# Tetrahedron atop cuboctahedron

Tetrahedron atop cuboctahedron
Rank4
TypeSegmentotope
Notation
Bowers style acronymTetaco
Coxeter diagramxx3oo3ox&#x
Elements
Cells1+4 tetrahedra, 4+6 triangular prisms, 1 cuboctahedron
Faces4+4+4+12 triangles, 6+12 squares
Edges6+12+12+12
Vertices4+12
Vertex figures4 triangular antiprisms, edge lengths 1 (base) and 2 (sides)
12 skewed rectangular pyramids, base edge lengths 1, 2, 1, 2, side edge lengths 1, 1, 2. 2
Measures (edge length 1)
Hypervolume${\displaystyle {\frac {35{\sqrt {5}}}{96}}\approx 0.81523}$
Dichoral anglesTrip–4–trip: ${\displaystyle \arccos \left(-{\frac {2}{3}}\right)\approx 131.81032^{\circ }}$
Tet–3–trip: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{4}}\right)\approx 127.76124^{\circ }}$
Tet–3–co: ${\displaystyle \arccos \left({\frac {1}{4}}\right)\approx 75.52249^{\circ }}$
Trip–4–co: ${\displaystyle \arccos \left({\frac {\sqrt {6}}{6}}\right)\approx 65.90516^{\circ }}$
Trip–3–co: ${\displaystyle \arccos \left({\frac {\sqrt {6}}{4}}\right)\approx 52.23876^{\circ }}$
Height${\displaystyle {\frac {\sqrt {10}}{4}}\approx 0.79057}$
Central density1
Related polytopes
ArmyTetaco
RegimentTetaco
DualTetrahedral-rhombic dodecahedral tegmoid
ConjugateNone
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryA3×I, order 24
ConvexYes
NatureTame

Tetrahedron atop cuboctahedron, or tetaco, is a CRF segmentochoron (designated K-4.23 on Richard Klitzing's list). As the name suggests, it consists of a tetrahedron and a cuboctahedron as bases, connected by 4 further tetrahedra and 4+6 triangular prisms.

It is also sometimes referred to as a tetrahedral cupola, as one generalization of the definition of a cupola is to have a polytope atop an expanded version.

Two tetrahedron atop cuboctahedron segmentochora can be attached at their cuboctahedral bases, such that the tetrahedral bases are in dual positions, to form the small prismatodecachoron.

## Vertex coordinates

The vertices of a tetrahedron atop cuboctahedron segmentochoron of edge length 1 are given by:

• ${\displaystyle \left({\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {10}}{4}}\right)}$ and all even sign changes of first three coordinates
• ${\displaystyle \left(\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,0,\,0\right)}$ and all permuations of first three coordinates

Alternative coordinates can be obtained from those of the small prismatodecachoron by removing the vertices of one of its tetrahedral cells:

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

## Related polychora

This segmentochoron can be split into a triangular cupofastegium and the segmentochoron tetrahedron atop triangular cupola, joining at a common triangular cupola cell.