Cube

Cubical graph
Cubical graph
Named after	Q3
Vertices	8
Edges	12
Radius	3
Diameter	3
Girth	4
Automorphisms	48
Chromatic number	2
Properties	Hamiltonian, regular, symmetric, distance-regular, distance-transitive, 3-vertex-connected, bipartite, planar graph
	Table of graphs and parameters

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}
Schläfli symbols	t{2,4} or {4}×{} tr{2,2} {}×{}×{} = {}³
Face configuration	V3.3.3.3
Wythoff symbol	3 \| 2 4
Coxeter diagram
Symmetry	O_h, B₃, [4,3], (*432)
Rotation group	O, [4,3]⁺, (432)
References	U₀₆, C₁₈, W₃
Properties	regular, convex zonohedron, Hanner polytope
Dihedral angle	90°
4.4.4 (Vertex figure)	Octahedron (dual polyhedron)
Net

3D model of a cube

In geometry, a cube^{[lower-alpha 1]} is a three-dimensional solid object bounded by six square faces, facets, or sides, with three meeting at each vertex. Viewed from a corner, it is a hexagon and its net is usually depicted as a cross.^[1]

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 dual to the octahedron. It has cubical or octahedral symmetry, and is the only convex polyhedron whose faces are all squares. Its generalization for higher-dimensional spaces is called a hypercube.

Orthogonal projections

The cube has four special orthogonal projections, centered, on a vertex, edges, face and normal to its vertex figure. The first and third correspond to the A₂ and B₂ 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 spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, 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.

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

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

\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 $a$ :

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

As the volume of a cube is the third power of its sides $a\times a\times a$ , 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]

{\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

Doubling the cube, or the Delian problem, was the problem posed by ancient Greek mathematicians of using only a compass and straightedge to start with the length of the edge of a given cube and to construct the length of the edge of a cube with twice the volume of the original cube. They were unable to solve this problem, which in 1837 Pierre Wantzel proved it to be impossible because the cube root of 2 is not a constructible number.

Uniform colorings and symmetry

Octahedral symmetry tree

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 vertex-transitive coloring the faces. The highest octahedral symmetry O_h has all the faces the same color. The dihedral symmetry D_4h comes from the cube being a solid, with all the six sides being different colors. The prismatic subsets D_2d has the same coloring as the previous one and D_2h has alternating colors for its sides for a total of three colors, paired by opposite sides. Each symmetry form has a different Wythoff symbol.

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	O_h [4,3] (*432)	D_4h [4,2] (*422)	D_2d [4,2⁺] (2*2)	D_2h [2,2] (*222)		D_3d [6,2⁺] (2*3)
Symmetry order	24	16	8	8		12
Image (uniform coloring)	(111)	(112)	(112)	(123)	(112)	(111), (112)

Geometric relations

The 11 nets of the cube

Net of a cube

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 dual of a cube is an octahedron, seen here with vertices at the center of the cube's square faces.

The hemicube is the 2-to-1 quotient of the cube.

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 $\scriptstyle {\sqrt {2}}/2$ .

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 tetrahedron; more generally this is referred to as a demicube. These two together form a regular compound, the stella octangula. 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}
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{3^1,1}	t{3,4} t{3^1,1}	{3,4} {3^1,1}	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{3^1,1}

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

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

With dihedral symmetry, 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
Symmetry *n42 [n,4]	Spherical		Euclidean	Compact hyperbolic				Paracomp.
Symmetry *n42 [n,4]	*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
*n32	*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
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
Symmetry: [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 uniform polychora:

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

Cubical graph

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.

Notes

↑ from Latin cubus, from Greek κύβος (kubos) 'a cube, a die, vertebra'. In turn from Proto-Indo-European *keu(b)-, "to bend, turn".

References

↑ "Nets of a Solids | Geometry |Nets of a Cube |Nets of a Cone & Cylinder".
↑ Coxeter 1973, p. 12, §1.8 Configurations.
↑ 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
↑ Uehara, Ryuhei (2020). "Figure 1.1". Introduction to Computational Origami: The World of New Computational Geometry. Singapore: Springer. p. 4. doi:10.1007/978-981-15-4470-5. ISBN 978-981-15-4469-9. MR 4215620. S2CID 220150682.
↑ "Symbolism of the Cube • Eve Out of the Garden". 30 October 2020.
↑ Harary, Frank; Hayes, John P.; Wu, Horng-Jyh (1988). "A survey of the theory of hypercube graphs" (PDF). Computers & Mathematics with Applications. 15 (4): 277–289. doi:10.1016/0898-1221(88)90213-1. hdl:2027.42/27522. 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)

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
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	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

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[1] rom Latin cubus, from Greek κύβος (kubos) 'a cube, a die, vertebra'. In turn from Proto-Indo-European *keu(b)-, "to bend, turn".

[2] "Nets of a Solids | Geometry |Nets of a Cube |Nets of a Cone & Cylinder".

[FOOTNOTECoxeter197312§1.8_Configurations-3] Coxeter 1973, p. 12, §1.8 Configurations.

[4] 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

[5] Uehara, Ryuhei (2020). "Figure 1.1". Introduction to Computational Origami: The World of New Computational Geometry. Singapore: Springer. p. 4. doi:10.1007/978-981-15-4470-5. ISBN 978-981-15-4469-9. MR 4215620. S2CID 220150682.

[6] "Symbolism of the Cube • Eve Out of the Garden". 30 October 2020.

[7] Harary, Frank; Hayes, John P.; Wu, Horng-Jyh (1988). "A survey of the theory of hypercube graphs" (PDF). Computers & Mathematics with Applications. 15 (4): 277–289. doi:10.1016/0898-1221(88)90213-1. hdl:2027.42/27522. MR 0949280.

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