Dodecahedron atop icosidodecahedron

Dodecahedron atop icosidodecahedron
(OFF file)
Rank	4
Type	Segmentotope
Notation
Bowers style acronym	Doaid
Coxeter diagram	xo5ox3oo&#x
Elements
Cells	20 tetrahedra, 12 pentagonal antiprisms, 1 dodecahedron, 1 icosidodecahedron
Faces	20+30+60 = 110 triangles, 12+12 = 24 pentagons
Edges	30+60+60 = 150
Vertices	20+30 = 50
Vertex figures	20 triangular frustums, edge lengths 1(top base and sides) and (1+√5)/2 (bottom base)
	30 wedges, edge lengths (1+√5)/2 (two base edges and top edge) and 1 (remaining edges)
Measures (edge length 1)
Circumradius	${\frac {1+{\sqrt {5}}}{2}}\approx 1.61803$
Hypervolume	${\frac {83+37{\sqrt {5}}}{96}}\approx 8.63201$
Dichoral angles	Tet–3–pap: $\arccos \left(-{\frac {\sqrt {10}}{4}}\right)\approx 142.23876^{\circ }$
	Pap–3–pap: 120°
	Doe–5–pap: 108°
	Id–3–tet: $\arccos \left({\frac {3{\sqrt {2}}-{\sqrt {10}}}{8}}\right)\approx 82.23876^{\circ }$
	Id–5–pap: 72°
Height	${\frac {1+{\sqrt {5}}}{4}}\approx 0.80902$
Central density	1
Related polytopes
Army	Doaid
Regiment	Doaid
Dual	Icosahedral-rhombic triacontahedral tegmoid
Conjugate	Great stellated dodecahedron atop great icosidodecahedron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H3×I, order 120
Convex	Yes
Nature	Tame

Dodecahedron atop icosi-dodecahedron, or doaid, is a convex regular-faced polytope segmentochoron (designated as K-4.77 on Richard Klitzing's list). As the name suggests, it consists of a dodecahedron and an icosidodecahedron as bases, connected by 20 tetrahedra and 12 pentagonal antiprisms.

It is a segment of the hexacosichoron, with the icosidodecahedral base lying on the hexacosichoron's equator.

Vertex coordinates

The vertices of a dodecahedron atop icosidodecahedron segmentochoron of edge length 1 are given by:

• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,0,\,{\frac {1+{\sqrt {5}}}{4}}\right),$ and all permutations of first three coordinates
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,{\frac {1+{\sqrt {5}}}{4}}\right)$
• $\left(\pm {\frac {1+{\sqrt {5}}}{2}},\,0,\,0,\,0\right)$ and all permutations of first three coordinates
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,0,\,0\right)$ and all permutations of first three coordinates

External links

• Klitzing, Richard. "doaid".