# Icosahedron atop dodecahedron

Icosahedron atop dodecahedron
Rank4
TypeSegmentotope
Notation
Coxeter diagramxo5oo3ox&#x
Elements
Cells20+30 tetrahedra, 12 pentagonal pyramids, 1 icosahedron, 1 dodecahedron
Faces20+30+30 triangles, 12 pentagons
Edges30+30+60
Vertices12+20
Vertex figures12 pentagonal antiprisms, edge length 1
20 triangular antipodiums, edge lengths (1+5)/2 (large base) and 1 (small base and sides)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {1+{\sqrt {5}}}{2}}\approx 1.61803}$
Hypervolume${\displaystyle {\frac {135+61{\sqrt {5}}}{48}}\approx 5.65417}$
Dichoral anglesTet–3–tet: ${\displaystyle \arccos \left(-{\frac {1+3{\sqrt {5}}}{8}}\right)\approx 164.47751^{\circ }}$
Tet–3–ike: ${\displaystyle \arccos \left(-{\frac {\sqrt {10}}{4}}\right)\approx 142.23876^{\circ }}$
Tet–3–peppy: ${\displaystyle \arccos \left(-{\frac {\sqrt {10}}{4}}\right)\approx 142.23876^{\circ }}$
Doe–5–peppy: 72°
Height${\displaystyle {\frac {1}{2}}=0.5}$
Central density1
Related polytopes
DualDodecahedral-icosahedral tegmoid
ConjugateGreat icosahedron atop great stellated dodecahedron
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryH3×I, order 120
ConvexYes
NatureTame

The icosahedron atop dodecahedron, or ikadoe, is a CRF segmentochoron (designated K-4.78 on Richard Klitzing's list). As the name suggests, it consists of a dodecahedron and an icosahedron as bases, connected by 12 pentagonal pyramids and 20+30 tetrahedra.

It is also commonly referred to as a dodecahedral or icosahedral antiprism, as the two bases are a pair of dual polyhedra.

The icosahedron atop dodecahedron can also be obtained from the hexacosichoron as a monostratic stack. This is more readily seen from the hexacosichoron's vertex-first projection (where the two bases are concentric) or its edge-first projection (where the two bases are flattened).

## Vertex coordinates

Coordinates for the vertices of an icosahedron atop dodecahedron of edge length 1 are given by:

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