# Quasitruncated hexahedron

(Redirected from Quith)
Quasitruncated hexahedron
Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymQuith
Coxeter diagramx4/3x3o ()
Elements
Faces8 triangles, 6 octagrams
Edges12+24
Vertices24
Vertex figureIsosceles triangle, edge lengths 1, 2–2, 2–2
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt{7-4\sqrt2}}{2} ≈ 0.57947}$
Volume${\displaystyle 7\frac{3-2\sqrt2}{3} ≈ 0.40034}$
Dihedral angles8/3–8/3: 90°
8/3–3: ${\displaystyle \arccos\left(\frac{\sqrt3}{3}\right) ≈ 54.73561^\circ}$
Central density7
Number of external pieces54
Level of complexity11
Related polytopes
ArmySirco, edge length ${\displaystyle \sqrt2-1}$
RegimentQuith
DualGreat triakis octahedron
ConjugateTruncated cube
Convex coreOctahedron
Abstract & topological properties
Flag count144
Euler characteristic2
OrientableYes
Genus0
Properties
SymmetryB3, order 120
ConvexNo
NatureTame

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

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

## 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 ()