Tetrahedron atop cuboctahedron
Tetrahedron atop cuboctahedron  

Rank  4 
Type  Segmentotope 
Notation  
Bowers style acronym  Tetaco 
Coxeter diagram  xx3oo3ox&#x 
Elements  
Cells  1+4 tetrahedra, 4+6 triangular prisms, 1 cuboctahedron 
Faces  4+4+4+12 triangles, 6+12 squares 
Edges  6+12+12+12 
Vertices  4+12 
Vertex figures  4 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)  
Circumradius  1 
Hypervolume  
Dichoral angles  Trip–4–trip: 
Tet–3–trip:  
Tet–3–co:  
Trip–4–co:  
Trip–3–co:  
Height  
Central density  1 
Related polytopes  
Army  Tetaco 
Regiment  Tetaco 
Dual  Tetrahedralrhombic dodecahedral tegmoid 
Conjugate  None 
Abstract & topological properties  
Euler characteristic  0 
Orientable  Yes 
Properties  
Symmetry  A_{3}×I, order 24 
Convex  Yes 
Nature  Tame 
Tetrahedron atop cuboctahedron, or tetaco, is a CRF segmentochoron (designated K4.23 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.
Segmentochoron display[edit  edit source]

Tetrahedron atop cuboctahedron
Vertex coordinates[edit  edit source]
The vertices of a tetrahedron atop cuboctahedron segmentochoron of edge length 1 are given by:
 and all even sign changes of first three coordinates
 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:
Related polychora[edit  edit source]
This segmentochoron can be split into a triangular cupofastegium and the segmentochoron tetrahedron atop triangular cupola, joining at a common triangular cupola cell.
External links[edit  edit source]
 Klitzing, Richard. "tetaco".
 Wikipedia contributors. "Tetrahedral cupola".