Tetrahedral-octahedral duoprism

Tetrahedral-octahedral duoprism
Rank	6
Type	Uniform
Notation
Bowers style acronym	Tetoct
Coxeter diagram	x3o3o o4o3x ()
Elements
Peta	8 triangular-tetrahedral duoprisms, 4 triangular-octahedral duoprisms
Tera	12 tetrahedral prisms, 32 triangular duoprisms, 6 octahedral prisms
Cells	6 tetrahedra, 48+48 triangular prisms, 4 octahedra
Faces	24+32 triangles, 72 squares
Edges	36+48
Vertices	24
Vertex figure	Square tettene, edge lengths 1 (base square and top triangle) and √2 (sides)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {14}}{4}}\approx 0.93541$
Hypervolume	${\frac {1}{18}}\approx 0.055556$
Dipetal angles	Tratet–tepe–tratet: $\arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }$
	Troct–triddip–tratet: 90°
	Troct–ope–troct: $\arccos \left({\frac {1}{3}}\right)\approx 70.52878^{\circ }$
Heights	Oct atop troct: ${\frac {\sqrt {6}}{3}}\approx 0.81650$
	Tratet atop trig-dual tratet: ${\frac {\sqrt {6}}{3}}\approx 0.81650$
	Ope atop ortho ope: ${\frac {\sqrt {2}}{2}}\approx 0.70711$
Central density	1
Number of external pieces	12
Level of complexity	20
Related polytopes
Army	Tetoct
Regiment	Tetoct
Dual	Tetrahedral-cubic duotegum
Conjugate	None
Abstract & topological properties
Flag count	23040
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A3×B3, order 1152
Convex	Yes
Nature	Tame

The tetrahedral-octahedral duoprism or tetoct is a convex uniform duoprism that consists of 4 triangular-octahedral duoprisms and 8 triangular-tetrahedral duoprisms. Each vertex joins 3 triangular-octahedral duoprisms and 4 triangular-tetrahedral duoprisms. It is a duoprism based on a tetrahedron and an octahedron, and is thus also a convex segmentopeton, as an octahedron atop triangular-octahedral duoprism.

Vertex coordinates

The vertices of a tetrahedral-octahedral duoprism of edge length 1 are given by all even sign changes of the first three coordinates, plus all permutations and sign changes of the last three coordinates, of:

• $\left({\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,\pm {\frac {\sqrt {2}}{2}},\,0,\,0\right).$

Representations

A triangular-tetrahedral duoprism has the following Coxeter diagrams:

• x3o3o o4o3x (full symmetry)
• x3o3o o3x3o (A3×A3 symmetry)
• xo3ox xx3oo3oo&#xt (A3×A2 axial, triangular-tetrahedral duoprism atop triangle-dual triangular-tetrahedral duoprism)
• ox3oo oo4oo3xx&#x (B3×A2 xaial, octahedron atop triangular-octahedral duoprism)
• xo ox oo4oo3xx&#x (B3×A1×A1 axial, octahedral prism atop orthogonal octahedral prism)

External links

• Klitzing, Richard. "tetoct".