# Tetrahedron atop truncated tetrahedron

Tetrahedron atop truncated tetrahedron
Rank4
TypeSegmentotope
Notation
Bowers style acronymTetatut
Coxeter diagramxx3ox3oo&#x
Elements
Cells1+4 tetrahedra, 4 triangular cupolas, 1 truncated tetrahedron
Faces4+4+12 triangles, 6 squares, 4 hexagons
Edges6+6+12+12
Vertices4+12
Vertex figures4 triangular prisms, edge lengths 1 (base) and 2 (sides)
12 skewed triangular pyramids, edge lengths 1 (base) and 2, 3, 3 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {6}}{2}}\approx 1.22475}$
Hypervolume${\displaystyle {\frac {3}{4}}=0.75}$
Dichoral anglesTet–3–tricu: 120°
Tricu–4–tricu: 90°
Tet–3–tut: 60°
Tut-6-tricu: 60°
Height${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Central density1
Related polytopes
ArmyTetatut
RegimentTetatut
DualTetrahedral-triakis tetrahedral tegmoid
ConjugateNone
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryA3×I, order 24
ConvexYes
NatureTame

Tetrahedron atop truncated tetrahedron, or tetatut, is a CRF segmentochoron (designated K-4.56 on Richard Klitzing's list). As the name suggests, it consists of a tetrahedron and a truncated tetrahedron as bases, connected by 4 further tetrahedra and 4 triangular cupolas.

It can be obtained as a segment of the rectified tesseract, which can be formed by attaching these segmentochora to both bases of the truncated tetrahedral cupoliprism.

## Vertex coordinates

The vertices of a tetrahedron atop truncated tetrahedronsegmentochoron of edge length 1 are given by:

• ${\displaystyle \left({\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{2}}\right)}$ and all even sign changes of first three coordinates
• ${\displaystyle \left({\frac {3{\sqrt {2}}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,0\right)}$ and all permutatoins and even sign changes of first three coordinates