# Truncated tetrahedral prism

Truncated tetrahedral prism
Rank4
TypeUniform
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${\displaystyle {\frac {\sqrt {26}}{4}}\approx 1.27475}$
Hypervolume${\displaystyle {\frac {23{\sqrt {2}}}{12}}\approx 2.71057}$
Dichoral anglesTrip–4–hip: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
Tut–6–hip: 90°
Tut–3–trip: 90°
Hip–4–hip: ${\displaystyle \arccos \left({\frac {1}{3}}\right)\approx 70.52877^{\circ }}$
Height1
Central density1
Number of external pieces10
Level of complexity12
Related polytopes
ArmyTuttip
RegimentTuttip
DualTriakis tetrahedral tegum
ConjugateNone
Abstract & topological properties
Flag count576
Euler characteristic0
OrientableYes
Properties
SymmetryA3×A1, order 48
ConvexYes
NatureTame

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:

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

## 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)