Truncated tetrahedral prism

Truncated tetrahedral prism
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Tuttip
Coxeter diagram	x x3x3o ()
Elements
Cells	4 triangular prisms, 4 hexagonal prisms, 2 truncated tetrahedra
Faces	8 triangles, 6+12 squares, 8 hexagons
Edges	12+12+24
Vertices	24
Vertex figure	Sphenoid, edge lengths 1, √3, √3 (base), √2 (legs)
Measures (edge length 1)
Circumradius	$\frac{\sqrt{26}}{4} ≈ 1.27475$
Hypervolume	$\frac{23\sqrt2}{12} ≈ 2.71057$
Dichoral angles	Trip–4–hip: $\arccos\left(-\frac13\right) ≈ 109.47122°$
	Tut–6–hip: 90°
	Tut–3–trip: 90°
	Hip–4–hip: $\arccos\left(\frac13\right) ≈ 70.52877°$
Height	1
Central density	1
Number of pieces	10
Level of complexity	12
Related polytopes
Army	Tuttip
Regiment	Tuttip
Dual	Triakis tetrahedral tegum
Conjugate	None
Abstract properties
Flag count	576
Euler characteristic	0
Topological properties
Orientable	Yes
Properties
Symmetry	A3×A1, order 48
Convex	Yes
Nature	Tame
Discovered by	{{{discoverer}}}

The truncated tetrahedral prism or tuttip is a prismatic uniform polychoron that consists of 2 truncated tetrahedra, 4 hexagonal prisms, and 4 triangular prisms. Each vertex joins 1 truncated tetrahedron, 1 triangular prism, and 2 hexagonal prisms. As the name suggests, it is a prism based on the truncated tetrahedron. As such it is also a convex segmentochoron (designated K-4.57 on Richard Klitzing's list).

Gallery

• 

  Card with cell counts, verf, and cross-sections

• 

  Segmentochoron display, tut atop tut

• 

  Net

Vertex coordinates

Coordinates for the vertices of a truncated tetrahedral prism of edge length 1 are given by all permutations and even sign changes of the first three coordinates of:

• $\left(\frac{3\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,±\frac12\right).$

Representations

A truncated tetrahedral prism has the following Coxeter diagrams:

• x x3x3o (full symmetry)
• xx3xx3oo&#x (bases considered separately)
• x2s4o3x () (bases as snub)
• xxx xux3oox&#xt (A₂×A₁ symmetry, triangular prism-first)
• x(xu)(xu)x-3-x(xo)(oo)o-&#xr (A₂ axial)
• xxxx xuxo oxux&#xt (A₁×A₁×A₁ axial, square-first)

External links

• Bowers, Jonathan. "Category 19: Prisms" (#898).

• Klitzing, Richard. "Tuttip".

• Wikipedia Contributors. "Truncated tetrahedral prism".