Cube

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Cube
Hexahedron.png
Rank3
TypeRegular
SpaceSpherical
Notation
Bowers style acronymCube
Coxeter diagramx4o3o (CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png)
Schläfli symbol{4,3}
Tapertopic notation111
Toratopic notationIII
Bracket notation[III]
Stewart notationP4
Elements
Faces6 squares
Edges12
Vertices8
Vertex figureTriangle, edge length 2
Cube vertfig.png
Measures (edge length 1)
Circumradius
Edge radius
Inradius
Volume1
Dihedral angle90°
Height1
Central density1
Number of pieces6
Level of complexity1
Related polytopes
ArmyCube
RegimentCube
DualOctahedron
Petrie dualPetrial cube
ConjugateNone
Abstract properties
Flag count48
Net count11[1]
Euler characteristic2
Topological properties
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryB3, order 48
ConvexYes
NatureTame

The cube or hexahedron is one of the five Platonic solids. It has 6 square faces, joining 3 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[edit | edit source]

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

Representations[edit | edit source]

A cube can be represented by the following Coxeter diagrams:

  • x4o3o (regular)
  • x x4o (CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png) (generally a square prism)
  • x x x (CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png) (generally a cuboid)
  • s2s4x (CDel node h.pngCDel 2.pngCDel node h.pngCDel 4.pngCDel node 1.png) (generally a rectangular trapezoprism)
  • x2s4s (CDel node 1.pngCDel 2.pngCDel node h.pngCDel 4.pngCDel node h.png)
  • x2s4x (CDel node 1.pngCDel 2.pngCDel node h.pngCDel 4.pngCDel node 1.png)
  • 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 (rhombi prism)

In vertex figures[edit | edit source]

Cubes in vertex figures
Name Picture Schläfli symbol Edge length
Icositetrachoron
Schlegel wireframe 24-cell.png
{3,4,3}
Square tiling honeycomb
H3 443 FC boundary.png
{4,4,3}

Variations[edit | edit source]

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 incllude:

  • 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[edit | edit source]

The cube can be augmented with a square pyramid to form the elongated square pyramid, 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:

o4o3o truncations
Name OBSA Schläfli symbol CD diagram Picture
Cube cube {4,3} x4o3o
Uniform polyhedron-43-t0.png
Truncated cube tic t{4,3} x4x3o
Uniform polyhedron-43-t01.png
Cuboctahedron co r{4,3} o4x3o
Uniform polyhedron-43-t1.png
Truncated octahedron toe t{3,4} o4x3x
Uniform polyhedron-43-t12.png
Octahedron oct {3,4} o4o3x
Uniform polyhedron-43-t2.png
Small rhombicuboctahedron sirco rr{4,3} x4o3x
Uniform polyhedron-43-t02.png
Great rhombicuboctahedron girco tr{4,3} x4x3x
Uniform polyhedron-43-t012.png
Snub cube snic sr{4,3} s4s3s
Uniform polyhedron-43-s012.png

External links[edit | edit source]

  • Hi.gher.Space Wiki Contributors. "Cube".


References[edit | edit source]

  1. Goldstone, Richard; Suzzi Valli, Robert (2016-04-18), Unfoldings of the Cube, doi:10.48550/arXiv.1604.05004 Check |doi= value (help)