Cube

Cube
(3D model) (OFF file)
Rank	3
Type	Regular
Space	Spherical
Notation
Bowers style acronym	Cube
Coxeter diagram	x4o3o ()
Schläfli symbol	{4,3}
Tapertopic notation	111
Toratopic notation	III
Bracket notation	[III]
Stewart notation	P4
Elements
Faces	6 squares
Edges	12
Vertices	8
Vertex figure	Triangle, edge length √2
Measures (edge length 1)
Circumradius	$\frac{\sqrt3}{2} ≈ 0.86603$
Edge radius	$\frac{\sqrt2}{2} ≈ 0.70711$
Inradius	$\frac12 = 0.5$
Volume	1
Dihedral angle	90°
Height	1
Central density	1
Number of pieces	6
Level of complexity	1
Related polytopes
Army	Cube
Regiment	Cube
Dual	Octahedron
Petrie dual	Petrial cube
Conjugate	None
Abstract properties
Flag count	48
Net count	11
Euler characteristic	2
Topological properties
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	B3, order 48
Convex	Yes
Nature	Tame

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

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

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

Representations

A cube can be represented by the following Coxeter diagrams:

• x4o3o (regular)
• x x4o () (generally a square prism)
• x x x () (generally a cuboid)
• s2s4x () (generally a rectangular trapezoprism)
• x2s4s ()
• x2s4x ()
• 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

Cubes in vertex figures

Name	Picture	Schläfli symbol	Edge length
Icositetrachoron		{3,4,3}	$1$
Square tiling honeycomb		{4,4,3}

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 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

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:

• Rhombihedron (5)
• Rhombihexahedron (3)
• Rhombisnub dishexahedron (6, with rotational freedom)
• An infinite number of prismatic compounds that are the prisms of compounds of squares

o4o3o truncations

Name	OBSA	Schläfli symbol	CD diagram	Picture
Cube	cube	{4,3}	x4o3o
Truncated cube	tic	t{4,3}	x4x3o
Cuboctahedron	co	r{4,3}	o4x3o
Truncated octahedron	toe	t{3,4}	o4x3x
Octahedron	oct	{3,4}	o4o3x
Small rhombicuboctahedron	sirco	rr{4,3}	x4o3x
Great rhombicuboctahedron	girco	tr{4,3}	x4x3x
Snub cube	snic	sr{4,3}	s4s3s

External links

• Bowers, Jonathan. "Polyhedron Category 1: Regulars" (#2).

• Klitzing, Richard. "Cube".

• Quickfur. "The Cube".

• Wikipedia Contributors. "Cube".
• McCooey, David. "Cube"

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