# Icosidodecahedron atop truncated icosahedron

Icosidodecahedron atop truncated icosahedron
Rank4
TypeSegmentotope
Notation
Bowers style acronymIdati
Coxeter diagramoo5xx3ox&#x
Elements
Cells12 pentagonal prisms, 20 triangular cupolas, 1 icosidodecahedron, 1 truncated icosahedron
Faces20+30 triangles, 60 squares, 12+12 pentagons, 20 hexagons
Edges30+60+60+60
Vertices30+60
Vertex figures30 wedges, edge lengths 1 (2 base edges and top edges), 2 (sides), and (1+5)/2 (other base edges)
60 sphenoids, edge lengths 1 (1), 2 (2), (1+5)/2 (1), and 3 (2)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {19+8{\sqrt {5}}}}\approx 6.07359}$
Hypervolume${\displaystyle {\frac {325+277{\sqrt {5}}}{96}}\approx 9.83740}$
Dichoral anglesPip–4–tricu: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{10}}}\right)\approx 166.71747^{\circ }}$
Tricu–3–tricu: ${\displaystyle \arccos \left(-{\frac {1+3{\sqrt {5}}}{8}}\right)\approx 164.47751^{\circ }}$
Id–5–pip: 162°
Id–3–tricu: ${\displaystyle \arccos \left(-{\frac {\sqrt {7+3{\sqrt {5}}}}{4}}\right)\approx 157.76124^{\circ }}$
Ti–6–tricu: ${\displaystyle \arccos \left({\frac {\sqrt {7+3{\sqrt {5}}}}{4}}\right)\approx 22.23876^{\circ }}$
Ti–5–pip: 18°
Height${\displaystyle {\frac {{\sqrt {5}}-1}{4}}\approx 0.30902}$
Central density1
Related polytopes
ArmyIdati
RegimentIdati
DualRhombic triacontahedral-pentakis dodecahedral tegmoid
ConjugateGreat icosidodecahedron atop truncated great icosahedron
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryH3×I, order 120
ConvexYes
NatureTame

Icosidodecahedron atop truncated icosahedron, or idati, is a CRF segmentochoron (designated K-4.158 on Richard Klitzing's list). As the name suggests, it consists of an icosidodecahedron and a truncated icosahedron as bases, connected by 12 pentagonal prisms and 20 triangular cupolas.

It can be obtained as an icosidodecahedron-first cap of the small rhombated hexacosichoron.

## Vertex coordinates

The vertices of an icosidodecahedron atop truncated icosahedron segmentochoron of edge length 1 are given by all permutations of the first three coordinates of:

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

Plus all even permutations of the first three coordinates of:

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