Truncated cube

(Redirected from Tic)
Truncated cube
Rank3
TypeUniform
Notation
Bowers style acronymTic
Coxeter diagramx4x3o ()
Conway notationtC
Elements
Faces8 triangles, 6 octagons
Edges12+24
Vertices24
Vertex figureIsosceles triangle, edge lengths 1, 2+2, 2+2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {7+4{\sqrt {2}}}}{2}}\approx 1.77882}$
Volume${\displaystyle 7{\frac {3+2{\sqrt {2}}}{3}}\approx 13.59966}$
Dihedral angles8–3: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
8–8: 90°
Central density1
Number of external pieces14
Level of complexity3
Related polytopes
ArmyTic
RegimentTic
DualTriakis octahedron
ConjugateQuasitruncated hexahedron
Abstract & topological properties
Flag count144
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryB3, order 48
Flag orbits3
ConvexYes
NatureTame

The truncated cube, the truncated hexahedron, or tic, is one of the 13 Archimedean solids. It consists of 8 triangles and 6 octagons. Each vertex joins one triangle and two octagons. As the name suggests, it can be obtained by truncation of the cube.

Vertex coordinates

A truncated cube of edge length 1 has vertex coordinates given by all permutations of:

• ${\displaystyle \left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}}\right)}$.

Representations

A truncated cube has the following Coxeter diagrams:

• x4x3o () (full symmetry)
• xwwx4xoox&#xt (BC2 axial, octagon-first)
• xwwxoo3ooxwwx&#xt (A2 axial, triangle-first)
• wx3oo3xw&#zx (A3 subsymmetry, as hull of 2 small rhombitetratetrahedra)
• wx xw4xo&#zx (BC2×A1 symmetry)
• wwx wxw xww&#zx (A1×A1×A1 symmetry)
• oxwUwxo xwwxwwx&#xt (A1×A1 axial)

Semi-uniform variant

The truncated cube has a semi-uniform variant of the form x4y3o that maintains its full symmetry. This variant has 8 triangles of size y and 6 ditetragons as faces.

With edges of length a (between two ditetragons) and b (between a ditetragon and a triangle), its circumradius is given by

${\displaystyle R={\frac {\sqrt {3a^{2}+4b^{2}+4ab{\sqrt {2}}}}{2}}}$

and its volume is given by

${\displaystyle V=a^{3}+6ab^{2}+(9a^{2}b+5b^{3}){\frac {\sqrt {2}}{3}}}$.

It has coordinates given by all permutations of:

• ${\displaystyle \left(\pm {\frac {a+b{\sqrt {2}}}{2}},\,\pm {\frac {a+b{\sqrt {2}}}{2}},\,\pm {\frac {a}{2}}\right)}$.

Related polyhedra

A truncated cube can be augmented by attaching a square cupola to one of its octagonal faces, forming the augmented truncated cube. If a second square cupola is attached to the opposite octagonal face, the result is the biaugmented truncated cube.

The truncated rhombihedron is a uniform polyhedron compound composed of 5 truncated cubes.