Quasitruncated hexahedron

Quasitruncated hexahedron
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Quith
Coxeter diagram	x4/3x3o ()
Elements
Faces	8 triangles, 6 octagrams
Edges	12+24
Vertices	24
Vertex figure	Isosceles triangle, edge lengths 1, √2–√2, √2–√2
Measures (edge length 1)
Circumradius	$\frac{\sqrt{7-4\sqrt2}}{2} ≈ 0.57947$
Volume	$7\frac{3-2\sqrt2}{3} ≈ 0.40034$
Dihedral angles	8/3–8/3: 90°
	8/3–3: $\arccos\left(\frac{\sqrt3}{3}\right) ≈ 54.73561^\circ$
Central density	7
Number of external pieces	54
Level of complexity	11
Related polytopes
Army	Sirco, edge length $\sqrt2-1$
Regiment	Quith
Dual	Great triakis octahedron
Conjugate	Truncated cube
Convex core	Octahedron
Abstract & topological properties
Flag count	144
Euler characteristic	2
Orientable	Yes
Genus	0
Properties
Symmetry	B3, order 120
Convex	No
Nature	Tame

The quasitruncated hexahedron, the quasitruncated cube, or quith, also called the stellated truncated hexahedron, is a uniform polyhedron. It consists of 8 triangles and 6 octagrams. Each vertex joins one triangle and two octagrams. As the name suggests, it can be obtained by quasitruncation of the cube.

Vertex coordinates

A quasitruncated hexahedron of edge length 1 has vertex coordinates given by all permutations and sign changes of

• $\left(±\frac{\sqrt2-1}{2},\,±\frac{\sqrt2-1}{2},\,±\frac12\right).$

Related polyhedra

The quasitruncated rhombihedron is a uniform polyhedron compound composed of 5 quasitruncated hexahedra.

o4/3o3o truncations

Name	OBSA	Schläfli symbol	CD diagram	Picture
Cube	cube	{4/3,3}	x4/3o3o ()
Quasitruncated hexahedron	quith	t{4/3,3}	x4/3x3o ()
Cuboctahedron	co	r{3,4/3}	o4/3x3o ()
Truncated octahedron	toe	t{3,4/3}	o4/3x3x ()
Octahedron	oct	{3,4/3}	o4/3o3x ()
Quasirhombicuboctahedron	querco	rr{3,4/3}	x4/3o3x ()
Quasitruncated cuboctahedron	quitco	tr{3,4/3}	x4/3x3x ()
(degenerate, oct+6(4))			o4/3o3ß ()
Icosahedron	ike	s{3,4/3}	o4/3s3s ()

External links

• Bowers, Jonathan. "Polyhedron Category 2: Truncates" (#17).

• Klitzing, Richard. "quith".

• Wikipedia Contributors. "Stellated truncated hexahedron".
• McCooey, David. "Stellated Truncated Hexahedron"