View source for Cube

{{Short description|Solid with six equal square faces}}
{{good article}}
{{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 [[square]]
 | 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>[[Rupert property]]: can pass through a hole with its copy,<br>[[vertex-transitive]]
 | surface area = 6 × side<sup>2</sup>
 | volume = side<sup>3</sup>
}}
A '''cube''' is a [[three-dimensional space|three-dimensional]] solid object in [[geometry]]. A cube has eight vertices and twelve straight edges of the same length, so that these edges form six [[square (geometry)|square]] faces of the same size. It is an example of a [[polyhedron]].

The cube is found in many popular cultures, including toys and games, the arts, optical illusions, and architectural buildings. Cubes can be found in crystal structures, science, and technological devices. It is also found in ancient texts, such as [[Plato]]'s work [[Timaeus (dialogue)|''Timaeus'']], which described a set of solids now called [[Platonic solid]]s, associating a cube with the [[classical element]] of [[Earth (classical element)|earth]]. A [[Unit cube|cube with unit length]] is the canonical unit of [[volume]] in three-dimensional space, relative to which other solid objects are measured.

The cube relates to the construction of many polyhedra, such as [[Truncation (geometry)|truncation]] and attaching to other polyhedra. It also represents geometrical shapes. The cube can be attached to its faces with its copy to [[Space-filling polyhedron|fill a space without leaving a gap]], which forms a [[Honeycomb (geometry)|honeycomb]].

The cube can be represented in many ways. One example is by drawing a [[Graph (discrete mathematics)|graph]], a structure in [[graph theory]] consisting of a set of [[Vertex (graph theory)|vertices]] that are connected with an [[Edge (graph theory)|edge]]. This graph also represents the family of a [[cuboid]], a polyhedron with six quadrilateral faces, which includes the cube as its special case. The cube and its graph are a three-dimensional [[hypercube]], a family of [[polytope]]s that also includes the two-dimensional square and four-dimensional [[tesseract]].

== Properties ==
[[File:Hexahedron.stl|thumb|3D model of a cube|alt=Cube modeled in 3D, where one can click and drag to rotate]]
A cube is a [[polyhedron]] with eight vertices and twelve equal-length edges, forming six [[square]]s as its faces. A cube is a special case of a [[rectangular cuboid]], which has six [[Rectangle|rectangular]] faces, each of which has a pair of opposite equal-length and parallel edges.{{r|dupuis}} Both polyhedra have the same [[dihedral angle]], the angle between two adjacent faces at a common edge, a [[right angle]] or 90°, obtained from the [[interior angle]] (an angle formed between two adjacent sides at a common point of a [[polygon]] within) of a square.{{r|johnson|bird}} More generally, the cube and the rectangular cuboid are special cases of a [[cuboid]], a polyhedron with six [[quadrilateral]]s (four-sided polygons).{{r|mk}} As for all convex polyhedra, the cube has [[Euler characteristic]] of 2, according to the formula <math> V - E + F = 2 </math>; the three letters denote respectively the number of vertices, edges, and faces.{{r|richeson}}

All three square faces surrounding a vertex are [[orthogonality|orthogonal]] to each other, meaning the planes are perpendicular, forming a right angle between two adjacent squares. Hence, the cube is classified as an [[orthogonal polyhedron]].{{r|jessen}} The cube is a special case of other cuboids. These include a [[parallelepiped]], a polyhedron with six [[parallelogram]]s faces, because its pairs of opposite faces are congruent;{{r|calter}} a [[rhombohedron]], as a special case of a parallelepiped with six [[rhombi]] faces, because the interior angle of all of the faces is right;{{sfnp|Hoffmann|2020|p=[http://books.google.com/books?id=16H0DwAAQBAJ&pg=PA83 83]}} and a [[trigonal trapezohedron]], a polyhedron with congruent quadrilateral faces, since its square faces are the special cases of rhombi.{{r|cc}}

The cube is a [[non-composite polyhedron|non-composite or an elementary polyhedron]]. That is, no plane intersecting its surface only along edges, thereby cutting into two or more convex, regular-faced polyhedra.{{r|timofeenko-2010}}

=== Measurement ===
[[File:Cube diagonals.svg|thumb|upright=0.8|A face diagonal is denoted as <math> AC </math> and a space diagonal is <math> AC' </math>|alt=A space diagonal AC' and a face diagonal AC]]
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 are not in the same face, formulated as {{nowrap|1=<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 rectangular 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.{{r|khattar}} This leads to the use of the term ''[[Cube (algebra)|cube]]'' as a [[verb]], to mean raising any number to the third power:{{r|mk}}
<math display="block"> V = a^3. </math>

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 congruent to 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.{{r|fuchs}}

=== Insphere, midsphere, circumsphere ===
An [[inscribed sphere|insphere]] of a cube <math> r_i </math> is a sphere tangent to the faces of a cube at their [[centroid]]s. Its [[midsphere]] <math> r_m </math> is a sphere tangent to the edges of a cube. Its [[circumscribed sphere|circumsphere]] <math> r_c </math> is a sphere tangent to the vertices of a cube. With edge length <math> a </math>, they are respectively:{{r|radii}}
<math display="block"> r_i = \frac{1}{2}a = 0.5a, \qquad r_m = \frac{\sqrt{2}}{2}a \approx 0.707a, \qquad r_c = \frac{\sqrt{3}}{2}a \approx 0.866a. </math>

=== Unit cube ===
{{main article|Unit cube|Prince Rupert's cube|Doubling the cube}}
[[File:Prince Ruperts cube.png|thumb|upright=0.8|[[Prince Rupert's cube]]|alt=Cube with a hole through which an equal cube can pass]]
A [[unit cube]] is a cube with [[unit of length|1 unit in 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 of the Rhine]], known for [[Prince Rupert's drop]], wagered whether [[Prince Rupert's cube|a cube could be passed through a hole made in another cube of the same size]]. The story recounted in 1693 by English mathematician [[John Wallis]] answered that it is possible, although there were some errors in Wallis's presentation. Roughly a century later, Dutch mathematician [[Pieter Nieuwland]] provided a better solution that the edges of a cube passing through the unit cube's hole could be as large as approximately 1.06 units in length.{{r|jw|rickey}} One way to obtain this result is by using the Pythagorean theorem or the formula for [[Euclidean distance]] in three-dimensional space.{{r|gardner}}

An ancient problem of [[doubling the cube]] requires the construction of a cube with a volume twice the original by using only a [[compass and straightedge]]. This was concluded by French mathematician [[Pierre Wantzel]] in 1837, proving that it is impossible to implement since a cube with twice the volume of the original&mdash;the [[cube root]] of 2, <math> \sqrt[3]{2} </math>&mdash;is not [[constructible number|constructible]].{{r|lutzen}} However, this problem was solved with [[Mathematics of paper folding#Doubling the cube|folding an origami paper]] by {{harvtxt|Messer|1986}}.{{r|messer86}}

=== Symmetry ===
[[File:Dual Cube-Octahedron.svg|thumb|upright=0.8|The dual polyhedron of a cube is the regular octahedron. Both have [[octahedral symmetry]].|alt=Cube, the dual of a regular octahedron]]
The cube has [[octahedral symmetry]] <math> \mathrm{O}_\mathrm{h} </math> of order 48. In other words, the cube has 48 [[isometry|isometries]] (including [[Identity element|identity]]), each of which transforms the cube to itself. These transformations include nine [[reflection symmetries]] (where two halves cut by a plane are identical): three cut the cube at the midpoints of its edges, and six cut diagonally. The cube also has thirteen axes of [[rotational symmetry]] (whereby rotation around the axis results in an identical appearance): three axes pass through the centroids of opposite faces, six through the midpoints of opposite edges, and four through 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|kane}}

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 have the same octahedral symmetry.{{r|erickson}}

The cube is [[face-transitive]], meaning its two square faces 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, and each pair of adjacent faces has 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> by [[Schläfli symbol]].{{r|wd}}
{{Clear}}

== Appearances ==
=== In popular cultures ===
{{multiple image
 | image1 = One-red-dice-01.jpg
 | caption1 = A six-sided [[dice|die]]
 | alt1 = A dice with six-sided square
 | image2 = Skewb.jpg
 | caption2 = A completed [[Skewb]]
 | alt2 = An alternative Rubik's Cube shaped toy
 | image3 = Just another Alamo afternoon.jpg
 | caption3 = The [[Alamo (sculpture)|''Alamo'' sculpture]]
 | alt3 = A sculpture with a cube shape
 | 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 that spins around its vertical axis.{{r|rz}} [[Optical illusions]] such as the [[impossible cube]] and [[Necker cube]] have been explored by artists such as [[M. C. Escher]].{{r|barrow}} 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 become the main floor.{{r|an}}

=== In nature and science ===
{{multiple image
 | image1 = Cubic.svg
 | caption1 = Simple cubic crystal structure
 | alt1 = Crystal structure in cubic shape
 | image2 = HALIT X NaCl Natriumchlorid WÜRFEL KUBUS 50P.jpg
 | caption2 = [[Table salt]] cubic crystals
 | alt2 = Table salt with cubic-shaped crystal
 | image3 = Cubane molecule ball.png
 | caption3 = [[Ball-and-stick model]] of [[cubane]]
 | alt3 = Cubane in ball-and-stick model
 | total_width = 360
}}
Cubes are also found in various fields of natural science and technology. It is applied to the [[unit cell]] of a crystal known as a [[cubic crystal system]].{{r|tisza}} [[Table salt]] is an example of a [[mineral]] with a commonly cubic shape.{{r|chieh}} Other examples are [[pyrite]] (although there are many variations){{r|hoffmann}} and [[uranium]] cubic-shaped in nuclear program.{{r|kratz}} The [[radiolarian]] ''Lithocubus geometricus'', discovered by [[Ernst Haeckel]], has a cubic shape.{{r|haeckel}}  [[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}}

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}}

{{multiple image
 | image1 = Ncube2.jpg
 | caption1 = The [[nCube (satellites)|Norwegian CubeSat]]
 | alt1 = A series of two Norwegian CubeSats
 | image2 = Corner reflector.svg
 | caption2 = [[Corner reflector]] redirecting two rays back to their source
 | alt2 = A cube's [[rotoflector]], a [[corner reflector]], redirects two rays back to theier source
 | total_width = 360
}}
Technological cubes include the spacecraft device [[CubeSat]],{{r|helvajian}} [[thermal radiation]] demonstration device [[Leslie cube]],{{r|vm}} and [[web server]] machine [[Cobalt Qube]].{{r|qube}} 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}} In various areas of [[engineering]], including [[traffic signs]] and [[radar]], the corner of a cube is useful as a [[retroreflector]], called a [[corner reflector]], which redirects any [[Ray (optics)|ray]] or [[wave]] back to its source.{{r|kraus}}

=== In antiquity ===
{{further|Platonic solids#History|Mysterium Cosmographicum#Shapes and the planets}}
{{multiple image
 | image1 = Kepler Hexahedron Earth.jpg
 | caption1 = Sketch of a cube by Johannes Kepler
 | alt1 = Sketch of trees on cube's face by Johannes Kepler
 | image2 = Mysterium Cosmographicum solar system model.jpg
 | caption2 = [[Johannes Kepler|Kepler's]] Platonic solid model of the [[Solar System]]
 | alt2 = Solar system, modeled with five Platonic solids by Johannes Kepler
 | align = right
 | total_width = 300
}}
The [[Platonic solid]]s are five polyhedra known since antiquity. The set is named for [[Plato]], who attributed these solids to nature in his dialogue [[Timaeus (dialogue)|''Timaeus'']]. One of them, the cube, represented the [[classical element]] of [[Earth (classical element)|earth]] because of the building blocks of Earth's foundation.{{r|djkt}} [[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 proposed the structure of [[Solar System]] and the relationships between its [[extraterrestrial planet]]s with the set of Platonic solids, inscribed and circumscribed by [[Orb (astronomy)|spherical orbs]]. Each solid encased in a sphere, within one another, would produce six layers, corresponding to the six known planets. [[Mercury (planet)|Mercury]], [[Venus]], [[Earth]], [[Mars]], [[Jupiter]], and [[Saturn]]. From innermost to outermost, these solids were arranged from [[Regular octahedron|octahedron]], followed by the [[Regular icosahedron|icosahedron]], [[Regular dodecahedron|dodecahedron]], [[Regular tetrahedron|tetrahedron]], and eventually the cube.{{r|livio}}

== Constructions ==
[[File:The 11 cubic nets 2.svg|thumb|upright=1|The eleven nets of a cube|alt=Cube' eleven different nets]]
The cube has eleven different [[Net (geometry)|nets]], each of which consists of an arrangement of edge-joined squares. If each boundary between squares is folded to a right angle, the squares become the faces of a cube.{{r|jeon|turney}}

A cube can be constructed by attaching six [[square pyramid]]s (each with height of half an edge-length) with their apices meeting at the center.{{r|barnes}}

In [[analytic geometry]], a cube can 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|alt=Cube projected in a plane as a graph]]

The cube can be drawn into a [[Graph (discrete mathematics)|graph]], a structure in [[graph theory]] consisting of a set of [[Vertex (graph theory)|vertices]] that are connected with an [[Edge (graph theory)|edge]]. It is attainable according to [[Steinitz's theorem]], which states that a graph can be represented as the [[Graph of a polytope|vertex-edge graph]] of a polyhedron, as long as it possesses the following two properties. These are [[Planar graph|planarity]] (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 <math> Q_2 \mathbin{\Box} Q_1 </math>, where <math> \mathbin{\Box} </math> denotes the Cartesian product of graphs. 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 ===
{{further|Platonic solid#As a configuration}}
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 configuration matrix of a cube 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 topics ==

=== Construction of polyhedra ===
{{multiple image
 | image1 = CubeAndStel.svg
 | alt1 = Facetting version of a cube
 | image2 = Tetrakishexahedron.jpg
 | alt2 = Attached from a cube and pyramids onto its square faces
 | image3 = Truncated rhombic dodecahedron.svg
 | alt3 = A chamfered cube
 | footer = Some of the polyhedra constructed based on a cube: the [[stellated octahedron]], [[tetrakis hexahedron]], and chamfered cube
 | total_width = 360
}}
Many polyhedra can be constructed based on a cube. Examples include:
* 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}}
* New convex polyhedra can be constructed by attaching less-regular polyhedra to a cube's faces.{{r|timofeenko-2010}} The cube is thus a component of two [[Johnson solid]]s, the  [[elongated square pyramid]] and [[elongated square bipyramid]], the latter being a cube with [[square pyramid]]s on opposite faces.{{r|rajwade}}
* Attaching a low pyramid to each face of a cube produces its [[Kleetope]], the [[tetrakis hexahedron]],{{r|sod}} [[dual polyhedron|dual]] to the [[truncated octahedron]].
* 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}}
* 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". The same figure can be derived in the same way from the cube's dual, the regular octahedron.{{r|vxac}}
* The chamfered cube is constructed from a cube by a truncating operator called [[Chamfer (geometry)|chamfer]]. The resulting polyhedron has twelve hexagonal and six square [[Zonogon|centrally symmetric faces]], a [[zonohedron]].{{r|gy|ddg}}
* Three mutually perpendicular [[golden rectangle]]s can be constructed from a pair of vertices located on the midpoints of the opposite edges on a cube's surface, drawing a segment line between those two, and dividing that segment line in a [[golden ratio]] from its midpoint. The corners of these rectangles are the vertices of a [[regular icosahedron]] with twenty equilateral triangles.{{sfnp|Cromwell|1997|p=[https://archive.org/details/polyhedra0000crom/page/70/mode/1up?view=theater 70]}}

=== Polycubes ===
{{main|Polycube|Dali cross}}
{{multiple image
 | image1 = AGK-pentacube.png
 | alt1 = A pentacube
 | image2 = Net of tesseract.gif
 | alt2 = Dali cross, animated as one of tesseract's nets
 | total_width = 300
 | footer = Two examples of polycubes are a polycube with five cubes attached face-to-face and [[Dalí cross]]. The latter is one of 261 nets of a [[tesseract]].
}}
A [[polycube]] is a [[solid figure]] formed by joining one or more equal cubes face-to-face. Polycubes are the three-dimensional analogues of two-dimensional [[polyomino]]es.{{r|lunnon}}

When four cubes are stacked vertically, and four others are attached to the second-from-top cube of the stack, the resulting polycube is the [[Dalí cross]], named after Spanish [[surrealist]] artist [[Salvador Dalí]], whose 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 Dalí cross can be folded in a fourth dimension to enclose a [[tesseract]].{{r|pucc}} A cube is a three-dimensional instance of a [[hypercube]] (also known as a 3-cube); the two-dimensional hypercube (2-cube) is a [[square]], and the four-dimensional hypercube (4-cube) is a tesseract.{{r|hall}}

=== Space-filling ===
[[File:Partial cubic honeycomb.png|thumb|upright=0.6|[[Cubic honeycomb]] is an example of [[Honeycomb (geometry)|honeycomb]] in Euclidean three-dimensional space]]
A cube can achieve a [[Honeycomb (geometry)|honeycomb]] by filling together with its copy in three-dimensional space without leaving a gap. Cubes are [[space-filling polyhedron|space-fillings]], where the phrase "space-filling" can be understood as a generalized [[tessellation]].{{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 parallelepiped.{{r|horvath}} Every three-dimensional parallelohedron is a [[zonohedron]], a [[centrally symmetric]] polyhedron whose faces are [[Zonogon|centrally symmetric polygons]].{{r|shephard}}

An example of a honeycomb with a cubic type only, called a [[Cell (mathematics)|cell]], is a [[cubic honeycomb]] that consists of four cubes around its edges in Euclidean three-dimensional space.{{r|twelveessay|ns}} More examples in three-dimensional non-Euclidean space are the honeycomb with three cubes around its edges in a [[3-sphere|three-dimensional sphere]] and [[Order-5 cubic honeycomb|the honeycomb with five cubes around its edges]] in [[hyperbolic space]].{{r|ns}}

Any parallelepiped, including a cube, can achieve a honeycomb if its [[Dehn invariant]] is zero.{{r|am}} The Dehn invariant's inception dates back to [[Hilbert's third problem]], 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. This problem was solved by [[Max Dehn]], inventing his [[Invariant (mathematics)|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}}

=== Miscellanea ===
{{multiple image
 | image1 = UC07-6 cubes.png
 | alt1 = A compound of six cubes with rotational freedom
 | image2 = UC08-3 cubes.png
 | alt2 = A compound of three cubes
 | image3 = UC09-5 cubes.png
 | alt3 = A compound of five cubes
 | 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&mdash;although possibly [[enantiomorphous]]&mdash;[[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 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
 | alt1 = Cube in spherical geometry
 | image2 = 3-Manifold 3-Torus.png
 | caption2 = A view in [[3-torus|three-dimensional torus]]
 | alt2 = A torus as a three-dimensional cube modeled
 | total_width = 300
}}
{{anchor|Spherical cube}}The spherical cube represents the [[spherical polyhedron]], which can be modeled with the [[Arc (geometry)|arcs]] 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 to 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}}

{{multiple image
 | image1 = Menger sponge rotation.gif
 | alt1 = Menger sponge
 | image2 = Cube de Jérusalem, itération 3.png
 | alt2 = Jerusalem cube
 | image3 = Sierpinskisnowflake.gif
 | alt3 = Mosley snowflake
 | footer = Fractal-shaped cubes are the [[Menger sponge]], Jerusalem cube, and [[Mosely snowflake]].
 | total_width = 400
}}
A cube can have [[Fractal geometry|fractal shapes]], which retain the pattern shape recursively regardless of the magnification. The [[Menger sponge]] is an example of a fractal-shaped cube, analogous to the two-dimensional version, the [[Sierpiński carpet]].{{r|bh}} Other varieties are the Jerusalem cube and [[Mosely snowflake]].{{r|husain|ra}}

== 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
* [[Kaaba]], cubic buildings of importance to [[Islam]]. 
* [[Magic cube]], a [[magic square]] in three-dimensional version
* [[Schläfli double six]], a configuration of 30 points and 12 lines in three-dimensional Euclidean space
* [[Squaring the square]]'s three-dimensional analogue, [[cubing the cube]]
* [[Superellipsoid]], a solid whose horizontal sections are of the same squareness
* [[Tychonoff cube]], generalization of a unit cube
{{Clear}}

== References ==
<references>
<ref name=am>{{cite book
 | last1 = Akiyama | first1 = Jin | author1-link = Jin Akiyama
 | last2 = Matsunaga | first2 = Kiyoko
 | contribution = 15.3 Hilbert's Third Problem and Dehn Theorem
 | doi = 10.1007/978-4-431-55843-9
 | isbn = 9784431558415
 | mr = 3380801
 | pages = 382–388
 | publisher = Springer Tokyo | title = Treks Into Intuitive Geometry
 | title-link = Treks Into Intuitive Geometry
 | year = 2015| doi-access = free
}}</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 = 9781614442165
 | 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 = 9781852332594
}}</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 = 9783642309649
 | doi = 10.1007/978-3-642-30964-9
}}</ref>

<ref name=barrow>{{cite book
 | last = Barrow | first = John D.| author-link = John D. Barrow
 | title = Impossibility: The Limits of Science and the Science of Limits
 | 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 = 9781493907816
}}</ref>

<ref name=bh>{{cite book
 | last1 = Bunde | first1 = Armin
 | last2 = Havlin | first2 = Shlomo
 | title = Fractals in Science
 | year = 2013
 | publisher = Springer
 | isbn = 9783642779534
 | page = 7
 | url = https://books.google.com/books?id=dh7rCAAAQBAJ&pg=PA7
}}</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=bird>{{cite book
 | last = Bird | first = John
 | year = 2020
 | title = Science and Mathematics for Engineering
 | edition = 6th
 | url = https://books.google.com/books?id=GVbGDwAAQBAJ&pg=PA143
 | pages = 143&ndash;144
 | publisher = Routledge
 | isbn = 9780429261701
}}</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 = 9780470534922
}}</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=chieh>{{cite book
 | last = Chieh | first = Chung
 | editor-last = Senechal | editor-first = Marjorie
 | year = 2013
 | title = Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination
 | contribution = Polyhedra and Crystal Structures
 | contribution-url = https://books.google.com/books?id=kZtCAAAAQBAJ&pg=PA141
 | page = 141
 | publisher = Springer
 | doi = 10.1007/978-0-387-92714-5_10
 | isbn = 9780387927138
}}</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
 | article-number = 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
 | 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 = 9780521554329
}}</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 = 9780486297309
}}</ref>

<ref name=ddg>{{cite journal
 | last1 = Deza  | first1 = Antoinea
 | last2 = Michel | first2 = Deza
 | last3 = Grishukhin | first3 = Viatcheslav
 | year = 1998
 | journal = Discrete Mathematics
 | volume = 192 | issue = 1–3 | pages = 41–80
 | title = Fullerenes and Coordination Polyhedra versus Half-cube Embeddings
 | doi = 10.1016/S0012-365X(98)00065-X
}}</ref>

<ref name=djkt>{{cite journal
 | title = Plato's Cube and the Natural Geometry of Fragmentation
 | last1 = Domokos | first1 = Gábor
 | last2 = Jerolmack | first2 = Douglas J.
 | last3 = Kun | first3 = Ferenc
 | last4 = Török | first4 = János
 | year = 2020
 | volume = 117 | issue = 31 | pages = 18178&ndash;18185
 | journal = [[Proceedings of the National Academy of Sciences of the United States of America]]
 | doi = 10.1073/pnas.2001037117
 | pmid = 32680966 | doi-access = free | arxiv = 1912.04628
| bibcode = 2020PNAS..11718178D }}</ref>

<ref name=dupuis>{{cite book
 | last = Dupuis | first = Nathan Fellowes
 | year = 1893
 | title = Elements of Synthetic Solid Geometry
 | publisher = Macmillan
 | page = 68
 | url = https://archive.org/details/elementssynthet01dupugoog/page/n68
}}</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 = 9781614445098
}}</ref>

<ref name=fowler>{{cite journal
 | last = Fowler | first = David
 | title = Mathematics in Science Fiction: Mathematics as Science Fiction
 | 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
 | url = https://web.archive.org/web/20180304110822id_/http://www.ams.org:80/distribution/mmj/vol7-2-2007/fuchs.pdf
}} See Figure 11, p. 273, for showing three types of cube's geodesics.</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 = 9780020653202
}}</ref>

<ref name=gardner>{{cite book
 | last = Gardner | first = Martin | author-link = Martin Gardner
 | title = The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems: Number Theory, Algebra, Geometry, Probability, Topology, Game Theory, Infinity, and Other Topics of Recreational Mathematics
 | publisher = W. W. Norton & Company
 | year = 2001
 | isbn = 9780393020236
 | pages = 172–173
 | url = https://books.google.com/books?id=orz0SDEakpYC&pg=PA172
}}</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
 | bibcode = 1986ICGA....6k..21G
 | 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 = 9783540711322
 | 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 = 0387404090
 | 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=gy>{{cite journal
 | last1 = Gelişgen | first1 = Özcan
 | last2 = Yavuz | first2 = Serhat
 | journal = Mathematical Sciences and Applications E-Notes
 | volume = 7 | issue = 2 | pages = 174&ndash;182
 | year = 2019
 | title = Isometry Groups of Chamfered Cube and Chamfered Octahedron Spaces
 | url = https://dergipark.org.tr/en/pub/mathenot/issue/49271/542272
 | doi = 10.36753/mathenot.542272
}}</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 =  9781938664304
 | 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
 | editor2-first = Siegfried W. | editor2-last = Janson
 | title = Small Satellites: Past, Present, and Future
 | year = 2008
 | publisher = Aerospace Press
 | isbn = 9781884989223
 | 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
 | issue = 4
 | pages = 277–289
 | 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 = 9783030351106
}}</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 = 9783642144417
 | doi = 10.1007/978-3-642-14441-7
}}</ref>

<ref name=horvath>{{cite journal
 | last = Horváth | first = Ákos G.
 | issue = 1
 | journal = Periodica Polytechnica Mechanical Engineering
 | pages = 25–42
 | title = On Dirichlet–Voronoi Cell, Part I: Classical Problems
 | url = https://pp.bme.hu/me/article/view/5495
 | volume = 39
 | year = 1995
}}</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=husain>{{cite journal
 | last = Husain | first = Sakhlaq
 | journal = Annals of Communications in Mathematics
 | volume = 8 | issue = 2 | year = 2025 | pages = 293&ndash;298
 | doi = 10.62072/acm.2025.080211
 | issn = 2582-0818
 | title = On Jerusalem Square Fractal, Its Construction and Properties
| doi-access = free}}</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 = 9780801890123
 | publisher = The Johns Hopkins University Press
}}</ref>

<ref name=jw>{{cite journal
 | last1 = Jerrard | first1 = Richard P.
 | last2 = Wetzel | first2 = John E.
 | doi = 10.2307/4145012
 | issue = 1
 | journal = The American Mathematical Monthly
 | mr = 2026310
 | pages = 22–31
 | title = Prince Rupert's rectangles
 | volume = 111
 | year = 2004| jstor = 4145012
}}</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 = 9780387989792
}}</ref>

<ref name=kemp>{{cite journal
 | last = Kemp | first = Martin
 | title = Dali's Dimensions
 | journal = [[Nature (journal)|Nature]]
 | volume = 391 | issue = 27 | page = 27
 | date = 1 January 1998
 | 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 = 9788131716823
}}</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 = 9783319258591
}}</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=kratz>{{cite book
 | last = Kratz | first = Jens-Volker
 | title = Nuclear and Radiochemistry: Fundamentals and Applications
 | year = 2021
 | publisher = [[John Wiley & Sons]]
 | url = https://books.google.com/books?id=fsZAEAAAQBAJ&pg=PA31
 | page = 31
 | isbn = 9783527831944
 | doi = 10.1002/9783527831944
}}</ref>

<ref name=kraus>{{Cite book
| last=Kraus 
| first=John D. 
| url=http://archive.org/details/electromagnetics0000krau 
| title=Electromagnetics With Applications 
| last2=Fleisch |first2=Daniel A. 
| date=1999 |publisher=[[McGraw Hill]] 
| isbn=0-07-289969-7 
| location=Boston 
| pages=324 
| lccn=98-34935
}}</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 = 9780080965291
 }}</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]]
 | isbn = 9780767908160
 | 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 = 9781483255125
 | 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 = 9781584885054
}}</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 = 9783031074424
}}</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=messer86>{{cite journal
 | last = Messer | first = Peter
 | year = 1986
 | title = Problem 1054
 | url = https://cms.math.ca/crux/backfile/Crux_v12n10_Dec.pdf
 | journal = [[Crux Mathematicorum]]
 | volume = 12 | issue = 10 | pages = 284–285
 | via = Canadian Mathematical Society
}}</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 = 9780435024741
 | 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
 | last = Padmanabhan | first = Thanu
 | title = Sleeping Beauties in Theoretical Physics
 | chapter = The Grand Cube of Theoretical Physics
 | publisher = Springer
 | year = 2015 | pages = 1–8
 | isbn = 9783319134420
}}</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 = 9780817683634
 | 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=qube>{{cite news
 | last = Sims | first = Ralph
 | title = Cobalt Qube Microserver
 | url = http://www.linuxjournal.com/article/3052
 | publisher = Linux Journal
 | date = Oct 1, 1998
}}</ref>

<ref name=ra>{{cite conference
 | last1 = Regueiro | first1 = Manuel Diaz
 | last2 = Allegue | first2 = Luis Diaz
 | chapter = An Algorithm for Extending Menger-type Fractal Structures
 | title = Hyperseeing: Proceedings of SMI-SCULPT 2023: Shape Modeling International 2023, Shape Creation Using Layouts, Programs, & Technology (SCULPT) Event: Twenty-second Interdisciplinary Conference of the International Society of the Arts, Mathematics, and Architecture
 | url = https://hyperseeing.viz.tamu.edu/2023/title.pdf
 | chapter-url = https://people.tamu.edu/~ergun/hyperseeing/2023/diaz2023.pdf
 | date = July 7, 2023
 | isbn = 9781387733309
 | pages = 55-62
}}</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 = 9789386279064
 | 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]
 | isbn = 9780691126777
}}</ref>

<ref name=rickey>{{cite web
 | url = http://www.math.usma.edu/people/Rickey/papers/ShortCourseAlbuquerque.pdf
 | title = Dürer's Magic Square, Cardano's Rings, Prince Rupert's Cube, and Other Neat Things
 | year = 2005
 | last = Rickey | first = V. Frederick
 | archive-url = https://web.archive.org/web/20100705014816/http://www.math.usma.edu/people/Rickey/papers/ShortCourseAlbuquerque.pdf
 | archive-date = 2010-07-05
}} Notes for “Recreational Mathematics: A Short Course in Honor of the 300th Birthday of Benjamin Franklin,” Mathematical Association of America, Albuquerque, NM, August 2–3, 2005</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 = 10.1017/9781316466919
 | isbn = 9781316466919
}}</ref>

<ref name=rz>{{cite book
 | last1 = Reaven | first1 = Marci
 | 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 = 9780813538907
 | 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 = 9781118031032
 }}</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=timofeenko-2010>{{cite journal
 | last = Timofeenko | first = A. V.
 | year = 2010
 | 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]]
 | isbn = 9781615032419
}}</ref>

<ref name="turney">{{cite journal
 | last = Turney | first = Peter D.
 | year = 1984–1985
 | title = Unfolding the Tesseract
 | journal = [[Journal of Recreational Mathematics]]
 | volume = 17 | issue = 1
 | url = https://unfolding.apperceptual.com/
}}</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 = 9780486409191
}} See table III.</ref>

<ref name=vm>{{cite book
 | last1 = Vollmer | first1 = Michael
 | last2 = Möllmann | first2 = Klaus-Peter
 | title = Infrared Thermal Imaging: Fundamentals, Research and Applications
 | publisher = [[John Wiley & Sons]]
 | year = 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 = 9783319955872
 }} 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 = 9783642018985
 | 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 = 9780966520200
}}</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 = 038794365X
 | pages = 103–126
 | publisher = Springer-Verlag
 | series = [[Graduate Texts in Mathematics]]
 | title = Lectures on Polytopes
 | volume = 152
 | year = 1995
}}</ref>
</references>

==External links==
*{{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}}
{{Authority control}}

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