# Truncated dodecahedral prism

Truncated dodecahedral prism
Rank4
TypeUniform
Notation
Bowers style acronymTiddip
Coxeter diagramx x5x3o ()
Elements
Cells20 triangular prisms, 12 decagonal prisms, 2 truncated dodecahedra
Faces40 triangles, 30+60 squares, 24 decagons
Edges60+60+120
Vertices120
Vertex figureSphenoid, edge lengths 1, 10+25/2, 10+25/2 (base), 2 (legs)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {39+15{\sqrt {5}}}{8}}}\approx 3.01125}$
Hypervolume${\displaystyle 5{\frac {99+47{\sqrt {5}}}{12}}\approx 85.03966}$
Dichoral anglesTrip–4–dip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
Dip–4–dip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
Tid–10–dip: 90°
Tid–3–trip: 90°
Height1
Central density1
Number of external pieces34
Level of complexity12
Related polytopes
ArmyTiddip
RegimentTiddip
DualTriakis icosahedral tegum
ConjugateQuasitruncated great stellated dodecahedral prism
Abstract & topological properties
Flag count2880
Euler characteristic0
OrientableYes
Properties
SymmetryH3×A1, order 240
ConvexYes
NatureTame

The truncated dodecahedral prism or tiddip is a prismatic uniform polychoron that consists of 2 truncated dodecahedra, 12 decagonal prisms, and 20 triangular prisms. Each vertex joins 1 truncated dodecahedron, 1 triangular prism, and 2 decagonal prisms. It is a prism based on the truncated dodecahedron. As such it is also a convex segmentochoron (designated K-4.130 on Richard Klitzing's list).

## Vertex coordinates

The vertices of a truncated dodecahedral prism of edge length 1 are given by all even permutations of the first three coordinates of:

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

## Representations

A truncated dodecahedral prism has the following Coxeter diagrams:

• x x5x3o (full symmetry)
• xx5xx3oo&#x (bases considered separately)