View source for Cube

{{Short description|Solid object with six equal square faces}}
{{other uses}}
{{infobox polyhedron
 | name = Cube
 | image = File:Cube-h.svg
 | type = [[Hanner polytope]],<br>[[orthogonal polyhedron]],<br>[[parallelohedron]],<br>[[Platonic solid]],<br>[[plesiohedron]],<br>[[regular polyhedron]],<br>[[zonohedron]]
 | faces = 6
 | edges = 12 
 | vertices = 8
 | euler = 2
 | vertex_config = <math> 8 \times (4^3) </math>
 | schläfli = <math> \{4,3\} </math>
 | symmetry = [[octahedral symmetry]] <math> \mathrm{O}_\mathrm{h} </math>
 | dual = [[regular octahedron]]
 | angle = 90°
 | properties = [[Convex set|convex]],<br>[[edge-transitive]],<br>[[face-transitive]],<br>[[non-composite polyhedron|non-composite]],<br>[[Orthogonality|orthogonal]] faces,<br>[[vertex-transitive]]
 | surface area = 6 × side<sup>2</sup>
 | volume = side<sup>3</sup>
}}
A '''cube''' or '''regular hexahedron'''{{r|trudeau}} is a [[three-dimensional space|three-dimensional]] solid object in [[geometry]], which is bounded by six congruent [[square (geometry)|square]] faces, a type of [[polyhedron]]. It has twelve congruent edges and eight vertices. It is a type of [[parallelepiped]], with pairs of parallel opposite faces, and more specifically a [[rhombohedron]], with congruent edges, and a [[rectangular cuboid]], with [[right angle]]s between pairs of intersecting faces and pairs of intersecting edges. It is an example of many classes of polyhedra: [[Platonic solid]], [[regular polyhedron]], [[parallelohedron]], [[zonohedron]], and [[plesiohedron]]. The [[dual polyhedron]] of a cube is the [[regular octahedron]].

The cube can be represented in many ways, one of which is the graph known as the '''cubical graph'''. It can be constructed by using the [[Cartesian product of graphs]]. The cube is the three-dimensional [[hypercube]], a family of [[polytope]]s also including the two-dimensional square and four-dimensional [[tesseract]]. A cube with [[1|unit]] side length is the canonical unit of [[volume]] in three-dimensional space, relative to which other solid objects are measured. Other related figures involve the construction of polyhedra, [[Space-filling polyhedron|space-filling]] and [[Honeycomb (geometry)|honeycomb]]s, [[polycube]]s, as well as cubes in compounds, spherical, and topological space.

The cube was discovered in antiquity, associated with the nature of [[Earth (classical element)|earth]] by [[Plato]], for whom the Platonic solids are named.  It can be derived differently to create more polyhedra, and it has applications to construct a new [[polyhedron]] by attaching others. Other applications include popular culture of toys and games, arts, optical illusions, architectural buildings, as well as natural science and technology.

== Properties ==
[[File:Hexahedron.stl|thumb|3D model of a cube]]
A cube is a special case of [[rectangular cuboid]] in which the edges are equal in length.{{r|mk}} Like other cuboids, every face of a cube has four vertices, each of which connects with three congruent lines. These edges form [[square]] faces, making the [[dihedral angle]] of a cube between every two adjacent squares the [[interior angle]] of a square, 90°. Hence, the cube has six faces, twelve edges, and eight vertices,{{r|johnson}} and its [[Euler characteristic]] is 2, as for any convex polyhedron.{{r|richeson}}

The cube is one of the five [[Platonic solid]]s—[[polyhedron]]s in which all the [[regular polygon]]s are [[Congruence (geometry)|congruent]] and the same number of faces meet at each vertex.{{r|hs}} Every three square faces surrounding a vertex is [[orthogonality|orthogonal]] to each other, so the cube is classified as an [[orthogonal polyhedron]].{{r|jessen}} The cube may also be considered a [[parallelepiped]] in which all of its edges are equal{{r|calter}} (or more specifically a [[rhombohedron]] with congruent edges),{{sfnp|Hoffmann|2020|p=[http://books.google.com/books?id=16H0DwAAQBAJ&pg=PA83 83]}} and a [[trigonal trapezohedron]] since its square faces are the special cases of [[rhombi]].{{r|cc}}

=== Measurement and other metric properties ===
[[File:Cube diagonals.svg|thumb|upright=0.6|A face diagonal in red and space diagonal in blue]]
Given a cube with edge length <math> a </math>, the [[face diagonal]] of the cube is the [[diagonal]] of a square <math> a\sqrt{2} </math>, and the [[space diagonal]] of the cube is a line connecting two vertices that is not in the same face, formulated as <math> a \sqrt{3} </math>. Both formulas can be determined by using the [[Pythagorean theorem]]. The surface area of a cube <math> A </math> is six times the area of a square:{{r|khattar}}
<math display="block"> A = 6a^2. </math>
The volume of a cuboid is the product of its length, width, and height. Because all the edges of a cube are equal in length, the formula for the volume of a cube is the third power of its side length, leading to the use of the term ''[[Cube (algebra)|cubic]]'' to mean raising any number to the third power:{{r|thomson|khattar}}
<math display="block"> V = a^3. </math>

[[File:Prince Ruperts cube.png|thumb|upright=0.6|[[Prince Rupert's cube]]]]
One special case is the [[unit cube]], so named for measuring a single [[unit of length]] along each edge. It follows that each face is a [[unit square]] and that the entire figure has a volume of 1 cubic unit.{{r|ball|hr-w}} [[Prince Rupert's cube]], named after [[Prince Rupert of the Rhine]], is the largest cube that can pass through a hole cut into the unit cube, despite having sides approximately 6% longer.{{r|sriraman}} A polyhedron that can pass through a copy of itself of the same size or smaller is said to have the [[Rupert property]].{{r|jwy}} A geometric problem of [[doubling the cube]]&mdash;alternatively known as the ''Delian problem''&mdash;requires the construction of a cube with a volume twice the original by using only a [[compass and straightedge]]. Ancient mathematicians could not solve this problem until the French mathematician [[Pierre Wantzel]] proved it was impossible in 1837.{{r|lutzen}}

The cube has three types of [[closed geodesic]]s, or paths on a cube's surface that are locally straight. In other words, they avoid the vertices, follow line segments across the faces that they cross, and form [[complementary angle]]s on the two incident faces of each edge that they cross. One type lies in a plane parallel to any face of the cube, forming a square, with the length being equal to the perimeter of a face, four times the length of each edge. Another type lies in a plane perpendicular to the long diagonal, forming a regular hexagon; its length is <math> 3 \sqrt 2 </math> times that of an edge. The third type is a non-planar hexagon<!--, with the length being <math> 20 </math> (How is this derived? What is the unit of length? -->.{{r|fuchs}}

=== Relation to the spheres ===
With edge length <math> a </math>, the [[inscribed sphere]] of a cube is the sphere tangent to the faces of a cube at their centroids, with radius <math display="inline"> \frac{1}{2}a </math>. The [[midsphere]] of a cube is the sphere tangent to the edges of a cube, with radius <math display="inline"> \frac{\sqrt{2}}{2}a </math>. The [[circumscribed sphere]] of a cube is the sphere tangent to the vertices of a cube, with radius <math display="inline"> \frac{\sqrt{3}}{2}a </math>.{{r|radii}}

For a cube whose circumscribed sphere has radius <math> R </math>, and for a given point in its three-dimensional space with distances <math> d_i </math> from the cube's eight vertices, it is:{{r|poo-sung}}
<math display="block"> \frac{1}{8}\sum_{i=1}^8 d_i^4 + \frac{16R^4}{9} = \left(\frac{1}{8}\sum_{i=1}^8 d_i^2 + \frac{2R^2}{3}\right)^2. </math>

=== Symmetry ===
The cube has [[octahedral symmetry]] <math> \mathrm{O}_\mathrm{h} </math>. There are nine [[reflection symmetries]] (where two halves cut by a plane are identical): five cut the cube from the midpoints of its edges, and four are cut diagonally. It also has octahedral [[rotational symmetry]] (whereby rotation around the axis results in an identical appearance) <math> \mathrm{O} </math>: three axes pass through the centroids of the cube's opposite faces, six through the midpoints of the cube's opposite edges, and four through the cube's opposite vertices; these axes are respectively four-fold rotational symmetry (0°, 90°, 180°, and 270°), two-fold rotational symmetry (0° and 180°), and three-fold rotational symmetry (0°, 120°, and 240°).{{r|french|cromwell|cp}} Its [[automorphism group]] has order 48; that is, the cube has 48 [[Isometry|isometries]].{{r|kane}}

[[File:Dual Cube-Octahedron.svg|thumb|upright=0.8|The dual polyhedron of a cube is the regular octahedron]]
The [[dual polyhedron]] can be obtained from each of the polyhedra's vertices tangent to a plane by a process known as [[polar reciprocation]].{{r|cr}} One property of dual polyhedra is that the polyhedron and its dual share their [[Point groups in three dimensions|three-dimensional symmetry point group]]. In this case, the dual polyhedron of a cube is the [[regular octahedron]], and both of these polyhedra has the same octahedral symmetry.{{r|erickson}}

The cube is [[face-transitive]], meaning its two squares are alike and can be mapped by rotation and reflection.{{r|mclean}} It is [[vertex-transitive]], meaning all of its vertices are equivalent and can be mapped [[Isometry|isometrically]] under its symmetry.{{r|grunbaum-1997}} It is also [[edge-transitive]], meaning the same kind of faces surround each of its vertices in the same or reverse order, all two adjacent faces have the same [[dihedral angle]]. Therefore, the cube is a [[regular polyhedron]].{{r|senechal}} Each vertex is surrounded by three squares, so the cube is <math> 4.4.4 </math> by [[vertex configuration]] or <math> \{4,3\} </math> in a [[Schläfli symbol]].{{r|wd}}
{{Clear}}

== Applications ==
{{multiple image
 | image1 = One-red-dice-01.jpg
 | caption1 = A six-sided [[dice]]
 | image2 = Skewb.jpg
 | caption2 = A completed [[Skewb]]
 | image3 = Just another Alamo afternoon.jpg
 | caption3 = The [[Alamo (sculpture)|''Alamo'' sculpture]]
 | total_width = 360
}}
Cubes have appeared in many roles in popular culture.  It is the most common form of [[dice]].{{r|mclean}}  Puzzle toys such as pieces of a [[Soma cube]],{{r|masalski}} [[Rubik's Cube]], and [[Skewb]] are built of cubes.{{r|joyner}} ''[[Minecraft]]'' is an example of a [[Sandbox game|sandbox video game]] of cubic blocks.{{r|moore}} The outdoor sculpture [[Alamo (sculpture)|''Alamo'']] (1967) is a cube standing on a vertex.{{r|rz}} [[Optical illusions]] such as the [[impossible cube]] and [[Necker cube]] have been explored by artists such as [[M. C. Escher]].{{r|barrow}}  [[Salvador Dalí]]'s painting ''[[Corpus Hypercubus]]'' (1954) contains a [[tesseract]] unfolding into a six-armed cross; a similar construction is central to [[Robert A. Heinlein]]'s short story "[[And He Built a Crooked House]]" (1940).{{r|kemp|fowler}} The cube was applied in [[Leon Battista Alberti|Alberti]]'s treatise on [[Renaissance architecture]], ''[[De re aedificatoria]]'' (1450).{{r|march}} [[Kubuswoningen|Cube houses]] in the Netherlands are a set of cubical houses whose [[hexagon]]al space diagonals becomes the main floor.{{r|an}}

{{multiple image
 | image1 = Cubic.svg
 | caption1 = Simple cubic crystal structure
 | image2 = 2780M-pyrite1.jpg
 | caption2 = [[Pyrite]] cubic crystals
 | image3 = Cubane molecule ball.png
 | caption3 = [[Ball-and-stick model]] of [[cubane]]
 | total_width = 360
}}
Cubes are also found in natural science and technology. It is applied to the [[unit cell]] of a crystal known as a [[cubic crystal system]].{{r|tisza}} [[Pyrite]] is an example of a [[mineral]] with a commonly cubic shape, although there are many varied shapes.{{r|hoffmann}} The [[radiolarian]] ''Lithocubus geometricus'', discovered by [[Ernst Haeckel]], has a cubic shape.{{r|haeckel}} A historical attempt to unify three physics ideas of [[Galilean relativity|relativity]], [[gravitation]], and [[quantum mechanics]] used the framework of a cube known as a [[cGh physics|''cGh'' cube]].{{r|padmanabhan}} [[Cubane]] is a synthetic [[hydrocarbon]] consisting of eight carbon [[atom]]s arranged at the corners of a cube, with one [[hydrogen]] atom attached to each carbon atom.{{r|biegasiewicz}}  

Other technological cubes include the spacecraft device [[CubeSat]],{{r|helvajian}} and [[thermal radiation]] demonstration device [[Leslie cube]].{{r|vm}} Cubical grids are usual in three-dimensional [[Cartesian coordinate system]]s.{{r|knstv}} In [[computer graphics]], [[Marching cubes|an algorithm]] divides the input volume into a discrete set of cubes known as the unit on [[isosurface]],{{r|cmsi}} and the faces of a cube can be used for [[Cube mapping|mapping a shape]].{{r|greene}}

{{multiple image
 | image1 = Kepler Hexahedron Earth.jpg
 | caption1 = Sketch of a cube by Johannes Kepler
 | image2 = Mysterium Cosmographicum solar system model.jpg
 | caption2 = [[Johannes Kepler|Kepler's]] Platonic solid model of the [[Solar System]]
 | align = right
 | total_width = 300
}}
The [[Platonic solid]]s are five polyhedra known since antiquity. The set is named for [[Plato]] who, in his dialogue [[Timaeus (dialogue)|''Timaeus'']], attributed these solids to nature. One of them, the cube, represented the [[classical element]] of [[Earth (classical element)|earth]] because of its stability.{{sfnp|Cromwell|1997|p=[https://archive.org/details/polyhedra0000crom/page/55 55]}} [[Euclid]]'s [[Euclid's Elements|''Elements'']] defined the Platonic solids, including the cube, and showed how to find the ratio of the circumscribed sphere's diameter to the edge length.{{r|heath}} Following Plato's use of the regular polyhedra as symbols of nature, [[Johannes Kepler]] in his ''[[Harmonices Mundi]]'' sketched each of the Platonic solids; he decorated the cube's side with a tree.{{sfnp|Cromwell|1997|p=[https://archive.org/details/polyhedra0000crom/page/55 55]}} In his ''[[Mysterium Cosmographicum]]'', Kepler also proposed that the ratios between sizes of the orbits of the planets are the ratios between the sizes of the [[inscribed sphere|inscribed]] and [[circumscribed sphere]]s of the Platonic solids. That is, if the orbits are great circles on spheres, the sphere of Mercury is tangent to a [[regular octahedron]], whose vertices lie on the sphere of Venus, which is in turn tangent to a [[regular icosahedron]], within the sphere of Earth, within a [[regular dodecahedron]], within the sphere of Mars, within a [[regular tetrahedron]], within the sphere of Jupiter, within a cube, within the sphere of Saturn. In fact, the orbits are not circles but ellipses (as Kepler himself later showed), and these relations are only approximate.{{r|livio}}

== Construction ==
[[File:The 11 cubic nets.svg|thumb|The eleven nets of a cube]]
An elementary way to construct a cube is using its [[Net (polyhedron)|net]], an arrangement of edge-joining polygons, by connecting the edges of those polygons. Eleven nets for the cube are possible (''see right'').{{r|jeon}}

In [[analytic geometry]], a cube may be constructed using the [[Cartesian coordinate systems]]. 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 <math> (\pm 1, \pm 1, \pm 1) </math>.{{r|smith}} Its interior consists of all points <math> (x_0, x_1, x_2) </math> with <math> -1 < x_i < 1 </math> for all <math> i </math>. A cube's surface with center <math> (x_0, y_0, z_0) </math> and edge length of <math> 2a </math> is the [[Locus (mathematics)|locus]] of all points <math> (x,y,z) </math> such that
<math display="block"> \max\{ |x-x_0|,|y-y_0|,|z-z_0| \} = a.</math>

The cube is a [[Hanner polytope]], because it can be constructed by using the [[Cartesian product]] of three line segments. Its dual polyhedron, the regular octahedron, is constructed by the [[direct sum]] of three line segments.{{r|kozachok}}

== Representation ==
=== As a graph ===
{{main|Hypercube graph}}
[[File:Cube skeleton.svg|upright=0.8|thumb|The graph of a cube]]

According to [[Steinitz's theorem]], a [[Graph (discrete mathematics)|graph]] can be represented as the [[Skeleton (topology)|skeleton]] of a polyhedron. Such a graph has two properties: [[Planar graph|planar]] (the edges of a graph are connected to every vertex without crossing other edges), and [[k-vertex-connected graph|3-connected]] (whenever a graph with more than three vertices, and two of the vertices are removed, the edges remain connected).{{r|grunbaum-2003|ziegler}} The skeleton of a cube, represented as the graph, is called the '''cubical graph''', a [[Platonic graph]]. It has the same number of vertices and edges as the cube, twelve vertices and eight edges.{{r|rudolph}} The cubical graph is also classified as a [[prism graph]], resembling the skeleton of a cuboid.{{r|ps}}

The cubical graph is a special case of [[hypercube graph]] or {{nowrap|1=<math>n</math>-}}cube&mdash;denoted as <math> Q_n </math>&mdash;because it can be constructed by using the [[Cartesian product of graphs]]: two graphs connecting the pair of vertices with an edge to form a new graph.{{r|hh}} In the case of the cubical graph, it is the product of two <math> Q_2 </math>; roughly speaking, it is a graph resembling a square. In other words, the cubical graph is constructed by connecting each vertex of two squares with an edge. Notationally, the cubical graph is <math> Q_3 </math>.{{r|cz}} Like any hypercube graph, it has a [[Cycle (graph theory)|cycle]] which visits [[Hamiltonian path|every vertex exactly once]],{{r|ly}} and it is also an example of a [[unit distance graph]].{{r|hp}}

The cubical graph is [[bipartite graph|bipartite]], meaning every [[Independent set (graph theory)|independent set]] of four vertices can be [[Disjoint set|disjoint]] and the edges connected in those sets.{{r|berman-graph}} However, every vertex in one set cannot connect all vertices in the second, so this bipartite graph is not [[complete bipartite graph|complete]].{{r|aw}} It is an example of both a [[crown graph]] and a [[bipartite Kneser graph]].{{r|kl|berman-graph}}

=== In orthogonal projection ===
An object illuminated by parallel rays of light casts a shadow on a plane perpendicular to those rays, called an [[Orthographic projection|orthogonal projection]]. A polyhedron is considered [[Equiprojective polyhedra|equiprojective]] if, for some position of the light, its orthogonal projection is a regular polygon. The cube is equiprojective because, if the light is parallel to one of the four lines joining a vertex to the opposite vertex, its projection is a [[regular hexagon]].{{r|hhlnqr}}

=== As a configuration matrix ===
The cube can be represented as a [[Platonic solid#As a configuration|configuration matrix]], a [[Matrix (mathematics)|matrix]] in which the rows and columns correspond to the elements of a polyhedron as the vertices, edges, and faces. The [[Main diagonal|diagonal]] of a matrix denotes the number of each element that appears in a polyhedron, whereas the non-diagonal of a matrix denotes the number of the column's elements that occur in or at the row's element. The cube's eight vertices, twelve edges, and six faces are denoted by each element in a matrix's diagonal (8, 12, and 6). The first column of the middle row indicates that there are two vertices on each edge, denoted as 2; the middle column of the first row indicates that three edges meet at each vertex, denoted as 3. The following matrix is:{{r|coxeter}}
<math display="block"> \begin{bmatrix}\begin{matrix}8 & 3 & 3 \\ 2 & 12 & 2 \\ 4 & 4 & 6 \end{matrix}\end{bmatrix}</math>

== Related figures ==
=== Construction of polyhedra ===
{{multiple image
 | image1 = CubeAndStel.svg
 | image2 = Tetrakishexahedron.jpg
 | footer = Some of the derived cubes, the [[stellated octahedron]] and [[tetrakis hexahedron]].
 | total_width = 320
}}
The cube can appear in the construction of a polyhedron, and some of its types can be derived differently in the following:
* When [[faceting]] a cube, meaning removing part of the polygonal faces without creating new vertices of a cube, the resulting polyhedron is the [[stellated octahedron]].{{r|inchbald}}
* The cube is a [[non-composite polyhedron]], meaning it is a convex polyhedron that cannot be separated into two or more regular polyhedra. The cube can be applied to construct a new convex polyhedron by attaching another.{{r|timofeenko-2010}} Attaching a [[square pyramid]] to each square face of a cube produces its [[Kleetope]], a polyhedron known as the [[tetrakis hexahedron]].{{r|sod}} If one and two equilateral square pyramids are attached to the square faces, the [[elongated square pyramid]] and [[elongated square bipyramid]] respectively, the [[Johnson solid]]'s examples, are constructed.{{r|rajwade}}
* Each of the cube's vertices can be [[Truncation (geometry)|truncated]], and the resulting polyhedron is the [[Archimedean solid]], the [[truncated cube]].{{sfnp|Cromwell|1997|pp=[https://books.google.com/books?id=OJowej1QWpoC&pg=PA81 81–82]}} When its edges are truncated, it is a [[rhombicuboctahedron]].{{r|linti}} Relatedly, the rhombicuboctahedron can also be constructed by separating the cube's faces and then spreading away, after which adding other triangular and square faces between them; this is known as the "expanded cube". Similarly, it is constructed by the cube's dual, the regular octahedron.{{r|vxac}}
* The [[barycentric subdivision]] of a cube (or its dual, the regular octahedron) is the [[disdyakis dodecahedron]], a [[Catalan solid]].{{r|ls}}
* The corner region of a cube can also be truncated by a plane (e.g., spanned by the three neighboring vertices), resulting in a [[trirectangular tetrahedron]].{{sfnp|Coxeter|1973|p=[http://books.google.com/books?id=2ee7AQAAQBAJ&pg=PA71 71]}}
* The [[snub cube]] is an Archimedean solid that can be constructed by separating the cube's faces, and filling the gaps with twisted angle equilateral triangles, a process known as a [[Snub (geometry)|snub]].{{r|holme}}
The cube can be constructed with six [[square pyramid]]s, tiling space by attaching their apices. In some cases, this produces the [[rhombic dodecahedron]] circumscribing a cube.{{r|barnes|cundy}}

=== Polycubes ===
{{main|Polycubes}}
[[File:Net of tesseract.gif|thumb|upright=0.6|[[Dali cross]], the net of a [[tesseract]]]]
The [[Polycube]] is a polyhedron in which the faces of many cubes are attached. Analogously, it can be interpreted as the [[polyominoes]] in three-dimensional space.{{r|lunnon}} When four cubes are stacked vertically, and the other four are attached to the second-from-top cube of the stack, the resulting polycube is the [[Dali cross]], named after [[Salvador Dali]]. The Dali cross is a tile space polyhedron,{{r|hut|pucc}} which can be represented as the net of a [[tesseract]]. A tesseract is a cube's analogous [[four-dimensional space]] bounded by twenty-four squares and eight cubes.{{r|hall}}

=== Space-filling and honeycombs ===
[[Hilbert's third problem]] asks whether every two equal-volume polyhedra can always be dissected into polyhedral pieces and reassembled into each other. If yes, then the volume of any polyhedron could be defined axiomatically as the volume of an equivalent cube into which it could be reassembled. [[Max Dehn]] solved this problem by inventing the [[Dehn invariant]], answering that not all polyhedra can be reassembled into a cube.{{r|gruber}} It showed that two equal volume polyhedra should have the same Dehn invariant, except for the two tetrahedra whose Dehn invariants were different.{{r|zeeman}}

[[File:Partial cubic honeycomb.png|thumb|upright=0.6|[[Cubic honeycomb]]]]
The cube has a Dehn invariant of zero, meaning that cubes can achieve a [[Honeycomb (geometry)|honeycomb]]. It is also a [[Space-filling polyhedron|space-filling tile]] in three-dimensional space in which the construction begins by attaching a polyhedron onto its faces without leaving a gap.{{r|lm}} The cube is a [[plesiohedron]], a special kind of space-filling polyhedron that can be defined as the [[Voronoi cell]] of a symmetric [[Delone set]].{{r|erdahl}} The plesiohedra include the [[parallelohedra]], which can be [[Translation (geometry)|translated]] without rotating to fill a space in which each face of any of its copies is attached to a like face of another copy. There are five kinds of parallelohedra, one of which is the cuboid.{{r|alexandrov}} Every three-dimensional parallelohedron is a [[zonohedron]], a [[centrally symmetric]] polyhedron whose faces are [[Zonogon|centrally symmetric polygons]].{{r|shephard}} In the case of the cube, it can be represented as a [[Cell (geometry)|cell]]. Some honeycombs have cubes as the only cells; one example is the [[cubic honeycomb]], the only regular honeycomb in Euclidean three-dimensional space, which has four cubes around each edge.{{r|twelveessay|ns}}

=== Miscellaneous ===
{{multiple image
 | image1 = UC07-6 cubes.png
 | image2 = UC08-3 cubes.png
 | image3 = UC09-5 cubes.png
 | footer = Enumeration according to {{harvtxt|Skilling|1976}}: compound of six cubes with rotational freedom <math> \mathrm{UC}_7 </math>, [[Compound of three cubes|three cubes]] <math> \mathrm{UC}_8 </math>, and [[Compound of five cubes|five cubes]] <math> \mathrm{UC}_9 </math>
 | total_width = 360
}}
{{anchor|Compound of cubes}}The [[polyhedral compound]]s, in which the cubes share the same centre, are [[uniform polyhedron compound]]s, meaning they are polyhedral compounds whose constituents are identical (although possibly [[enantiomorphous]]) [[uniform polyhedra]], in an arrangement that is also uniform. Respectively, the list of compounds enumerated by {{harvtxt|Skilling|1976}} in the seventh to ninth uniform compounds for the compound of six cubes with rotational freedom, [[Compound of three cubes|three cubes]], and [[Compound of five cubes|five cubes]].{{r|skilling}} Two compounds, consisting of [[compound of two cubes|two]] and three cubes were found in [[M. C. Escher|Escher]]'s [[wood engraving]] print [[Stars (M. C. Escher)|''Stars'']] and [[Max Brückner]]'s book ''Vielecke und Vielflache''.{{r|hart}}

{{multiple image
 | image1 = Square on sphere.svg
 | caption1 = Spherical cube
 | image2 = 3-Manifold 3-Torus.png
 | caption2 = A view in [[3-torus|three-dimensional torus]]
 | total_width = 300
}}
{{anchor|Spherical cube}}The spherical cube represents the [[spherical polyhedron]], which can be modeled by the [[Arc (geometry)|arc]] of [[great circle]]s, creating bounds as the edges of a [[spherical polygon|spherical square]].{{r|yackel}}
Hence, the spherical cube consists of six spherical squares with 120° interior angles on each vertex. It has [[vector equilibrium]], meaning that the distance from the centroid and each vertex is the same as the distance from that and each edge.{{r|popko|fuller}} Its dual is the [[spherical octahedron]].{{r|yackel}}

The topological object [[3-torus|three-dimensional torus]] is a topological space defined to be [[homeomorphic]] to the Cartesian product of three circles. It can be represented as a three-dimensional model of the cube shape.{{r|marar}}

== See also ==
* [[Bhargava cube]], a configuration to study the law of [[binary quadratic form]] and other such forms, of which the cube's vertices represent the [[integer]].
* [[Chazelle polyhedron]], a notched opposite faces of a cube.
* [[Cubism]], an [[art movement]] of revolutionized painting and the visual arts.
* [[Hemicube (geometry)|Hemicube]], an abstract polyhedron produced by identifying opposite faces of a cube
* [[Squaring the square]]'s three-dimensional analogue, [[cubing the cube]]
{{Clear}}

== References ==
{{reflist|refs=

<ref name=alexandrov>{{cite book
 | last = Alexandrov | first = A. D. | author-link = Aleksandr Danilovich Aleksandrov
 | contribution = 8.1 Parallelohedra
 | contribution-url = https://books.google.com/books?id=R9vPatr5aqYC&pg=PA349
 | pages = 349–359
 | publisher = Springer
 | title = Convex Polyhedra | title-link = Convex Polyhedra (book)
 | year = 2005
}}</ref>

<ref name=an>{{cite book
 | last1 = Alsina | first1 = Claudi
 | last2 = Nelsen | first2 = Roger B.
 | year = 2015
 | title = A Mathematical Space Odyssey: Solid Geometry in the 21st Century
 | publisher = [[Mathematical Association of America]]
 | url = https://books.google.com/books?id=FEl2CgAAQBAJ&pg=PA85
 | page = 85
 | isbn = 978-1-61444-216-5
 | volume = 50
}}</ref>

<ref name=aw>{{cite book
 | last1 = Aldous | first1 = Joan
 | last2 = Wilson | first2 = Robin
 | year = 2000
 | title = Graphs and Applications: An Introductory Approach
 | url = https://books.google.com/books?id=1qRvTI_oWUAC&pg=PA382
 | publisher = Springer
 | isbn = 978-1-85233-259-4
}}</ref>

<ref name=ball>{{cite book
 | last = Ball | first = Keith
 | year = 2010
 | editor-last = Gowers | editor-first = Timothy | editor-link = Timothy Gowers
 | contribution = High-dimensional geometry and its probabilistic analogues
 | title = The Princeton Companion to Mathematics | title-link = The Princeton Companion to Mathematics
 | publisher = Princeton University Press
 | isbn = 9781400830398
 | page = [https://books.google.com/books?id=ZOfUsvemJDMC&pg=PA671 671]
}}</ref>

<ref name=barnes>{{cite book
 | last = Barnes | first = John
 | year = 2012
 | title = Gems of Geometry
 | edition = 2nd
 | url = https://books.google.com/books?id=7YCUBUd-4BQC&pg=PA82
 | page = 82
 | publisher = Springer
 | isbn = 978-3-642-30964-9
 | doi = 10.1007/978-3-642-30964-9
}}</ref>

<ref name=barrow>{{cite book|title= Impossibility: The Limits of Science and the Science of Limits|author=John D. Barrow|author-link=John D. Barrow|publisher=Oxford University Press|year=1999|isbn=9780195130829|page=14|url=https://books.google.com/books?id=0jRa1a4pD5IC&pg=PA14}}</ref>

<ref name=berman-graph>{{cite book
 | last = Berman | first = Leah
 | editor-last1 = Connelly | editor-first1 = Robert
 | editor-last2 = Weiss | editor-first2 = Asia
 | editor-last3 = Whiteley | editor-first3 = Walter
 | title = Rigidity and Symmetry
 | series = Fields Institute Communications
 | contribution = Geometric Constructions for Symmetric 6-Configurations
 | contribution-url = https://books.google.com/books?id=_n4eBAAAQBAJ&pg=PA84
 | page = 84
 | year = 2014
 | volume = 70
 | publisher = Springer
 | doi = 10.1007/978-1-4939-0781-6
 | isbn = 978-1-4939-0781-6
}}</ref>

<ref name=biegasiewicz>{{cite journal| last1 = Biegasiewicz | first1 = Kyle | last2 = Griffiths | first2 = Justin | last3 = Savage | first3 = G. Paul | last4 = Tsanakstidis | first4 = John | last5 = Priefer | first5 = Ronny | year = 2015 | title = Cubane: 50 years later | journal = Chemical Reviews| volume = 115 | issue = 14 | pages = 6719–6745 | doi=10.1021/cr500523x | pmid=26102302}}</ref>

<ref name=calter>{{cite book
 | last1 = Calter | first1 = Paul
 | last2 = Calter | first2 = Michael
 | year = 2011
 | title = Technical Mathematics
 | url = https://books.google.com/books?id=4fHwTZK3JEIC&pg=PA197
 | page = 197
 | publisher = [[John Wiley & Sons]]
| isbn = 978-0-470-53492-2
 }}</ref>

<ref name=cc>{{cite journal
 | last1 = Chilton | first1 = B. L.
 | last2 = Coxeter | first2 = H. S. M. | author2-link = Harold Scott MacDonald Coxeter
 | doi = 10.1080/00029890.1963.11992147
 | journal = [[The American Mathematical Monthly]]
 | jstor = 2313051
 | mr = 157282
 | pages = 946–951
 | title = Polar zonohedra
 | volume = 70
 | year = 1963
| issue = 9
 }}</ref>

<ref name=cmsi>{{cite conference
 | last1 = Chin | first1 = Daniel Jie Yuan Chin
 | last2 = Mohamed | first2 = Ahmad Sufril Azlan
 | last3 = Shariff | first3 = Khairul Anuar
 | last4 = Ishikawa | first4 = Kunio
 | editor-last1 = Zaman | editor-first1 = Halimah Badioze
 | editor-last2 = Smeaton | editor-first2 = Alan
 | editor-last3 = Shih | editor-first3 = Timothy
 | editor-last4 = Velastin | editor-first4 = Sergio
 | editor-last5 = Terutoshi | editor-first5 = Tada
 | editor-last6 = Jørgensen | editor-first6 = Bo Nørregaard
 | editor-last7 = Aris | editor-first7 = Hazleen Aris
 | editor-last8 = Ibrahim | editor-first8 = Nazrita Ibrahim
 | date = 23–25 November 2021
 | conference = 7th International Visual Informatics Conference
 | title = Advances in Visual Informatics
 | location = [[Kajang, Malaysia]]
 | contribution = GPU-Accelerated Enhanced Marching Cubes 33 for Fast 3D Reconstruction of Large Bone Defect CT Images
 | contribution-url = https://books.google.com/books?id=GclOEAAAQBAJ&pg=PA376
 | page = 376
}}</ref>

<ref name=coxeter>{{cite book
 | last = Coxeter | first = H.S.M. | author-link = Harold Scott MacDonald Coxeter
 | year = 1973
 | title = Regular Polytopes | title-link = Regular Polytopes (book)
 | publisher = [[Dover Publications]]
 | place = New York
 | edition = 3rd
 | pages = [http://books.google.com/books?id=2ee7AQAAQBAJ&pg=PA122 122–123]
}} See §1.8 Configurations.</ref>

<ref name=cp>{{cite journal
 | last1 = Cunningham | first1 = Gabe
 | last2 = Pellicer | first2 = Daniel
 | year = 2024
 | title = Finite 3-orbit polyhedra in ordinary space, II
 | journal = Boletín de la Sociedad Matemática Mexicana
 | volume = 30 | issue = 32
 | doi = 10.1007/s40590-024-00600-z
| doi-access = free
 }} See p. 276.</ref>

<ref name=cr>{{cite book
 | last1 = Cundy | first1 = H. Martyn | author1-link = Martyn Cundy
 | last2 = Rollett | first2 = A.P.
 | edition = 2nd
 | location = Oxford
 | mr = 0124167
 | publisher = Clarendon Press
 | title = Mathematical models
 | title-link = Mathematical Models (Cundy and Rollett)
 | year = 1961
 | contribution = 3.2 Duality
 | pages = 78–79
}}</ref>

<ref name=cromwell>{{cite book
 | last = Cromwell | first = Peter R.
 | year = 1997
 | title = Polyhedra
 | publisher = Cambridge University Press
 | url = https://archive.org/details/polyhedra0000crom/page/309
 | page = 309
 | isbn = 978-0-521-55432-9
}}</ref>

<ref name=cundy>{{cite journal
 | last = Cundy | first = H. Martyn
 | year = 1956
 | title = 2642. Unitary Construction of Certain Polyhedra
 | journal = [[The Mathematical Gazette]]
 | volume = 40 | issue = 234 | pages = 280&ndash;282
 | jstor = 3609622
 | doi = 10.2307/3609622
}}</ref>

<ref name=cz>{{cite book
 | last1 = Chartrand | first1 = Gary
 | last2 = Zhang | first2 = Ping
 | year = 2012
 | title = A First Course in Graph Theory
 | publisher = [[Dover Publications]]
 | url = https://books.google.com/books?id=zA_CAgAAQBAJ&pg=PA25
 | page = 25
| isbn = 978-0-486-29730-9
 }}</ref>

<ref name=erdahl>{{cite journal
 | last = Erdahl | first = R. M.
 | doi = 10.1006/eujc.1999.0294
 | issue = 6
 | journal = European Journal of Combinatorics
 | mr = 1703597
 | pages = 527–549
 | title = Zonotopes, dicings, and Voronoi's conjecture on parallelohedra
 | volume = 20
 | year = 1999| doi-access = free
 }}. Voronoi conjectured that all tilings of higher-dimensional spaces by translates of a single [[convex polytope]] are combinatorially equivalent to Voronoi tilings, and Erdahl proves this in the special case of [[zonotope]]s. But as he writes (p.&nbsp;429), Voronoi's conjecture for dimensions at most four was already proven by Delaunay. For the classification of three-dimensional parallelohedra into these five types, see {{cite journal
 | last1 = Grünbaum | first1 = Branko | author1-link = Branko Grünbaum
 | last2 = Shephard | first2 = G. C. | author2-link = Geoffrey Colin Shephard
 | doi = 10.1090/S0273-0979-1980-14827-2
 | issue = 3
 | journal = [[Bulletin of the American Mathematical Society]]
 | mr = 585178
 | pages = 951–973
 | series = New Series
 | title = Tilings with congruent tiles
 | volume = 3
 | year = 1980| doi-access = free
}}</ref>

<ref name=erickson>{{cite book
 | last = Erickson | first = Martin
 | year = 2011
 | title = Beautiful Mathematics
 | publisher = [[Mathematical Association of America]]
 | url = https://books.google.com/books?id=LgeP62-ZxikC&pg=PA62
 | page = 62
 | isbn = 978-1-61444-509-8
}}</ref>

<ref name=fowler>{{cite journal|title=Mathematics in Science Fiction: Mathematics as Science Fiction|first=David|last=Fowler|journal=World Literature Today|volume=84|issue=3|year=2010|pages=48–52|doi=10.1353/wlt.2010.0188 |jstor=27871086|s2cid=115769478 |quote=Robert Heinlein's "And He Built a Crooked House," published in 1940, and Martin Gardner's "The No-Sided Professor," published in 1946, are among the first in science fiction to introduce readers to the Moebius band, the Klein bottle, and the hypercube (tesseract).}}</ref>

<ref name=french>{{cite journal
 | last = French | first = Doug
 | year = 1988
 | title = Reflections on a Cube
 | journal = Mathematics in School
 | volume = 17 | issue = 4 | pages = 30&ndash;33
 | jstor = 30214515
}}</ref>

<ref name=fuchs>{{cite journal
 | last1 = Fuchs | first1 = Dmitry
 | last2 = Fuchs | first2 = Ekaterina
 | journal = Moscow Mathematical Journal
 | volume = 7 | issue = 2 | year = 2007 | pages = 265–279
 | title = Closed Geodesics on Regular Polyhedra
 | doi = 10.17323/1609-4514-2007-7-2-265-279
| doi-broken-date = 9 June 2025
 }}</ref>

<ref name=fuller>{{cite book
 | last = Fuller | first = Buckimster
 | title = Synergetics: Explorations in the Geometry of Thinking
 | year = 1975
 | publisher = MacMillan Publishing
 | url = https://books.google.com/books?id=AKDgDQAAQBAJ&pg=PA173
 | page = 173
| isbn = 978-0-02-065320-2
 }}</ref>

<ref name=greene>{{cite journal | last = Greene | first = N | year = 1986 | title = Environment mapping and other applications of world projections | journal = IEEE Computer Graphics and Applications | volume = 6 | issue = 11| pages = 21–29 | doi = 10.1109/MCG.1986.276658 | s2cid = 11301955 }}</ref>

<ref name=gruber>{{cite book
 | last = Gruber | first = Peter M.
 | contribution = Chapter 16: Volume of Polytopes and Hilbert's Third Problem
 | doi = 10.1007/978-3-540-71133-9
 | isbn = 978-3-540-71132-2
 | mr = 2335496
 | pages = 280–291
 | publisher = Springer, Berlin
 | series = Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]
 | title = Convex and Discrete Geometry
 | volume = 336
 | year = 2007
}}</ref>

<ref name=grunbaum-1997>{{cite journal
 | last = Grünbaum | first = Branko | authorlink = Branko Grünbaum 
 | year = 1997
 | title = Isogonal Prismatoids
 | journal = Discrete & Computational Geometry
 | volume = 18 | issue = 1 | pages = 13&ndash;52
 | doi = 10.1007/PL00009307
}}</ref>

<ref name=grunbaum-2003>{{cite book
 | last = Grünbaum | first = Branko | author-link = Branko Grünbaum
 | contribution = 13.1 Steinitz's theorem
 | edition = 2nd
 | isbn = 0-387-40409-0
 | pages = 235–244
 | publisher = Springer-Verlag
 | series = [[Graduate Texts in Mathematics]]
 | title = Convex Polytopes
 | title-link = Convex Polytopes
 | volume = 221
 | year = 2003
}}</ref>

<ref name=haeckel>{{cite book
 | last = Haeckel | first = E. | author-link = Ernst Haeckel
 | year = 1904
 | title = [[Kunstformen der Natur]]
 | language = de
}} See [http://www.biolib.de/haeckel/kunstformen/index.html here] for an online book.</ref>

<ref name=hall>{{cite journal
 | last = Hall | first = T. Proctor | authorlink = T. Proctor Hall
 | year = 1893
 | jstor = 2369565
 | title = The projection of fourfold figures on a three-flat
 | journal = [[American Journal of Mathematics]]
 | volume = 15
 | issue = 2 | pages = 179–189
| doi = 10.2307/2369565 }}</ref>

<ref name=hart>{{cite conference
 | last = Hart | first = George
 | editor-last1 = Goldstein | editor-first1 = Susan
 | editor-last2 = McKenna | editor-first2 = Douglas
 | editor-last3 = Fenyvesi | editor-first3 = Kristóf
 | date = 16–20 July 2019
 | location = [[Linz, Austria]] 
 | title = Bridges 2019 Conference Proceedings
 | contribution = Max Brücknerʼs Wunderkammer of Paper Polyhedra
 | contribution-url = https://archive.bridgesmathart.org/2019/bridges2019-59.pdf
 | isbn =  978-1-938664-30-4
 | publisher = Tessellations Publishing, [[Phoenix, Arizona]]
}}</ref>

<ref name=heath>{{cite book
 | last = Heath | first = Thomas L. | authorlink = Thomas Little Heath
 | year = 1908
 | title = The Thirteen Books of Euclid's Elements
 | edition = 3rd
 | publisher = [[Cambridge University Press]]
 | url = https://archive.org/details/thirteenbookseu03heibgoog/page/480
 | pages = 262, 478, 480
}}</ref>

<ref name=helvajian>{{cite book|editor1-last= Helvajian|editor1-first= Henry|title= Small Satellites: Past, Present, and Future|year= 2008|publisher= Aerospace Press|location= El Segundo, Calif.|isbn= 978-1-884989-22-3|editor2-first= Siegfried W.|editor2-last= Janson|page=159}}</ref>

<ref name=hh>{{cite journal
 | last1 = Harary | first1 = F. | authorlink1 = Frank Harary
 | last2 = Hayes | first2 = J. P. | last3 = Wu | first3 = H.-J.
 | title = A survey of the theory of hypercube graphs
 | journal = Computers & Mathematics with Applications
 | volume = 15
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 | year = 1988
 | doi = 10.1016/0898-1221(88)90213-1| hdl = 2027.42/27522
 | hdl-access = free
}}</ref>

<ref name=hhlnqr>{{cite arXiv
 | last1 = Hasan | first1 = Masud 
 | last2 = Hossain | first2 = Mohammad M.
 | last3 = López-Ortiz | first3 = Alejandro
 | last4 = Nusrat | first4 = Sabrina
 | last5 = Quader | first5 = Saad A.
 | last6 = Rahman | first6 = Nabila
 | title = Some New Equiprojective Polyhedra
 | date = 2010 
 | class = cs.CG 
 | eprint = 1009.2252 
 }}</ref>

<ref name=hoffmann>{{cite book
 | last = Hoffmann | first = Frank
 | year = 2020
 | title = Introduction to Crystallography
 | url = https://books.google.com/books?id=16H0DwAAQBAJ&pg=PA35
 | page = 35
 | publisher = Springer
 | doi = 10.1007/978-3-030-35110-6
 | isbn = 978-3-030-35110-6
}}</ref>

<ref name=holme>{{cite book
 | last = Holme | first = A.
 | year = 2010
 | title = Geometry: Our Cultural Heritage
 | publisher = [[Springer Science+Business Media|Springer]]
 | url = https://books.google.com/books?id=zXwQGo8jyHUC
 | isbn = 978-3-642-14441-7
 | doi = 10.1007/978-3-642-14441-7
}}</ref>

<ref name=hp>{{cite journal
 | last1 = Horvat | first1 = Boris
 | last2 = Pisanski | first2 = Tomaž | author2-link = Tomaž Pisanski
 | doi = 10.1016/j.disc.2009.11.035 | doi-access = free
 | issue = 12
 | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
 | mr = 2610282
 | pages = 1783–1792
 | title = Products of unit distance graphs
 | volume = 310
 | year = 2010
}}</ref>

<ref name=hr-w>{{cite book
 | title = Geometry: Reteaching Masters
 | publisher = Holt Rinehart & Winston
 | isbn = 9780030543289
 | year = 2001
 | page = 74
}}</ref>

<ref name=hs>{{cite book
 | last1 = Herrmann | first1 = Diane L.
 | last2 = Sally | first2 = Paul J.
 | year = 2013
 | title = Number, Shape, & Symmetry: An Introduction to Number Theory, Geometry, and Group Theory
 | publisher = Taylor & Francis
 | isbn = 978-1-4665-5464-1
 | url = https://books.google.com/books?id=b2fjR81h6yEC&pg=PA252
 | page = 252
}}</ref>

<ref name="hut">{{cite arXiv
 | last1 = Diaz | first1 = Giovanna
 | last2 = O'Rourke | first2 = Joseph | author2-link = Joseph O'Rourke (professor)
 | title = Hypercube unfoldings that tile <math>\mathbb{R}^3</math> and <math>\mathbb{R}^2</math>
 | year = 2015
 | class = cs.CG
 | eprint = 1512.02086
}}</ref>

<ref name=inchbald>{{cite journal
 | last = Inchbald | first = Guy
 | year = 2006
 | title = Facetting Diagrams
 | journal = [[The Mathematical Gazette]]
 | volume = 90 | issue = 518 | pages = 253&ndash;261
 | doi = 10.1017/S0025557200179653
 | jstor = 40378613
}}</ref>

<ref name=jessen>{{cite journal
 | last = Jessen | first = Børge | author-link = Børge Jessen
 | issue = 2
 | journal = Nordisk Matematisk Tidskrift
 | jstor = 24524998
 | mr = 0226494
 | pages = 90–96
 | title = Orthogonal icosahedra
 | volume = 15
 | year = 1967
}}</ref>

<ref name=jeon>{{cite journal
 | last = Jeon | first = Kyungsoon
 | year = 2009
 | title = Mathematics Hiding in the Nets for a CUBE
 | journal = Teaching Children Mathematics
 | volume = 15 | issue = 7 | pages = 394&ndash;399
 | doi = 10.5951/TCM.15.7.0394
 | jstor = 41199313
}}</ref>

<ref name=johnson>{{cite journal
 | last = Johnson | first = Norman W. | authorlink = Norman W. Johnson
 | year = 1966
 | title = Convex polyhedra with regular faces
 | journal = [[Canadian Journal of Mathematics]]
 | volume = 18
 | pages = 169–200
 | doi = 10.4153/cjm-1966-021-8
 | mr = 0185507
 | s2cid = 122006114
 | zbl = 0132.14603| doi-access = free
}} See table II, line 3.</ref>

<ref name=joyner>{{cite book
 | last = Joyner | first = David
 | year = 2008
 | title = Adventures in Group Theory: Rubik's Cube, Merlin's Machine, and Other Mathematical Toys
 | url = https://books.google.com/books?id=6an_DwAAQBAJ&pg=PA76
 | page = 76
 | edition = 2nd
 | isbn = 978-0-8018-9012-3
 | publisher = The Johns Hopkins University Press
}}</ref>

<ref name=jwy>{{cite journal
 | last1 = Jerrard | first1 = Richard P.
 | last2 = Wetzel | first2 = John E.
 | last3 = Yuan | first3 = Liping
 | title = Platonic passages
 | journal = [[Mathematics Magazine]]
 | date = April 2017
 | volume = 90 | issue = 2 | pages = 87–98
 | publisher = [[Mathematical Association of America]]
 | location = Washington, DC
 | doi = 10.4169/math.mag.90.2.87
 | s2cid = 218542147
}}</ref>

<ref name=kane>{{cite book
 | last = Kane | first = Richard
 | year = 2001
 | title = Reflection Groups and Invariant Theory
 | url = https://books.google.com/books?id=KmL1uuiMyFUC&pg=PA16
 | page = 16
| publisher = Springer
 | isbn = 978-0-387-98979-2
 }}</ref>

<ref name=kemp>{{cite journal|title=Dali's dimensions|first=Martin|last=Kemp|journal=[[Nature (journal)|Nature]]|volume=391|issue=27|date=1 January 1998|page=27|doi=10.1038/34063|bibcode=1998Natur.391...27K|doi-access=free}}</ref>

<ref name=khattar>{{cite book
 | last = Khattar | first = Dinesh
 | year = 2008
 | title = Guide to Objective Arithmetic
 | url = https://books.google.com/books?id=BoBCOYuOlzkC&pg=PA377
 | page = 377
 | publisher = [[Pearson Education]]
 | edition = 2nd
 | isbn = 978-81-317-1682-3
}}</ref>

<ref name=kl>{{cite book
 | last1 = Kitaev | first1 = Sergey
 | last2 = Lozin | first2 = Vadim
 | title = Words and Graphs
 | url = https://books.google.com/books?id=EXH_CgAAQBAJ&pg=PA171
 | page = 171
 | year = 2015 
 | doi = 10.1007/978-3-319-25859-1
 | isbn = 978-3-319-25859-1
}}</ref>

<ref name=knstv>{{cite journal
 | last1 = Kov´acs | first1 = Gergely
 | last2 = Nagy | first2 = Benedek Nagy
 | last3 = Stomfai | first3 = Gergely
 | last4 = Turgay | first4 = Nes¸et Deni̇z
 | last5 = Vizv´ari | first5 = B´ela
 | year = 2021
 | title = On Chamfer Distances on the Square and Body-Centered CubicGrids: An Operational Research Approach
 | doi = 10.1155/2021/5582034
 | journal = Mathematical Problems in Engineering
 | pages = 1&ndash;9
 | doi-access = free
| hdl = 10831/111490
 | hdl-access = free
 }}</ref>

<ref name=kozachok>{{cite conference
 | last = Kozachok | first = Marina
 | contribution = Perfect prismatoids and the conjecture concerning with face numbers of centrally symmetric polytopes
 | pages = 46–49
 | publisher = P.G. Demidov Yaroslavl State University, International B.N. Delaunay Laboratory
 | title = Yaroslavl International Conference "Discrete Geometry" dedicated to the centenary of A.D.Alexandrov (Yaroslavl, August 13-18, 2012)
 | url = https://www.dcglab.uniyar.ac.ru/sites/default/files/papers/Alexandrov2012Thesis.pdf#page=46
 | year = 2012
}}</ref>

<ref name=linti>{{cite book
 | last = Linti | first = G.
 | editor-last1 = Reedijk | editor-first1 = J.
 | editor-last2 = Poeppelmmeier | editor-first2 = K.
 | year = 2013
 | title = Comprehensive Inorganic Chemistry II: From Elements to Applications
 | contribution = Catenated Compounds - Group 13 [Al, Ga, In, Tl]
 | publisher = Newnes
 | url = https://books.google.com/books?id=_4C7oid1kQQC&pg=RA7-PA41
 | page = 41
| isbn = 978-0-08-096529-1
 }}</ref>

<ref name=livio>{{cite book
 | last = Livio | first = Mario | author-link = Mario Livio
 | title = The Golden Ratio: The Story of Phi, the World's Most Astonishing Number
 | url = https://books.google.com/books?id=bUARfgWRH14C
 | orig-year = 2002
 | edition = 1st trade paperback
 | year = 2003
 | publisher = [[Random House|Broadway Books]]
 | location = New York City
 | isbn = 978-0-7679-0816-0
 | page = 147
}}</ref>

<ref name=lm>{{cite journal
 | last1 = Lagarias | first1 = J. C. | author1-link = Jeffrey Lagarias
 | last2 = Moews | first2 = D.
 | doi = 10.1007/BF02574064
 | issue = 3–4
 | journal = [[Discrete & Computational Geometry]]
 | mr = 1318797
 | pages = 573–583
 | title = Polytopes that fill <math>\mathbb{R}^n</math> and scissors congruence
 | volume = 13
 | year = 1995| doi-access = free
}}</ref>

<ref name=ls>{{cite journal
 | last1 = Langer | first1 = Joel C.
 | last2 = Singer | first2 = David A.
 | journal = [[Milan Journal of Mathematics]]
 | title = Reflections on the Lemniscate of Bernoulli: The Forty-Eight Faces of a Mathematical Gem
 | volume = 78 | pages = 643–682 | year = 2010
 | issue = 2
 | doi = 10.1007/s00032-010-0124-5
}}</ref>

<ref name=lunnon>{{cite book
 | last = Lunnon | first = W. F.
 | contribution = Symmetry of Cubical and General Polyominoes
 | editor-last = Read | editor-first = Ronald C.
 | title = Graph Theory and Computing
 | place = New York
 | publisher = [[Academic Press]]
 | year = 1972
 | pages = 101–108
 | isbn = 978-1-48325-512-5
 | url = https://books.google.com/books?id=ja7iBQAAQBAJ&pg=PA101
}}</ref>

<ref name=lutzen>{{cite journal
 | last = Lützen | first = Jesper
 | year = 2010
 | title = The Algebra of Geometric Impossibility: Descartes and Montucla on the Impossibility of the Duplication of the Cube and the Trisection of the Angle
 | url = https://onlinelibrary.wiley.com/doi/10.1111/j.1600-0498.2009.00160.x
 | journal = Centaurus
 | language = en
 | volume = 52 | issue = 1 | pages = 4–37
 | doi = 10.1111/j.1600-0498.2009.00160.x
| url-access = subscription
 }}</ref>

<ref name=ly>{{cite book
 | last1 = Gross | first1 = Jonathan L.
 | last2 = Yellen | first2 = Yellen
 | year = 2006
 | title = Graph Theory and Its Applications, Second Edition
 | url = https://books.google.com/books?id=-7Q_POGh-2cC&pg=PA273
 | page = 273
 | publisher = Taylor & Francis
| isbn = 978-1-58488-505-4
 }}</ref>

<ref name=marar>{{cite book
 | last = Marat | first = Ton
 | title = A Ludic Journey into Geometric Topology
 | year = 2022
 | url = https://books.google.com/books?id=aPGGEAAAQBAJ&pg=PA112
 | page = 112
 | publisher = Springer
 | doi = 10.1007/978-3-031-07442-4
 | isbn = 978-3-031-07442-4
}}</ref>

<ref name=march>{{cite journal
 | last = March | first = Lionel
 | title = Renaissance mathematics and architectural proportion in Alberti's De re aedificatoria
 |journal=Architectural Research Quarterly
 |date=1996
 |volume=2 |issue=1 |pages=54–65
 |doi=10.1017/S135913550000110X
 |s2cid=110346888
}}</ref>

<ref name=masalski>{{cite journal
 | last = Masalski | first = William J.
 | title = Polycubes
 | journal = The Mathematics Teacher
 | volume = 70 | issue = 1 | year = 1977 | pages = 46–50
 | doi = 10.5951/MT.70.1.0046
 | jstor = 27960702
}}</ref>

<ref name=mclean>{{cite journal
 | last = McLean | first = K. Robin
 | year = 1990
 | title = Dungeons, dragons, and dice
 | journal = [[The Mathematical Gazette]]
 | volume = 74 | issue = 469 | pages = 243–256
 | doi = 10.2307/3619822
 | jstor = 3619822
 | s2cid = 195047512
}} See p. 247.</ref>

<ref name=moore>{{cite journal
 | last = Moore | first = Kimberly
 | year = 2018
 | title = Minecraft Comes to Math Class
 | journal = Mathematics Teaching in the Middle School
 | volume = 23 | issue = 6 | pages = 334–341
 | doi = 10.5951/mathteacmiddscho.23.6.0334
 | jstor = 10.5951/mathteacmiddscho.23.6.0334
}}</ref>

<ref name=mk>{{cite book
 | last1 = Mills | first1 = Steve
 | last2 = Kolf | first2 = Hillary
 | year = 1999
 | title = Maths Dictionary
 | publisher = Heinemann
 | url = https://books.google.com/books?id=dvFfTAR6XwEC&pg=PA16
 | isbn = 978-0-435-02474-1
 | page = 16
}}</ref>

<ref name=ns>{{cite journal
 | last1 = Nelson | first1 = Roice
 | last2 = Segerman | first2 = Henry
 | title = Visualizing hyperbolic honeycombs
 | journal = Journal of Mathematics and the Arts
 | year = 2017
 | volume = 11 | issue = 1 | pages = 4–39
 | doi = 10.1080/17513472.2016.1263789
| arxiv = 1511.02851
 }}</ref>

<ref name=padmanabhan>{{cite book |title=Sleeping Beauties in Theoretical Physics |last=Padmanabhan |first=Thanu |chapter=The Grand Cube of Theoretical Physics |publisher=Springer |year=2015 |pages=1–8 |isbn=978-3319134420 }}</ref>

<ref name=poo-sung>{{cite journal
 | last = Poo-Sung
 | first = Park, Poo-Sung
 | year = 2016
 | title = Regular polytope distances
 | journal = [[Forum Geometricorum]]
 | volume = 16
 | pages = 227–232
 | url = http://forumgeom.fau.edu/FG2016volume16/FG201627.pdf
 | archive-date = 2016-10-10
 | access-date = 2016-05-24
 | archive-url = https://web.archive.org/web/20161010184811/http://forumgeom.fau.edu/FG2016volume16/FG201627.pdf
 | url-status = dead
 }}</ref>

<ref name=popko>{{cite book|title=Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere|first=Edward S.|last=Popko|publisher=CRC Press|year=2012|isbn=9781466504295|pages=100–101|url=https://books.google.com/books?id=WLAFlr1_2S4C&pg=PA100}}</ref>

<ref name=ps>{{cite book
 | last1 = Pisanski | first1 = Tomaž
 | last2 = Servatius | first2 = Brigitte
 | title = Configuration from a Graphical Viewpoint
 | year = 2013
 | url = https://books.google.com/books?id=3vnEcMCx0HkC&pg=PA21
 | page = 21
 | publisher = Springer
 | isbn = 978-0-8176-8363-4
 | doi = 10.1007/978-0-8176-8364-1
}}</ref>

<ref name="pucc">{{cite conference
 | contribution = Polycube unfoldings satisfying Conway's criterion
 | last1 = Langerman | first1 = Stefan | author1-link = Stefan Langerman
 | last2 = Winslow | first2 = Andrew
 | title = 19th Japan Conference on Discrete and Computational Geometry, Graphs, and Games (JCDCG^3 2016)
 | year = 2016
 | contribution-url = http://andrewwinslow.com/papers/polyunfold-jcdcggg16.pdf
}}</ref>

<ref name=radii>{{harvtxt|Coxeter|1973}} Table I(i), pp. 292–293. See the columns labeled <math>{}_0\!\mathrm{R}/\ell</math>, <math>{}_1\!\mathrm{R}/\ell</math>, and <math>{}_2\!\mathrm{R}/\ell</math>, Coxeter's notation for the circumradius, midradius, and inradius, respectively, also noting that Coxeter uses <math>2\ell</math> as the edge length (see p. 2).</ref>

<ref name=rajwade>{{cite book
 | last = Rajwade | first = A. R.
 | title = Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem
 | series = Texts and Readings in Mathematics
 | year = 2001
 | url = https://books.google.com/books?id=afJdDwAAQBAJ&pg=PA84
 | publisher = Hindustan Book Agency
 | page = 84&ndash;89
 | isbn = 978-93-86279-06-4
 | doi = 10.1007/978-93-86279-06-4
}}</ref>

<ref name="richeson">{{cite book
 | last = Richeson | first = D. S. | author-link = David Richeson
 | year = 2008
 | title-link = Euler's Gem
 | title = Euler's Gem: The polyhedron formula and the birth of topology
 | publisher = Princeton University Press
 | at = [https://books.google.com/books?id=kv2EDwAAQBAJ&pg=PA2 pp. 1&ndash;2]
}}</ref>

<ref name=rudolph>{{cite book
 | last = Rudolph | first = Michael
 | year = 2022
 | title = The Mathematics of Finite Networks: An Introduction to Operator Graph Theory
 | url = https://books.google.com/books?id=NIdoEAAAQBAJ&pg=PA25
 | page = 25
 | publisher = [[Cambridge University Press]]
 | doi = 10.1007/9781316466919
 | doi-broken-date = 9 June 2025
 | isbn = 9781316466919
}}</ref>

<ref name=rz>{{cite book
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 | last2 = Zeilten | first2 = Steve
 | year = 2006
 | title = Hidden New York: A Guide to Places that Matter
 | url = https://books.google.com/books?id=xgOtV4Nynd0C&pg=PA77
 | page = 77
 | isbn = 978-0-8135-3890-7
 | publisher = Rutgers University Press
}}</ref>

<ref name=senechal>{{cite book
 | last = Senechal | first = Marjorie
 | year = 1989
 | contribution = A Brief Introduction to Tilings
 | contribution-url = https://books.google.com/books?id=OToVjZW9CKMC&pg=PA12
 | editor-last = Jarić | editor-first = Marko
 | title = Introduction to the Mathematics of Quasicrystals
 | publisher = [[Academic Press]]
 | page = 12
}}</ref>

<ref name=shephard>In higher dimensions, however, there exist parallelopes that are not zonotopes. See e.g. {{cite journal
 | last = Shephard | first = G. C.
 | doi = 10.1112/S0025579300008652
 | journal = Mathematika
 | mr = 365332
 | pages = 261–269
 | title = Space-filling zonotopes
 | volume = 21
 | year = 1974| issue = 2
 }}</ref>

<ref name=skilling>{{cite journal|first=John|last=Skilling|title=Uniform Compounds of Uniform Polyhedra|journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]]|volume=79|issue=3|pages=447–457|year=1976|doi=10.1017/S0305004100052440|bibcode=1976MPCPS..79..447S |mr=0397554}}</ref>

<ref name=smith>{{cite book
 | last = Smith | first = James
 | year = 2000
 | title = Methods of Geometry
 | url = https://books.google.com/books?id=B0khWEZmOlwC&pg=PA392
 | page = 392
 | publisher = [[John Wiley & Sons]]
| isbn = 978-1-118-03103-2
 }}</ref>

<ref name=sod>{{cite journal
 | last1 = Slobodan | first1 = Mišić
 | last2 = Obradović | first2 = Marija
 | last3 = Ðukanović | first3 = Gordana
 | title = Composite Concave Cupolae as Geometric and Architectural Forms
 | year = 2015
 | journal = Journal for Geometry and Graphics
 | volume = 19
 | issue = 1
 | pages = 79–91
 | url = https://www.heldermann-verlag.de/jgg/jgg19/j19h1misi.pdf
}}</ref>

<ref name=sriraman>{{cite book
 | last = Sriraman | first = Bharath
 | editor1-first = Bharath | editor1-last = Sriraman
 | editor2-first = Viktor | editor2-last = Freiman
 | editor3-first = Nicole | editor3-last = Lirette-Pitre
 | title = Interdisciplinarity, Creativity, and Learning: Mathematics With Literature, Paradoxes, History, Technology, and Modeling
 | volume = 7 | series = The Montana Mathematics Enthusiast: Monograph Series in Mathematics Education
 | year = 2009
 | isbn = 9781607521013
 | publisher = Information Age Publishing, Inc.
 | contribution = Mathematics and literature (the sequel): imagination as a pathway to advanced mathematical ideas and philosophy
 | pages = 41–54
}}</ref>

<ref name=thomson>{{cite book|first=James|last=Thomson|author-link=James Thomson (mathematician)|title=An Elementary Treatise on Algebra: Theoretical and Practical|year=1845|location=London|publisher=Longman, Brown, Green, and Longmans|page=4|url=https://archive.org/details/anelementarytre01thomgoog/page/n15}}</ref>

<ref name=timofeenko-2010>{{cite journal
 | last = Timofeenko | first = A. V.
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 | title = Junction of Non-composite Polyhedra
 | journal = St. Petersburg Mathematical Journal
 | volume = 21 | issue = 3 | pages = 483–512
 | doi = 10.1090/S1061-0022-10-01105-2
 | url = https://www.ams.org/journals/spmj/2010-21-03/S1061-0022-10-01105-2/S1061-0022-10-01105-2.pdf
}}</ref>

<ref name=tisza>{{cite book
 | last = Tisza | first = Miklós
 | year = 2001
 | title = Physical Metallurgy for Engineers
 | url = https://books.google.com/books?id=y1eTDQRdI2wC&pg=PA45
 | page = 45
 | publisher = [[ASM International (society)|ASM International]]
 | location = [[Materials Park, Ohio]]
 | isbn = 978-1-61503-241-9
}}</ref>

<ref name=trudeau>{{cite book
 | last = Trudeau | first = Richard J.
 | title = Introduction to Graph Theory
 | year = 1976
 | isbn = 978-0-486-67870-2
 | publisher = Dover Publications
 | url = https://books.google.com/books?id=eRLEAgAAQBAJ&pg=PA122
 | page = 122
}}</ref>

<ref name=twelveessay>{{cite book
 | last = Coxeter | first = H. S. M. | authorlink = Harold Scott MacDonald Coxeter
 | year = 1968
 | title = The Beauty of Geometry: Twelve Essays
 | url = https://books.google.com/books?id=p4o-Uf-i-IUC&pg=PA212
 | page = 167
 | publisher = [[Dover Publications]]
 | isbn = 978-0-486-40919-1
}} See table III.</ref>

<ref name=vm>{{cite book |title=Infrared Thermal Imaging: Fundamentals, Research and Applications |first1=Michael |last1=Vollmer |first2=Klaus-Peter |last2=Möllmann |publisher=[[John Wiley & Sons]] |date=2011 |isbn=9783527641550 |pages=36–38 |url=https://books.google.com/books?id=b-MqbyPwAuoC&pg=PA36}}</ref>

<ref name=vxac>{{cite book
 | last1 = Viana | first1 = Vera
 | last2 = Xavier | first2 = João Pedro
 | last3 = Aires | first3 = Ana Paula
 | last4 = Campos | first4 = Helena
 | contribution = Interactive Expansion of Achiral Polyhedra
 | editor-last = Cocchiarella | editor-first = Luigi
 | year = 2019
 | title = ICGG 2018 - Proceedings of the 18th International Conference on Geometry and Graphics 40th Anniversary - Milan, Italy, August 3-7, 2018
 | series = Advances in Intelligent Systems and Computing
 | volume = 809
 | publisher = Springer
 | doi = 10.1007/978-3-319-95588-9
 | page = 1123
| isbn = 978-3-319-95587-2
 }} See Fig. 6.</ref>

<ref name=wd>{{cite book
 | last1 = Walter | first1 = Steurer
 | last2 = Deloudi | first2 = Sofia
 | year = 2009
 | title = Crystallography of Quasicrystals: Concepts, Methods and Structures
 | series = Springer Series in Materials Science
 | volume = 126
 | url = https://books.google.com/books?id=nVx-tu596twC&pg=PA50
 | page = 50
 | isbn = 978-3-642-01898-5
 | doi = 10.1007/978-3-642-01899-2
}}</ref>

<ref name=yackel>{{cite conference
 | last = Yackel | first = Carolyn
 | title = Proceedings of Bridges 2009: Mathematics, Music, Art, Architecture, Culture
 | editor-last1 = Kaplan | editor-first1 = Craig | editor-link1 = Craig S. Kaplan
 | editor-last2 = Sarhangi | editor-first2 = Reza
 | location = [[Banff, Alberta]], Canada
 | date = 26–30 July 2013
 | contribution = Marking a Physical Sphere with a Projected Platonic Solid
 | contribution-url = https://archive.bridgesmathart.org/2009/bridges2009-123.pdf
 | pages = 123&ndash;130
 | isbn = 978-0-96652-020-0
}}</ref>

<ref name=zeeman>{{cite journal
 | last = Zeeman | first = E. C. | author-link = Christopher Zeeman
 | date = July 2002
 | doi = 10.2307/3621846
 | issue = 506
 | journal = [[The Mathematical Gazette]]
 | jstor = 3621846
 | pages = 241–247
 | title = On Hilbert's third problem
 | volume = 86
}}</ref>

<ref name=ziegler>{{cite book
 | last = Ziegler | first = Günter M. | author-link = Günter M. Ziegler
 | contribution = Chapter 4: Steinitz' Theorem for 3-Polytopes
 | isbn = 0-387-94365-X
 | pages = 103–126
 | publisher = Springer-Verlag
 | series = [[Graduate Texts in Mathematics]]
 | title = Lectures on Polytopes
 | volume = 152
 | year = 1995
}}</ref>

}}

==External links==
{{General geometry}}
*{{mathworld |urlname=Cube |title=Cube}}
*[https://web.archive.org/web/20071009235233/http://polyhedra.org/poly/show/1/cube Cube: Interactive Polyhedron Model]*
*[http://www.mathopenref.com/cubevolume.html Volume of a cube], with interactive animation
*[http://www.software3d.com/Cube.php Cube] (Robert Webb's site)
{{Convex polyhedron navigator|state=collapsed}}
{{Authority control}}

[[Category:Cubes| ]]
[[Category:Cuboids]]
[[Category:Elementary shapes]]
[[Category:Platonic solids]]
[[Category:Prismatoid polyhedra]]
[[Category:Space-filling polyhedra]]
[[Category:Zonohedra]]