# Truncated cubic prism

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).

Truncated cubic prism
Rank4
TypeUniform
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${\displaystyle {\sqrt {2+{\sqrt {2}}}}\approx 1.84776}$
Hypervolume${\displaystyle 7{\frac {3+2{\sqrt {2}}}{3}}\approx 13.59966}$
Dichoral anglesTrip–4–op: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Op–4–op: 90°
Tic–8–op: 90°
Tic–3–trip: 90°
Height1
Central density1
Number of external pieces16
Level of complexity12
Related polytopes
ArmyTiccup
RegimentTiccup
DualTriakis octahedral tegum
ConjugateQuasitruncated hexahedral prism
Abstract & topological properties
Flag count1152
Euler characteristic0
OrientableYes
Properties
SymmetryB3×A1, order 96
ConvexYes
NatureTame

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:

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

## 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)