# Truncated cubic prism

Truncated cubic prism
Rank4
TypeUniform
SpaceSpherical
Notation
Bowers style acronymTiccup
Coxeter diagramx x4x3o ()
Elements
Cells8 triangular prisms, 6 octagonal prisms, 2 truncated cubes
Faces16 triangles, 12+24 squares, 12 octagons
Edges24+24+48
Vertices48
Vertex figureSphenoid, edge lengths 1, 2+2, 2+2 (base), 2 (legs)
Measures (edge length 1)
Circumradius${\displaystyle \sqrt{2+\sqrt2} ≈ 1.84776}$
Hypervolume${\displaystyle 7\frac{3+2\sqrt2}{3} ≈ 13.59966}$
Dichoral anglesTrip–4–op: ${\displaystyle \arccos\left(-\frac{\sqrt3}{3}\right) ≈ 125.26439°}$
Op–4–op: 90°
Tic–8–op: 90°
Tic–3–trip: 90°
Height1
Central density1
Number of pieces16
Level of complexity12
Related polytopes
ArmyTiccup
RegimentTiccup
DualTriakis octahedral tegum
ConjugateQuasitruncated hexahedral prism
Abstract properties
Flag count1152
Euler characteristic0
Topological properties
OrientableYes
Properties
SymmetryB3×A1, order 96
ConvexYes
NatureTame
Discovered by{{{discoverer}}}

The truncated cubic prism or ticcup, is a prismatic uniform polychoron that consists of 2 truncated cubes, 6 octagonal prisms, and 8 triangular prisms. Each vertex joins 1 truncated cube, 1 octagonal prism, and 2 triangular prisms. As the name suggests, it is a prism based on the truncated cube. As such it is also a convex segmentochoron (designated K-4.99 on Richard Klitzing's list).

The truncated cubic prism can be obtained from a small rhombated tesseract by removing two small rhombicuboctahedron atop truncated cube segmentochora. This diminishing cuts the lateral small rhombicuboctahedra into their equatorial octagonal prisms only. It is also a central segment of the small prismatotetracontoctachoron.

## Vertex coordinates

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

• ${\displaystyle \left(±\frac{1+\sqrt2}{2},\,±\frac{1+\sqrt2}{2},\,±\frac12,\,±\frac12\right).}$

## Representations

The truncated cubic prism has the following Coxeter diagrams:

• x x4x3o (full symmetry)
• xx4xx3oo&#x (bases considered separately)
• xxxx xwwx4xoox&#xt (BC2×A1 axial, octagonal prism-first)