# Icosidodecahedral prism

(Redirected from Iddip)
Icosidodecahedral prism
Rank4
TypeUniform
Notation
Bowers style acronymIddip
Coxeter diagramx o5x3o ()
Elements
Cells20 triangular prisms, 12 pentagonal prisms, 2 icosidodecahedra
Faces40 triangles, 60 squares, 24 pentagons
Edges30+120
Vertices60
Vertex figureRectangular pyramid, edge lengths 1, (1+5)/2 (base), 2 (legs)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {7+2{\sqrt {5}}}}{2}}\approx 1.69353}$
Hypervolume${\displaystyle {\frac {45+17{\sqrt {5}}}{6}}\approx 13.83552}$
Dichoral anglesTrip–4–pip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
Id–3–trip: 90°
Id–5–pip: 90°
Height1
Central density1
Number of external pieces34
Level of complexity8
Related polytopes
ArmyIddip
RegimentIddip
DualRhombic triacontahedral tegum
ConjugateGreat icosidodecahedral prism
Abstract & topological properties
Flag count1920
Euler characteristic0
OrientableYes
Properties
SymmetryH3×A1, order 240
ConvexYes
NatureTame

The icosidodecahedral prism or iddip, is a prismatic uniform polychoron that consists of 2 icosidodecahedra, 12 pentagonal prisms, and 20 triangular prisms. Each vertex joins 1 icosidodecahedron, 2 pentagonal prisms, and 2 triangular prisms. It is a prism based on the icosidodecahedron. As such it is also a convex segmentochoron (designated K-4.90 on Richard Klitzing's list).

## Vertex coordinates

The vertices of an icosidodecahedral prism of edge length 1 are given by all permutations and sign changes of the first three coordinates of:

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

along with all even permutations and all sign changes of the first three coordinates of:

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

## Representations

An icosidodecahedral prism has the following Coxeter diagrams:

• x o5x3o (full symmetry)
• oo5xx3oo&#x (bases seen separately)
• xxxxx xoxfo5ofxox&#xt (H2×A1 axial, pentagonal prism-first)

## Related polychora

The regiment of the icosidodecahedral prism also includes the small icosihemidodecahedral prism and small dodecahemidodecahedral prism.