# Octahedron atop truncated tetrahedron

Octahedron atop truncated tetrahedron
Rank4
TypeSegmentotope
Notation
Bowers style acronymOctatut
Coxeter diagramox3xx3oo&#x
Elements
Cells4 triangular prisms, 4 triangular cupolas, 1 octahedron, 1 truncated tetrahedron
Faces4+4+4+6 triangles, 12 squares, 4 hexagons
Edges6+12+12+12
Vertices6+12
Vertex figures6 square wedges, edge lengths 1 (base square and top edge) and 2 (sides)
12 sphenoids, edge lengths 1, 2, and 3 (2 each)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {35}}{5}}\approx 1.18322}$
Hypervolume${\displaystyle {\frac {49{\sqrt {5}}}{96}}\approx 1.14133}$
Dichoral anglesOct–3–trip: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{4}}\right)\approx 127.76125^{\circ }}$
Tricu–4–trip: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{6}}\right)\approx 114.09484^{\circ }}$
Tricu–3–oct: ${\displaystyle \arccos \left(-{\frac {1}{4}}\right)\approx 104.47751^{\circ }}$
Tricu–3–tricu: ${\displaystyle \arccos \left({\frac {1}{4}}\right)\approx 75.52249^{\circ }}$
Tricu–6–tut: ${\displaystyle \arccos \left({\frac {1}{4}}\right)\approx 75.52249^{\circ }}$
Tut–3–trip: ${\displaystyle \arccos \left({\frac {\sqrt {6}}{4}}\right)\approx 52.23876^{\circ }}$
Height${\displaystyle {\frac {\sqrt {10}}{4}}\approx 0.79057}$
Central density1
Related polytopes
ArmyOctatut
RegimentOctatut
DualCubic-triakis tetrahedral tegmoid
ConjugateNone
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryA3×I, order 24
ConvexYes
NatureTame

Octahedron atop truncated tetrahedron, or octatut, is a CRF segmentochoron (designated K-4.52 on Richard Klitzing's list). As the name suggests, it consists of an octahedron and a truncated tetrahedron as bases, connected by 4 triangular prisms and 4 triangular cupolas.

It can be obtained as a segment of the small rhombated pentachoron, which can be constructed by joining this segmentochoron to a cuboctahedron atop truncated tetrahedron segmentochoron at their common truncated tetrahedral base.

## Vertex coordinates

The vertices of an octahedron atop truncated tetrahedronsegmentochoron of edge length 1 are given by:

• ${\displaystyle \left(\pm {\frac {\sqrt {2}}{2}},\,0,\,0,\,{\frac {\sqrt {10}}{4}}\right)}$ and all permutations 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

Alternative coordinates can be obtained from those of the small rhombated pentachoron by removing the vertices of one of its cuboctahedral cells:

• ${\displaystyle \left(-{\frac {3{\sqrt {10}}}{10}},\,-{\frac {\sqrt {6}}{6}},\,-{\frac {\sqrt {3}}{3}},\,0\right),}$
• ${\displaystyle \left(-{\frac {3{\sqrt {10}}}{10}},\,{\frac {\sqrt {6}}{6}},\,{\frac {\sqrt {3}}{3}},\,0\right),}$
• ${\displaystyle \left(-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {2{\sqrt {3}}}{3}},\,0\right),}$
• ${\displaystyle \left(-{\frac {\sqrt {10}}{20}},\,-{\frac {5{\sqrt {6}}}{12}},\,-{\frac {\sqrt {3}}{3}},\,0\right),}$
• ${\displaystyle \left(-{\frac {\sqrt {10}}{20}},\,{\frac {\sqrt {6}}{4}},\,0,\,\pm 1\right),}$
• ${\displaystyle \left(-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {6}}{12}},\,{\frac {\sqrt {3}}{3}},\,\pm 1\right),}$
• ${\displaystyle \left(-{\frac {\sqrt {10}}{20}},\,-{\frac {5{\sqrt {6}}}{12}},\,{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(-{\frac {\sqrt {10}}{20}},\,{\frac {\sqrt {6}}{4}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(-{\frac {3{\sqrt {10}}}{10}},\,-{\frac {\sqrt {6}}{6}},\,{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(-{\frac {3{\sqrt {10}}}{10}},\,{\frac {\sqrt {6}}{6}},\,-{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}}\right).}$