Cube

Regular hexahedron
(Click here for rotating model)
Type	Platonic solid
Elements	F = 6, E = 12 V = 8 (χ = 2)
Faces by sides	6{4}
Conway notation	C
Schläfli symbols	{4,3}
	t{2,4} or {4}×{} tr{2,2} {}×{}×{} = {}3
Face configuration	V3.3.3.3
Wythoff symbol	3 | 2 4
Coxeter diagram
Symmetry	Oh, B3, [4,3], (*432)
Rotation group	O, [4,3]+, (432)
References	U06, C18, W3
Properties	convexzonohedron, Hanner polytope
Dihedral angle	90°
4.4.4 (Vertex figure)	Octahedron (dual polyhedron)
Net

In

The cube is the only regular hexahedron and is one of the five Platonic solids. It has 6 faces, 12 edges, and 8 vertices.

The cube is also a square parallelepiped, an equilateral cuboid, a right rhombohedron, and a 3-zonohedron. It is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations.

The cube is

.

Orthogonal projections

The cube has four special

Coxeter planes

.

Orthogonal projections

Centered by	Face	Vertex
Coxeter planes	B2	A2
Projective symmetry	[4]	[6]
Tilted views

Spherical tiling

The cube can also be represented as a

, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.

Orthographic projection	Stereographic projection

Cartesian coordinates

For a cube centered at the origin, with edges parallel to the axes and with an edge length of 2, the

Cartesian coordinates

of the vertices are

(±1, ±1, ±1)

while the interior consists of all points (x₀, x₁, x₂) with −1 < x_i < 1 for all i.

As a configuration

This configuration matrix represents the cube. The rows and columns correspond to vertices, edges, and faces. The diagonal numbers say how many of each element occur in the whole cube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.^[2] For example, the 2 in the first column of the middle row indicates that there are 2 vertices in (i.e., at the extremes of) each edge; the 3 in the middle column of the first row indicates that 3 edges meet at each vertex.

${\displaystyle {\begin{bmatrix}{\begin{matrix}8&3&3\\2&12&2\\4&4&6\end{matrix}}\end{bmatrix}}}$

Equation in three dimensional space

In analytic geometry, a cube's surface with center (x₀, y₀, z₀) and edge length of 2a is the locus of all points (x, y, z) such that

${\displaystyle \max\{|x-x_{0}|,|y-y_{0}|,|z-z_{0}|\}=a.}$

A cube can also be considered the limiting case of a 3D superellipsoid as all three exponents approach infinity.

Formulas

For a cube of edge length ${\displaystyle a}$:

surface area	${\displaystyle 6a^{2}\,}$	volume	${\displaystyle a^{3}\,}$
face diagonal	${\displaystyle {\sqrt {2}}a}$	space diagonal	${\textstyle {\sqrt {3}}a}$
radius of circumscribed sphere	${\displaystyle {\frac {\sqrt {3}}{2}}a}$	radius of sphere tangent to edges	${\displaystyle {\frac {a}{\sqrt {2}}}}$
radius of inscribed sphere	${\displaystyle {\frac {a}{2}}}$	angles between faces (in radians)	${\displaystyle {\frac {\pi }{2}}}$

As the volume of a cube is the third power of its sides ${\displaystyle a\times a\times a}$, third powers are called cubes, by analogy with squares and second powers.

A cube has the largest volume among cuboids (rectangular boxes) with a given surface area. Also, a cube has the largest volume among cuboids with the same total linear size (length+width+height).

Point in space

For a cube whose circumscribing sphere has radius R, and for a given point in its 3-dimensional space with distances d_i from the cube's eight vertices, we have:^[3]

${\displaystyle {\frac {\sum _{i=1}^{8}d_{i}^{4}}{8}}+{\frac {16R^{4}}{9}}=\left({\frac {\sum _{i=1}^{8}d_{i}^{2}}{8}}+{\frac {2R^{2}}{3}}\right)^{2}.}$

Doubling the cube

.

Uniform colorings and symmetry

The cube has three uniform colorings, named by the unique colors of the square faces around each vertex: 111, 112, 123.

The cube has four classes of symmetry, which can be represented by

.

Name	Regular hexahedron	Square prism	Rectangular trapezoprism	Rectangular cuboid	Rhombic prism	Trigonal trapezohedron
Coxeter diagram
Schläfli symbol	{4,3}	{4}×{ } rr{4,2}	s2{2,4}	{ }3 tr{2,2}	{ }×2{ }
Wythoff symbol	3 | 4 2	4 2 | 2		2 2 2 |
Symmetry	Oh [4,3] (*432)	D4h [4,2] (*422)	D2d [4,2+] (2*2)	D2h [2,2] (*222)		D3d [6,2+] (2*3)
Symmetry order	24	16	8	8		12
Image (uniform coloring)	(111)	(112)	(112)	(123)	(112)	(111), (112)

Geometric relations

A cube has eleven nets: that is, there are eleven ways to flatten a hollow cube by cutting seven edges.^[4] To color the cube so that no two adjacent faces have the same color, one would need at least three colors.

The cube is the cell of the only regular tiling of three-dimensional Euclidean space. It is also unique among the Platonic solids in having faces with an even number of sides and, consequently, it is the only member of that group that is a zonohedron (every face has point symmetry).

The cube can be cut into six identical square pyramids. If these square pyramids are then attached to the faces of a second cube, a rhombic dodecahedron is obtained (with pairs of coplanar triangles combined into rhombic faces).

In theology

Cubes appear in Abrahamic religions. The Kaaba (Arabic for 'cube') in Mecca is one example. Cubes also appear in Judaism as tefillin, and the New Jerusalem is described in the New Testament as a cube.^[5]

Other dimensions

The analogue of a cube in four-dimensional Euclidean space has a special name—a tesseract or hypercube. More properly, a hypercube (or n-dimensional cube or simply n-cube) is the analogue of the cube in n-dimensional Euclidean space and a tesseract is the order-4 hypercube. A hypercube is also called a measure polytope.

There are analogues of the cube in lower dimensions too: a point in dimension 0, a line segment in one dimension and a square in two dimensions.

Related polyhedra

The quotient of the cube by the antipodal map yields a projective polyhedron, the hemicube.

If the original cube has edge length 1, its dual polyhedron (an octahedron) has edge length ${\displaystyle \scriptstyle {\sqrt {2}}/2}$.

The cube is a special case in various classes of general polyhedra:

Name	Equal edge-lengths?	Equal angles?	Right angles?
Cube	Yes	Yes	Yes
Rhombohedron	Yes	Yes	No
Cuboid	No	Yes	Yes
Parallelepiped	No	Yes	No
quadrilaterally faced hexahedron	No	No	No

The vertices of a cube can be grouped into two groups of four, each forming a regular

. The intersection of the two forms a regular octahedron. The symmetries of a regular tetrahedron correspond to those of a cube which map each tetrahedron to itself; the other symmetries of the cube map the two to each other.

One such regular tetrahedron has a volume of 1/3 of that of the cube. The remaining space consists of four equal irregular tetrahedra with a volume of 1/6 of that of the cube, each.

The rectified cube is the cuboctahedron. If smaller corners are cut off we get a polyhedron with six octagonal faces and eight triangular ones. In particular we can get regular octagons (truncated cube). The rhombicuboctahedron is obtained by cutting off both corners and edges to the correct amount.

A cube can be inscribed in a dodecahedron so that each vertex of the cube is a vertex of the dodecahedron and each edge is a diagonal of one of the dodecahedron's faces; taking all such cubes gives rise to the regular compound of five cubes.

If two opposite corners of a cube are truncated at the depth of the three vertices directly connected to them, an irregular octahedron is obtained. Eight of these irregular octahedra can be attached to the triangular faces of a regular octahedron to obtain the cuboctahedron.

The cube is topologically related to a series of spherical polyhedral and tilings with order-3 vertex figures.

*n32 symmetry mutation of regular tilings: {n,3} v t e
Spherical				Euclidean	Compact hyperb.		Paraco.	Noncompact hyperbolic
{2,3}	{3,3}	{4,3}	{5,3}	{6,3}	{7,3}	{8,3}	{∞,3}	{12i,3}	{9i,3}	{6i,3}	{3i,3}

The cuboctahedron is one of a family of uniform polyhedra related to the cube and regular octahedron.

Uniform octahedral polyhedra
Symmetry: [4,3], (*432)							[4,3]+ (432)	[1+,4,3] = [3,3] (*332)		[3+,4] (3*2)
{4,3}	t{4,3}	r{4,3} r{31,1}	t{3,4} t{31,1}	{3,4} {31,1}	rr{4,3} s2{3,4}	tr{4,3}	sr{4,3}	h{4,3} {3,3}	h2{4,3} t{3,3}	s{3,4} s{31,1}
		=	=	=				= or	= or	=
Duals to uniform polyhedra
V43	V3.82	V(3.4)2	V4.62	V34	V3.43	V4.6.8	V34.4	V33	V3.62	V35

The cube is topologically related as a part of sequence of regular tilings, extending into the

: {4,p}, p=3,4,5...

*n42 symmetry mutation of regular tilings: {4,n} v t e
Spherical	Euclidean	Compact hyperbolic				Paracompact
{4,3}	{4,4}	{4,5}	{4,6}	{4,7}	{4,8}...	{4,∞}

With

, Dih₄, the cube is topologically related in a series of uniform polyhedral and tilings 4.2n.2n, extending into the hyperbolic plane:

*n42 symmetry mutation of truncated tilings: 4.2n.2n v t e
Symmetry *n42 [n,4]	Spherical		Euclidean	Compact hyperbolic				Paracomp.
	*242 [2,4]	*342 [3,4]	*442 [4,4]	*542 [5,4]	*642 [6,4]	*742 [7,4]	*842 [8,4]...	*∞42 [∞,4]
Truncated figures
Config.	4.4.4	4.6.6	4.8.8	4.10.10	4.12.12	4.14.14	4.16.16	4.∞.∞
n-kis figures
Config.	V4.4.4	V4.6.6	V4.8.8	V4.10.10	V4.12.12	V4.14.14	V4.16.16	V4.∞.∞

All these figures have octahedral symmetry.

The cube is a part of a sequence of rhombic polyhedra and tilings with [n,3] Coxeter group symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are squares.

Symmetry mutations of dual quasiregular tilings: V(3.n)2
*n32	Spherical			Euclidean	Hyperbolic
	*332	*432	*532	*632	*732	*832...	*∞32
Tiling
Conf.	V(3.3)2	V(3.4)2	V(3.5)2	V(3.6)2	V(3.7)2	V(3.8)2	V(3.∞)2

The cube is a square prism:

Family of uniform n-gonal prisms v t e
Prism name	Digonal prism	(Trigonal) Triangular prism	(Tetragonal) Square prism	Pentagonal prism	Hexagonal prism	Heptagonal prism	Octagonal prism	Enneagonal prism	Decagonal prism	Hendecagonal prism	Dodecagonal prism	...	Apeirogonal prism
Polyhedron image												...
Spherical tiling image												Plane tiling image
Vertex config.	2.4.4	3.4.4	4.4.4	5.4.4	6.4.4	7.4.4	8.4.4	9.4.4	10.4.4	11.4.4	12.4.4	...	∞.4.4
Coxeter diagram												...

As a trigonal trapezohedron, the cube is related to the hexagonal dihedral symmetry family.

Uniform hexagonal dihedral spherical polyhedra
[6,2] , (*622)							[6,2]+, (622)	[6,2+], (2*3)
{6,2}	t{6,2}	r{6,2}	t{2,6}	{2,6}	rr{6,2}	tr{6,2}	sr{6,2}	s{2,6}
Duals to uniforms
V62	V122	V62	V4.4.6	V26	V4.4.6	V4.4.12	V3.3.3.6	V3.3.3.3

Regular and uniform compounds of cubes

Compound of three cubes

Compound of five cubes

In uniform honeycombs and polychora

It is an element of 9 of 28 convex uniform honeycombs:

Cubic honeycomb	Truncated square prismatic honeycomb	Snub square prismatic honeycomb	Elongated triangular prismatic honeycomb	Gyroelongated triangular prismatic honeycomb
Cantellated cubic honeycomb	Cantitruncated cubic honeycomb	Runcitruncated cubic honeycomb	Runcinated alternated cubic honeycomb

It is also an element of five four-dimensional

:

Tesseract	Cantellated 16-cell	Runcinated tesseract	Cantitruncated 16-cell	Runcitruncated 16-cell

Cubical graph

Cubical graph
Hamiltonian, regular, symmetric, distance-regular, distance-transitive, 3-vertex-connected, bipartite, planar graph
Table of graphs and parameters

The skeleton of the cube (the vertices and edges) forms a graph with 8 vertices and 12 edges, called the cube graph. It is a special case of the hypercube graph.^[6] It is one of 5 Platonic graphs, each a skeleton of its Platonic solid.

An extension is the three dimensional k-ARY Hamming graph, which for k = 2 is the cube graph. Graphs of this sort occur in the theory of parallel processing in computers.

See also

• Pyramid
• Tesseract
• Trapezohedron
• Prince Rupert's cube

Notes

References

1. ^ "Nets of a Solids | Geometry |Nets of a Cube |Nets of a Cone & Cylinder".
2. ^ Coxeter 1973, p. 12, §1.8 Configurations.
3. ^ Park, Poo-Sung. "Regular polytope distances", Forum Geometricorum 16, 2016, 227-232. http://forumgeom.fau.edu/FG2016volume16/FG201627.pdf Archived 2016-10-10 at the Wayback Machine
4. S2CID 220150682
   .
5. ^ "Symbolism of the Cube • Eve Out of the Garden". 30 October 2020.
6. MR 0949280
   .

Works cited

• Coxeter, H.S.M. (1973). Regular Polytopes (3rd ed.). New York: Dover. pp. 122–123.

External links

• Weisstein, Eric W. "Cube". MathWorld.
• Cube: Interactive Polyhedron Model*
• Volume of a cube, with interactive animation
• Cube (Robert Webb's site)

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	An	Bn	I2(p) / Dn	E6 / E7 / E8 / F4 / G2	Hn
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	122 • 221
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	132 • 231 • 321
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	142 • 241 • 421
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1k2 • 2k1 • k21	n-pentagonal polytope
Topics: List of regular polytopes and compounds