Truncated tetrahedral alterprism
Truncated tetrahedral alterprism  

Rank  4 
Type  Scaliform 
Notation  
Bowers style acronym  Tuta 
Coxeter diagram  
Elements  
Cells  6 tetrahedra, 8 triangular cupolas, 2 truncated tetrahedra 
Faces  8+24 triangles, 12 squares, 8 hexagons 
Edges  12+24+24 
Vertices  24 
Vertex figure  Skew rectangular pyramid, base edge lengths 1 and √2, side edge lengths 1 and √3 
Measures (edge length 1)  
Circumradius  
Hypervolume  
Dichoral angles  Tet–3–tricu: 120° 
Tricu–6–tut: 120°  
Tricu–4–tricu: 90°  
Tricu–3–tut: 60°  
Height  
Central density  1 
Related polytopes  
Army  Tuta 
Regiment  Tuta 
Dual  Triakis tetrahedral altertegum 
Conjugate  None 
Abstract & topological properties  
Euler characteristic  0 
Orientable  Yes 
Properties  
Symmetry  (A_{3}×2×A_{1})/2, order 48 
Convex  Yes 
Nature  Tame 
The truncated tetrahedral alterprism or truncated tetrahedral cupoliprism, also known as the runcic snub cubic hosochoron, is a convex scaliform polychoron. It consists of two truncated tetrahedra as bases, joined by 8 triangular cupolas and 6 tetrahedra. 1 truncated tetrahedron, 1 tetrahedron, and 3 triangular cupolas join at each vertex. It can be formed by tetrahedrally alternating the small rhombicuboctahedral prism's triangles in such a way that the bases turn into truncated tetrahedra in opposite orientations.
It is also a convex segmentochoron (designated K4.55 in Richard Klitzing's list), as truncated tetrahedron atop truncated tetrahedron.
The two truncated tetrahedra are in opposite orientation, so that the hexagonal faces of one base are parallel to the triangular faces of the other.
It can also be seen as a diminishing of the rectified tesseract, specifically one where two tetrahedron atop truncated tetrahedron caps are removed.
This polychoron was discovered in 2000 by Richard Klitzing while he was searching for convex segmentochora. After its discovery he came up with the concept of scaliform polytopes, so this polychoron can in fact be said to be the first nonuniform scaliform polytope discovered.
Gallery[edit  edit source]

Alternate render, with the triangular cupolas hidden

Net
Vertex coordinates[edit  edit source]
The vertices of a truncated tetrahedral alterprism of edge length 1, centered at the origin, are given by all even changes of sign, and all permutations in the first three coordinates of:
Representations[edit  edit source]
The truncated tetrahedral alterprism has the following Coxeter diagrams:
 s2s4o3x (as snub derivation)
 xo3xx3ox&#x (as segmentochoron)
External links[edit  edit source]
 Wikipedia contributors. "Runcic snub cubic hosochoron".
 Bowers, Jonathan. "Category S1: Simple Scaliforms" (#S1).
 Klitzing, Richard. "Tuta".
 Quickfur. "Truncated Tetrahedral Cupoliprism".