# Cube

Last updated
Regular hexahedron

Type Platonic solid
Elements F = 6, E = 12
V = 8 (χ = 2)
Faces by sides6{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 U 06, C 18, W 3
Properties regular, convex zonohedron, Hanner polytope
Dihedral angle 90°

4.4.4
(Vertex figure)

Octahedron
(dual polyhedron)

Net

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]

## Contents

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 byFaceVertex
Coxeter planesB2
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.

## 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 < xi < 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 di 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

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

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 Oh has all the faces the same color. The dihedral symmetry D4h comes from the cube being a solid, with all the six sides being different colors. The prismatic subsets D2d has the same coloring as the previous one and D2h has alternating colors for its sides for a total of three colors, paired by opposite sides. Each symmetry form has a different Wythoff symbol.

NameRegular
hexahedron
Square prismRectangular
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 24 2 | 22 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
24168812
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.

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:

NameEqual edge-lengths?Equal angles?Right angles?
CubeYesYesYes
Rhombohedron YesYesNo
Cuboid NoYesYes
Parallelepiped NoYesNo

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

*n42 symmetry mutation of regular tilings: {4,n}
SphericalEuclideanCompact hyperbolicParacompact

{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 hyperbolicParacomp.
*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 V4.10.10V4.12.12V4.14.14V4.16.16V4..

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 EuclideanHyperbolic
*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. 2.4.43.4.44.4.45.4.46.4.47.4.48.4.49.4.410.4.411.4.412.4.4...∞.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
 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

Cubical graph
Named after Q3
Vertices 8
Edges 12
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

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

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

## Related Research Articles

A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral.

In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.

In geometry, an octahedron is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.

In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent regular polygons, and the same number of faces meet at each vertex. There are only five such polyhedra:

In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is a polyhedron with eight triangular, six square, and twelve rectangular faces. There are 24 identical vertices, with one triangle, one square, and two rectangles meeting at each one. If all the rectangles are themselves square, it is an Archimedean solid. The polyhedron has octahedral symmetry, like the cube and octahedron. Its dual is called the deltoidal icositetrahedron or trapezoidal icositetrahedron, although its faces are not really true trapezoids.

In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid with 38 faces: 6 squares and 32 equilateral triangles. It has 60 edges and 24 vertices.

In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces, 36 edges, and 24 vertices.

In geometry, the truncated cuboctahedron or great rhombicuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices, and 72 edges. Since each of its faces has point symmetry, the truncated cuboctahedron is a 9-zonohedron. The truncated cuboctahedron can tessellate with the octagonal prism.

In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. In particular, all its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are themselves regular polytopes of dimension jn.

In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.

In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric. Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in three-dimensional space, or as a three-dimensional projection of a hypercube. Zonohedra were originally defined and studied by E. S. Fedorove, a Russian crystallographer. More generally, in any dimension, the Minkowski sum of line segments forms a polytope known as a zonotope.

In geometry, a disdyakis dodecahedron,, is a Catalan solid with 48 faces and the dual to the Archimedean truncated cuboctahedron. As such it is face-transitive but with irregular face polygons. It resembles an augmented rhombic dodecahedron. Replacing each face of the rhombic dodecahedron with a flat pyramid creates a polyhedron that looks almost like the disdyakis dodecahedron, and is topologically equivalent to it.

In Euclidean geometry, rectification, also known as critical truncation or complete-truncation, is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded by vertex figure facets and the rectified facets of the original polytope.

In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. The term originates from Kepler's names for the Archimedean solids.

In geometry and topology, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations.

A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, and 160 diagonals. It is represented by the Schläfli symbol {5,3}.

In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive.

In geometry, chamfering or edge-truncation is a topological operator that modifies one polyhedron into another. It is similar to expansion: it moves the faces apart (outward), and adds a new face between each two adjacent faces; but contrary to expansion, it maintains the original vertices. For a polyhedron, this operation adds a new hexagonal face in place of each original edge.

In geometry, an icosahedron is a polyhedron with 20 faces. The name comes from Ancient Greek εἴκοσι (eíkosi) 'twenty', and ἕδρα (hédra) 'seat'. The plural can be either "icosahedra" or "icosahedrons".

In three-dimensional hyperbolic geometry, an ideal polyhedron is a convex polyhedron all of whose vertices are ideal points, points "at infinity" rather than interior to three-dimensional hyperbolic space. It can be defined as the convex hull of a finite set of ideal points. An ideal polyhedron has ideal polygons as its faces, meeting along lines of the hyperbolic space.

## 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. 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.
5. "Symbolism of the Cube • Eve Out of the Garden". 30 October 2020.
6. 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:. MR   0949280.

### Works cited

Family An Bn I2(p) / Dn E6 / E7 / E8 / / Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds