# Truncated tetrahedral prism

Truncated tetrahedral prism
Rank4
TypeUniform
SpaceSpherical
Notation
Bowers style acronymTuttip
Coxeter diagramx x3x3o ()
Elements
Cells4 triangular prisms, 4 hexagonal prisms, 2 truncated tetrahedra
Faces8 triangles, 6+12 squares, 8 hexagons
Edges12+12+24
Vertices24
Vertex figureSphenoid, edge lengths 1, 3, 3 (base), 2 (legs)
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt{26}}{4} ≈ 1.27475}$
Hypervolume${\displaystyle \frac{23\sqrt2}{12} ≈ 2.71057}$
Dichoral anglesTrip–4–hip: ${\displaystyle \arccos\left(-\frac13\right) ≈ 109.47122°}$
Tut–6–hip: 90°
Tut–3–trip: 90°
Hip–4–hip: ${\displaystyle \arccos\left(\frac13\right) ≈ 70.52877°}$
Height1
Central density1
Number of pieces10
Level of complexity12
Related polytopes
ArmyTuttip
RegimentTuttip
DualTriakis tetrahedral tegum
ConjugateNone
Abstract properties
Flag count576
Euler characteristic0
Topological properties
OrientableYes
Properties
SymmetryA3×A1, order 48
ConvexYes
NatureTame
Discovered by{{{discoverer}}}

The truncated tetrahedral prism or tuttip is a prismatic uniform polychoron that consists of 2 truncated tetrahedra, 4 hexagonal prisms, and 4 triangular prisms. Each vertex joins 1 truncated tetrahedron, 1 triangular prism, and 2 hexagonal prisms. As the name suggests, it is a prism based on the truncated tetrahedron. As such it is also a convex segmentochoron (designated K-4.57 on Richard Klitzing's list).

## Vertex coordinates

Coordinates for the vertices of a truncated tetrahedral prism of edge length 1 are given by all permutations and even sign changes of the first three coordinates of:

• ${\displaystyle \left(\frac{3\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,±\frac12\right).}$

## Representations

A truncated tetrahedral prism has the following Coxeter diagrams:

• x x3x3o (full symmetry)
• xx3xx3oo&#x (bases considered separately)
• x2s4o3x () (bases as snub)
• xxx xux3oox&#xt (A2×A1 symmetry, triangular prism-first)
• x(xu)(xu)x-3-x(xo)(oo)o-&#xr (A2 axial)
• xxxx xuxo oxux&#xt (A1×A1×A1 axial, square-first)