# Cube

(Redirected from Cubes)
Cube
Rank3
TypeRegular
Notation
Bowers style acronymCube
Coxeter diagramx4o3o ()
Schläfli symbol{4,3}
Tapertopic notation111
Toratopic notationIII
Bracket notation[III]
Conway notationC
Stewart notationP4
Elements
Faces6 squares
Edges12
Vertices8
Vertex figureTriangle, edge length 2
Petrie polygons4 skew hexagons
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {3}}{2}}\approx 0.86603}$
Edge radius${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Inradius${\displaystyle {\frac {1}{2}}=0.5}$
Volume1
Dihedral angle90°
Height1
Central density1
Number of external pieces6
Level of complexity1
Related polytopes
ArmyCube
RegimentCube
DualOctahedron
Petrie dualPetrial cube
κ ?Petrial tetrahedron
ConjugateNone
Abstract & topological properties
Flag count48
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
SkeletonQ 3
Properties
SymmetryB3, order 48
Flag orbits1
ConvexYes
Net count11[1]
NatureTame

The cube or hexahedron is one of the five Platonic solids. It has 6 square faces, joining 3 faces to a triangular vertex. It is the 3-dimensional hypercube.

It is the only Platonic solid that can tile 3-dimensional Euclidean space. This results in the cubic honeycomb. It also forms the cells of the 4D tesseract.

It is also the uniform square prism.

## Vertex coordinates

Coordinates for the vertices of a cube of edge length 1, centered at the origin, are:

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

## Representations

A cube can be represented by the following Coxeter diagrams:

• x4o3o () (regular)
• x2x4o () (generally a square prism)
• x2x2x () (generally a cuboid)
• s2s4x () (generally a rectangular trapezoprism)
• x2s4s ()
• x2s4x ()
• p2p6o () (generally a triangular antitegum)
• p2p3p ()
• xx4oo&#x (B2 axial, generally a square frustum)
• xx xx&#x (A1×A1 axial, rectangular frustum)
• oqoo3ooqo&#xt (A2 axial, generally a trigonal trapezohedron
• xx oqo&#xt (A1×A1, edge-first)
• oqooqo&#xt (A1 only)
• xxxx&#xr (A1 only, edge first)
• qo3oo3oq&#zx (A3 subsymmetry, hull of two tetrahedra)
• xx qo oq&#zx (rhombic prism)

## In vertex figures

Cubes in vertex figures
Name Picture Schläfli symbol Edge length
Icositetrachoron {3,4,3} ${\displaystyle 1}$
Square tiling honeycomb {4,4,3} ${\displaystyle {\sqrt {2}}}$

## Variations

A cube can be considered as the prism product of three mutually orthogonal dyads with the same length. By adjusting the sizes of these edges, we can create variations with different symmetry. Further notable variations of the cube arise from taking subgroups of the full B3 symmetry.

All of these double as colorings of the cube, when their symmetry is transferred to the regular cube.

The most notable variations include:

• Square prism - two squares and 4 rectangles, identical vertices
• Cuboid or rectangular prism - 3 pairs of rectangles, identical vertices
• Rectangular trapezoprism - 2 rectangles and 4 isosceles trapezoids with digonal antiprism symmetry, identical vertices
• Triangular antitegum - 6 identical rhombic faces, trigonal antiprism symmetry
• Square frustum - 2 base squares of different sizes, 4 isosceles trapezoid sides
• Triangular gyrotegum: 6 identical tetragonal faces, chiral triangular prismatic symmetry
• Rectangular frustum - 2 different base rectangles, with 2 pairs of isosceles trapezoid sides, digonal pyramid symmetry

## Related polyhedra

The cube can be augmented with a square pyramid to form the elongated square pyramid, which is a Johnson solid. If the opposite face is also augmented with a square pyramid, the result is the elongated square bipyramid.

A number of uniform polyhedron compounds are composed of cubes:

It is nontrivial to find a realizable polychoron with an odd number of cells, all of which are combinatorial cubes. The first such polychoron to be found has 16,533 cells. See cubical polytope.