Truncated tetrahedral alterprism

Truncated tetrahedral alterprism
(OFF file)
Rank	4
Type	Scaliform
Space	Spherical
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	$\frac{\sqrt6}{2} ≈ 1.22475$
Hypervolume	$\frac73 ≈ 2.33333$
Dichoral angles	Tet–3–tricu: 120°
	Tricu–6–tut: 120°
	Tricu–4–tricu: 90°
	Tricu–3–tut: 60°
Height	$\frac{\sqrt2}{2} ≈ 0.70711$
Central density	1
Related polytopes
Army	Tuta
Regiment	Tuta
Dual	Triakis tetrahedral altertegum
Conjugate	None
Abstract properties
Euler characteristic	0
Topological properties
Orientable	Yes
Properties
Symmetry	(A3×2×A1)/2, order 48
Convex	Yes
Nature	Tame

The truncated tetrahedral alterprism, truncated tetrahedral cupoliprism, or tuta, 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 K-4.55 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 non-uniform scaliform polytope discovered.

Gallery

• 

  Alternate render, with the triangular cupolas hidden

• 

  Net

Vertex coordinates

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:

• $\left(\frac{3\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4}\right).$

Representations

The truncated tetrahedral alterprism has the following Coxeter diagrams:

• s2s4o3x (as snub derivation)
• xo3xx3ox&#x (as segmentochoron)

External links

• Wikipedia Contributors. "Runcic snub cubic hosochoron".
• Bowers, Jonathan. "Category S1: Simple Scaliforms" (#S1).

• Klitzing, Richard. "Tuta".

• Quickfur. "Truncated Tetrahedral Cupoliprism".