edges	30+60+60+60
Vertices	20+60
Vertex figures	20 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	$3+{\sqrt {5}}\approx 5.23607$
Hypervolume	${\frac {335+143{\sqrt {5}}}{96}}\approx 6.82039$
Dichoral angles	Tet–3–trip: $\arccos \left(-{\frac {{\sqrt {6}}+{\sqrt {30}}}{8}}\right)\approx 172.23876^{\circ }$
	Pip–4–trip: $\arccos \left(-{\sqrt {\frac {10+2{\sqrt {5}}}{15}}}\right)\approx 169.18768^{\circ }$
	Doe–5–pip: 162°
	Srid–3–tet: $\arccos \left({\frac {\sqrt {7+3{\sqrt {5}}}}{4}}\right)\approx 22.23876^{\circ }$
	Srid–4–trip: $\arccos \left({\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 20.90516^{\circ }$
	Srid–5–pip: 18°
Height	${\frac {{\sqrt {5}}-1}{4}}\approx 0.30902$
Central density	1
Related polytopes
Army	Doasrid
Regiment	Doasrid
Dual	Icosahedral-deltoidal hexecontahedral tegmoid
Conjugate	Great stellated dodecahedron atop quasirhombicosidodecahedron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H3×I, order 120
Convex	Yes
Nature	Tame

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.

Segmentochoron display

• 

  Dodecahedron atop small rhombicosidodecahedron

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:

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

Plus all even permutations of:

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

External links

• Klitzing, Richard. "doasrid".
• Wikipedia contributors. "Dodecahedral cupola".