Cube: Difference between revisions

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Template:Short description Template:Good article Script error: No such module "other uses". Script error: No such module "Infobox".Template:Template other A cube is a 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 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, which described a set of solids now called Platonic solids, associating a cube with the classical element of earth. A 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 and attaching to other polyhedra. It also represents geometrical shapes. The cube can be attached to its faces with its copy to fill a space without leaving a gap, which forms a honeycomb.

The cube can be represented in many ways. One example is by drawing a graph, a structure in graph theory consisting of a set of vertices that are connected with an 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 polytopes that also includes the two-dimensional square and four-dimensional tesseract.

Properties

Cube modeled in 3D, where one can click and drag to rotate

3D model of a cube

A cube is a polyhedron with eight vertices and twelve equal-length edges, forming six squares as its faces. A cube is a special case of a rectangular cuboid, which has six rectangular faces, each of which has a pair of opposite equal-length and parallel edges.Template:R 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.Template:R More generally, the cube and the rectangular cuboid are special cases of a cuboid, a polyhedron with six quadrilaterals (four-sided polygons).Template:R As for all convex polyhedra, the cube has Euler characteristic of 2, according to the formula $V - E + F = 2$ ; the three letters denote respectively the number of vertices, edges, and faces.Template:R

All three square faces surrounding a vertex are 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.Template:R The cube is a special case of other cuboids. These include a parallelepiped, a polyhedron with six parallelograms faces, because its pairs of opposite faces are congruent;Template:R a rhombohedron, as a special case of a parallelepiped with six rhombi faces, because the interior angle of all of the faces is right;Template:Sfnp and a trigonal trapezohedron, a polyhedron with congruent quadrilateral faces, since its square faces are the special cases of rhombi.Template:R

The cube is a 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.Template:R

Measurement

A space diagonal AC' and a face diagonal AC

A face diagonal is denoted as

A C

and a space diagonal is

A C^{'}

Given a cube with edge length $a$ , the face diagonal of the cube is the diagonal of a square $a \sqrt{2}$ , and the space diagonal of the cube is a line connecting two vertices that are not in the same face, formulated as $a \sqrt{3}$ . Both formulas can be determined by using the Pythagorean theorem. The surface area of a cube $A$ is six times the area of a square:Template:R $A = 6 a^{2} .$ 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.Template:R This leads to the use of the term cube as a verb, to mean raising any number to the third power:Template:R $V = a^{3} .$

The cube has three types of closed geodesics, 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 angles 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 $3 \sqrt{2}$ times that of an edge. The third type is a non-planar hexagon.Template:R

Insphere, midsphere, circumsphere

An insphere of a cube $r_{i}$ is a sphere tangent to the faces of a cube at their centroids. Its midsphere $r_{m}$ is a sphere tangent to the edges of a cube. Its circumsphere $r_{c}$ is a sphere tangent to the vertices of a cube. With edge length $a$ , they are respectively:Template:R $r_{i} = \frac{1}{2} a = 0.5 a, r_{m} = \frac{\sqrt{2}}{2} a \approx 0.707 a, r_{c} = \frac{\sqrt{3}}{2} a \approx 0.866 a .$

Unit cube

Template:Main article

Cube with a hole through which an equal cube can pass

Prince Rupert's cube

A unit cube is a cube with 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.Template:R Prince Rupert of the Rhine, known for Prince Rupert's drop, wagered whether 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.Template:R One way to obtain this result is by using the Pythagorean theorem or the formula for Euclidean distance in three-dimensional space.Template:R

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—the cube root of 2, $\sqrt[3]{2}$ —is not constructible.Template:R However, this problem was solved with folding an origami paper by Template:Harvtxt.Template:R

Symmetry

Cube, the dual of a regular octahedron

The dual polyhedron of a cube is the regular octahedron. Both have octahedral symmetry.

The cube has octahedral symmetry $O_{h}$ of order 48. In other words, the cube has 48 isometries (including 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°).Template:R

The dual polyhedron can be obtained from each of the polyhedra's vertices tangent to a plane by a process known as polar reciprocation.Template:R One property of dual polyhedra is that the polyhedron and its dual share their 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.Template:R

The cube is face-transitive, meaning its two square faces are alike and can be mapped by rotation and reflection.Template:R It is vertex-transitive, meaning all of its vertices are equivalent and can be mapped isometrically under its symmetry.Template:R 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.Template:R Each vertex is surrounded by three squares, so the cube is $4.4.4$ by vertex configuration or ${4, 3}$ by Schläfli symbol.Template:R

Appearances

In popular cultures

Template:Multiple image Cubes have appeared in many roles in popular culture. It is the most common form of dice.Template:R Puzzle toys such as pieces of a Soma cube,Template:R Rubik's Cube, and Skewb are built of cubes.Template:R Minecraft is an example of a sandbox video game of cubic blocks.Template:R The outdoor sculpture Alamo (1967) is a cube that spins around its vertical axis.Template:R Optical illusions such as the impossible cube and Necker cube have been explored by artists such as M. C. Escher.Template:R The cube was applied in Alberti's treatise on Renaissance architecture, De re aedificatoria (1450).Template:R Cube houses in the Netherlands are a set of cubical houses whose hexagonal space diagonals become the main floor.Template:R

In nature and science

Template:Multiple image 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.Template:R Table salt is an example of a mineral with a commonly cubic shape.Template:R Other examples are pyrite (although there are many variations)Template:R and uranium cubic-shaped in nuclear program.Template:R The radiolarian Lithocubus geometricus, discovered by Ernst Haeckel, has a cubic shape.Template:R Cubane is a synthetic hydrocarbon consisting of eight carbon atoms arranged at the corners of a cube, with one hydrogen atom attached to each carbon atom.Template:R

A historical attempt to unify three physics ideas of relativity, gravitation, and quantum mechanics used the framework of a cube known as a cGh cube.Template:R

Template:Multiple image Technological cubes include the spacecraft device CubeSat,Template:R thermal radiation demonstration device Leslie cube,Template:R and web server machine Cobalt Qube.Template:R Cubical grids are usual in three-dimensional Cartesian coordinate systems.Template:R In computer graphics, an algorithm divides the input volume into a discrete set of cubes known as the unit on isosurface,Template:R and the faces of a cube can be used for mapping a shape.Template:R 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 or wave back to its source.Template:R

In antiquity

Script error: No such module "labelled list hatnote". Template:Multiple image The Platonic solids are five polyhedra known since antiquity. The set is named for Plato, who attributed these solids to nature in his dialogue Timaeus. One of them, the cube, represented the classical element of earth because of the building blocks of Earth's foundation.Template:R Euclid's 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.Template:R

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.Template:Sfnp In his Mysterium Cosmographicum, Kepler proposed the structure of Solar System and the relationships between its extraterrestrial planets with the set of Platonic solids, inscribed and circumscribed by spherical orbs. Each solid encased in a sphere, within one another, would produce six layers, corresponding to the six known planets. Mercury, Venus, Earth, Mars, Jupiter, and Saturn. From innermost to outermost, these solids were arranged from octahedron, followed by the icosahedron, dodecahedron, tetrahedron, and eventually the cube.Template:R

Constructions

Cube' eleven different nets

The eleven nets of a cube

The cube has eleven different 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.Template:R

A cube can be constructed by attaching six square pyramids (each with height of half an edge-length) with their apices meeting at the center.Template:R

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 $(\pm 1, \pm 1, \pm 1)$ .Template:R Its interior consists of all points $(x_{0}, x_{1}, x_{2})$ with $- 1 < x_{i} < 1$ for all $i$ . A cube's surface with center $(x_{0}, y_{0}, z_{0})$ and edge length of $2 a$ is the locus of all points $(x, y, z)$ such that $\max {| x - x_{0} |, | y - y_{0} |, | z - z_{0} |} = a .$

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.Template:R

Representation

As a graph

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Cube projected in a plane as a graph

The graph of a cube

The cube can be drawn into a graph, a structure in graph theory consisting of a set of vertices that are connected with an edge. It is attainable according to Steinitz's theorem, which states that a graph can be represented as the vertex-edge graph of a polyhedron, as long as it possesses the following two properties. These are planarity (the edges of a graph are connected to every vertex without crossing other edges), and 3-connected (whenever a graph with more than three vertices, and two of the vertices are removed, the edges remain connected).Template:R 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.Template:R The cubical graph is also classified as a prism graph, resembling the skeleton of a cuboid.Template:R

The cubical graph is a special case of hypercube graph or $n$ -cube—denoted as $Q_{n}$ —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.Template:R In the case of the cubical graph, it is the product of $Q_{2} ◻ Q_{1}$ , where $◻$ 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 $Q_{3}$ .Template:R Like any hypercube graph, it has a cycle which visits every vertex exactly once,Template:R and it is also an example of a unit distance graph.Template:R

The cubical graph is bipartite, meaning every independent set of four vertices can be disjoint and the edges connected in those sets.Template:R However, every vertex in one set cannot connect all vertices in the second, so this bipartite graph is not complete.Template:R It is an example of both a crown graph and a bipartite Kneser graph.Template:R

In orthogonal projection

An object illuminated by parallel rays of light casts a shadow on a plane perpendicular to those rays, called an orthogonal projection. A polyhedron is considered 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.Template:R

As a configuration matrix

Script error: No such module "labelled list hatnote". The cube can be represented as a configuration matrix, a matrix in which the rows and columns correspond to the elements of a polyhedron as the vertices, edges, and faces. The 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:Template:R $[\begin{matrix} \begin{matrix} 8 & 3 & 3 \\ 2 & 12 & 2 \\ 4 & 4 & 6 \end{matrix} \end{matrix}]$

Related topics

Construction of polyhedra

Template:Multiple image 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.Template:R
New convex polyhedra can be constructed by attaching less-regular polyhedra to a cube's faces.Template:R The cube is thus a component of two Johnson solids, the elongated square pyramid and elongated square bipyramid, the latter being a cube with square pyramids on opposite faces.Template:R
Attaching a low pyramid to each face of a cube produces its Kleetope, the tetrakis hexahedron,Template:R dual to the truncated octahedron.
The barycentric subdivision of a cube (or its dual, the regular octahedron) is the disdyakis dodecahedron, a Catalan solid.Template:R
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.Template:Sfnp
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.Template:R
Each of the cube's vertices can be truncated, and the resulting polyhedron is the Archimedean solid, the truncated cube.Template:Sfnp When its edges are truncated, it is a rhombicuboctahedron.Template:R 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.Template:R
The chamfered cube is constructed from a cube by a truncating operator called chamfer. The resulting polyhedron has twelve hexagonal and six square centrally symmetric faces, a zonohedron.Template:R
Three mutually perpendicular golden rectangles 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.Template:Sfnp

Polycubes

Script error: No such module "Labelled list hatnote". Template:Multiple image 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 polyominoes.Template:R

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).Template:R The Dalí cross can be folded in a fourth dimension to enclose a tesseract.Template:R 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.Template:R

Space-filling

File:Partial cubic honeycomb.png

Cubic honeycomb is an example of honeycomb in Euclidean three-dimensional space

A cube can achieve a honeycomb by filling together with its copy in three-dimensional space without leaving a gap. Cubes are space-fillings, where the phrase "space-filling" can be understood as a generalized tessellation.Template:R 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.Template:R The plesiohedra include the parallelohedra, which can be 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.Template:R Every three-dimensional parallelohedron is a zonohedron, a centrally symmetric polyhedron whose faces are centrally symmetric polygons.Template:R

An example of a honeycomb with a cubic type only, called a cell, is a cubic honeycomb that consists of four cubes around its edges in Euclidean three-dimensional space.Template:R More examples in three-dimensional non-Euclidean space are the honeycomb with three cubes around its edges in a three-dimensional sphere and the honeycomb with five cubes around its edges in hyperbolic space.Template:R

Any parallelepiped, including a cube, can achieve a honeycomb if its Dehn invariant is zero.Template:R 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, answering that not all polyhedra can be reassembled into a cube.Template:R It showed that two equal volume polyhedra should have the same Dehn invariant, except for the two tetrahedra whose Dehn invariants were different.Template:R

Miscellanea

Template:Multiple image Script error: No such module "anchor".The polyhedral compounds, in which the cubes share the same centre, are uniform polyhedron compounds, 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 Template:Harvtxt in the seventh to ninth uniform compounds for the compound of six cubes with rotational freedom, three cubes, and five cubes.Template:R Two compounds, consisting of two and three cubes were found in Escher's wood engraving print Stars and Max Brückner's book Vielecke und Vielflache.Template:R

Template:Multiple image Script error: No such module "anchor".The spherical cube represents the spherical polyhedron, which can be modeled with the arcs of great circles, creating bounds as the edges of a spherical square.Template:R 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.Template:R Its dual is the spherical octahedron.Template:R

The topological object 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.Template:R

Template:Multiple image A cube can have 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.Template:R Other varieties are the Jerusalem cube and Mosely snowflake.Template:R

References

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External links

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Cube: Interactive Polyhedron Model*
Volume of a cube, with interactive animation
Cube (Robert Webb's site)

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Cube: Difference between revisions

Latest revision as of 13:10, 18 November 2025

Contents

Properties

Measurement

Insphere, midsphere, circumsphere

Unit cube

Symmetry

Appearances

In popular cultures

In nature and science

In antiquity

Constructions

Representation

As a graph

In orthogonal projection

As a configuration matrix

Related topics

Construction of polyhedra

Polycubes

Space-filling

Miscellanea

See also

References

External links

Navigation menu

@@ Line 1: / Line 1: @@
-{{Short description|Solid object with six equal square faces}}
+{{Short description|Solid with six equal square faces}}
+{{good article}}
 {{other uses}}
 {{infobox polyhedron
@@ Line 5: / Line 6: @@
   | 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
+  | faces = 6 [[square]]
   | edges = 12
   | vertices = 8
+ | euler = 2
   | vertex_config = <math> 8 \times (4^3) </math>
   | schläfli = <math> \{4,3\} </math>
@@ Line 13: / Line 15: @@
   | 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]]
+  | 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''' 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]].
+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 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 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 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.
+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]]
+[[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 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}} Because of such properties, it is categorized as one of the five [[Platonic solid]]s, a [[polyhedron]] 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]] each other, so the cube is classified as [[orthogonal polyhedron]].{{r|jessen}} The cube may also be considered as the [[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 as the [[trigonal trapezohedron]] since its square faces are the [[rhombi]]' special case.{{r|cc}}
+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 and other metric properties ===
+=== Measurement ===
-[[File:Cube diagonals.svg|thumb|upright=0.6|A face diagonal in red and space diagonal in blue]]
+[[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 a cube is the [[diagonal]] of a square <math> a\sqrt{2} </math>, and the [[space diagonal]] of a 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 [[Pythagorean theorem]]. The surface area of a cube <math> A </math> is six times the area of a square:{{r|khattar}}
+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 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 as 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}}
+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>
-[[File:Prince Ruperts cube.png|thumb|upright=0.6|[[Prince Rupert's cube]]]]
+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}}
-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 a [[compass and straightedge]] solely. Ancient mathematicians could not solve this old problem until the French mathematician [[Pierre Wantzel]] in 1837 proved it was impossible.{{r|lutzen}}
-The cube has three types of [[closed geodesic]]s. The closed geodesics are paths on a cube's surface that are locally straight. In other words, they avoid the vertices of the polyhedron, 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. Two of its types are planar. The first 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. The second 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}}
+=== 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>
-=== Relation to the spheres ===
+=== Unit cube ===
-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}}
+{{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}}
-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}}
+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}}
-<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>. It is composed of [[reflection symmetry]], a symmetry by cutting into two halves by a plane. There are nine reflection symmetries: the five are cut the cube from the midpoints of its edges, and the four are cut diagonally. It is also composed of [[rotational symmetry]], a symmetry by rotating it around the axis, from which the appearance is interchangeable. It has octahedral rotation symmetry <math> \mathrm{O} </math>: three axes pass through the cube's opposite faces centroid, six through the cube's opposite edges midpoints, and four through the cube's opposite vertices; each of these axes is 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]] is the order of 48.{{r|kane}}
+[[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}}
-[[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 have the same octahedral symmetry.{{r|erickson}}
-The [[dual polyhedron]] can be obtained from each of the polyhedra's vertices tangent to a plane by the 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 polyhedron has the same symmetry, the 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 [[regular polyhedron]] because it requires those properties.{{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 [[Schläfli symbol]].{{r|wd}}
+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}}
-== Applications ==
+== Appearances ==
+=== In popular cultures ===
 {{multiple image
   | image1 = One-red-dice-01.jpg
-  | caption1 = A six-sided [[dice]]
+  | caption1 = A six-sided [[dice|die]]
+ | alt1 = A dice with six-sided square
   | image2 = Skewb.jpg
   | caption2 = A completed [[Skewb]]
-  | image3 = St Marks Place, East Village, Downtown New York City, Recover Reputation.jpg
+ | alt2 = An alternative Rubik's Cube shaped toy
-  | caption3 = A sculpture [[Alamo (sculpture)|''Alamo'']]
+  | 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 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 an unfolding of a [[tesseract]] 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]]'' is known for a set of cubical houses in which its [[hexagon]]al space diagonal becomes the main floor.{{r|an}}
+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
-  | image2 = 2780M-pyrite1.jpg
+ | alt1 = Crystal structure in cubic shape
-  | caption2 = [[Pyrite]] cubic crystals
+  | 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
 }}
-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}}
+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}}
-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}}
+=== 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, 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 ane side of the cube 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}}
+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}}
-== Construction ==
+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}}
-[[File:The 11 cubic nets.svg|thumb|Nets of a cube]]
-An elementary way to construct is using its [[Net (polyhedron)|net]], an arrangement of edge-joining polygons, constructing a polyhedron by connecting along the edges of those polygons. Eleven nets for the cube are shown here.{{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
+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 [[Hanner polytope]], because it can be constructed by using [[Cartesian product]] of three line segments. Its dual polyhedron, the regular octahedron, is constructed by [[direct sum]] of three line segments.{{r|kozachok}}
+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]]
+[[File:Cube skeleton.svg|upright=0.8|thumb|The graph of a cube|alt=Cube projected in a plane as a graph]]
-According to [[Steinitz's theorem]], the [[Graph (discrete mathematics)|graph]] can be represented as the [[Skeleton (topology)|skeleton]] of a polyhedron; roughly speaking, a framework 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 can be represented as the graph, and it 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 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 operation known as the [[Cartesian product of graphs]]: it involves 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]] 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 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 [[crown graph]] and [[bipartite Kneser graph]].{{r|kl|berman-graph}}
+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 ===
@@ Line 112: / Line 150: @@
 === As a configuration matrix ===
-The cube can be represented as [[Platonic solid#As a configuration|configuration matrix]]. A configuration matrix is a [[Matrix (mathematics)|matrix]] in which the rows and columns correspond to the elements of a polyhedron as in 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. As mentioned above, the cube has eight vertices, twelve edges, and six faces; each element in a matrix's diagonal is denoted as 8, 12, and 6. The first column of the middle row indicates that there are two vertices in (i.e., at the extremes of) 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}}
+{{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 figures ==
+== Related topics ==
 === Construction of polyhedra ===
 {{multiple image
   | image1 = CubeAndStel.svg
+ | alt1 = Facetting version of a cube
   | image2 = Tetrakishexahedron.jpg
-  | footer = Some of the derived cubes, the [[stellated octahedron]] and [[tetrakis hexahedron]].
+ | alt2 = Attached from a cube and pyramids onto its square faces
-  | total_width = 320
+ | 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
 }}
-The cube can appear in the construction of a polyhedron, and some of its types can be derived differently in the following:
+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}}
-* The cube is [[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}} Suppose one and two equilateral square pyramids are attached to their square faces. In that case, they are the construction of an [[elongated square pyramid]] and [[elongated square bipyramid]] respectively, the [[Johnson solid]]'s examples.{{r|rajwade}}
+* 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}}
-* 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}}
+* 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 away the cube square's face, and filling their gaps with twisted angle equilateral triangles, a process known as [[Snub (geometry)|snub]].{{r|holme}}
+* 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}}
+* 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|Polycubes}}
+{{main|Polycube|Dali cross}}
-[[File:Net of tesseract.gif|thumb|upright=0.6|[[Dali cross]], the net of a [[tesseract]]]]
+{{multiple image
-[[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 [[Dali cross]], after [[Salvador Dali]]. In addition to popular cultures, 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 analogous' [[four-dimensional space]] bounded by twenty-four squares and eight cubes.{{r|hall}}
+ | 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}}
-=== Space-filling and honeycombs ===
+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}}
-[[Hilbert's third problem]] asked whether every two equal-volume polyhedra could always be dissected into polyhedral pieces and reassembled into each other. If it were, 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 in an invention [[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]]]]
+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}}
-The cube has a Dehn invariant of zero. This indicates the cube is applied for [[Honeycomb (geometry)|honeycomb]]. More strongly, the cube is 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 [[zonohedron]], a [[centrally symmetric]] polyhedron whose faces are [[Zonogon|centrally symmetric polygons]].{{r|shephard}} In the case of cube, it can be represented as the [[Cell (geometry)|cell]]. Some honeycombs have cubes as the only cells; one example is [[cubic honeycomb]], the only regular honeycomb in Euclidean three-dimensional space, having four cubes around each edge.{{r|twelveessay|ns}}
-=== Miscellaneous ===
+=== 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}}Compound of cubes is the [[polyhedral compound]]s in which the cubes share the same centre. They belong to the [[uniform polyhedron compound]], 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 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}}
+{{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 by the [[Arc (geometry)|arc]] of [[great circle]]s, creating bounds as the edges of a [[spherical polygon|spherical square]].{{r|yackel}}
+{{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 and each edge.{{r|popko|fuller}} Its dual is the [[spherical octahedron]].{{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 ==
@@ Line 170: / Line 244: @@
 * [[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 ==
-{{reflist|refs=
+<references>
+<ref name=am>{{cite book
-<ref name=alexandrov>{{cite book
+  | last1 = Akiyama | first1 = Jin | author1-link = Jin Akiyama
-  | last = Alexandrov | first = A. D. | author-link = Aleksandr Danilovich Aleksandrov
+ | last2 = Matsunaga | first2 = Kiyoko
-  | contribution = 8.1 Parallelohedra
+  | contribution = 15.3 Hilbert's Third Problem and Dehn Theorem
-  | contribution-url = https://books.google.com/books?id=R9vPatr5aqYC&pg=PA349
+  | doi = 10.1007/978-4-431-55843-9
-  | pages = 349–359
+ | isbn = 9784431558415
-  | publisher = Springer
+ | mr = 3380801
- | title = Convex Polyhedra | title-link = Convex Polyhedra (book)
+  | pages = 382–388
-  | year = 2005
+  | publisher = Springer Tokyo | title = Treks Into Intuitive Geometry
+ | title-link = Treks Into Intuitive Geometry
+  | year = 2015| doi-access = free
 }}</ref>
@@ Line 194: / Line 275: @@
   | url = https://books.google.com/books?id=FEl2CgAAQBAJ&pg=PA85
   | page = 85
-  | isbn = 978-1-61444-216-5
+  | isbn = 9781614442165
   | volume = 50
 }}</ref>
@@ Line 205: / Line 286: @@
   | url = https://books.google.com/books?id=1qRvTI_oWUAC&pg=PA382
   | publisher = Springer
-  | isbn = 978-1-85233-259-4
+  | isbn = 9781852332594
 }}</ref>
@@ Line 212: / Line 293: @@
   | year = 2010
   | editor-last = Gowers | editor-first = Timothy | editor-link = Timothy Gowers
-  | contribution = High-dimensional geometry and its probabilistic analogues
+  | 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
@@ Line 227: / Line 308: @@
   | page = 82
   | publisher = Springer
-  | isbn = 978-3-642-30964-9
+  | isbn = 9783642309649
   | 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=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
@@ Line 247: / Line 336: @@
   | publisher = Springer
   | doi = 10.1007/978-1-4939-0781-6
-  | isbn = 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=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
@@ Line 260: / Line 383: @@
   | page = 197
   | publisher = [[John Wiley & Sons]]
-| isbn = 978-0-470-53492-2
+ | isbn = 9780470534922
- }}</ref>
+}}</ref>
 <ref name=cc>{{cite journal
@@ Line 271: / Line 394: @@
   | mr = 157282
   | pages = 946–951
-  | title = Polar zonohedra
+  | title = Polar Zonohedra
   | volume = 70
   | year = 1963
-| issue = 9
+ | issue = 9
-  }}</ref>
+}}</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
@@ Line 313: / Line 449: @@
   | last2 = Pellicer | first2 = Daniel
   | year = 2024
-  | title = Finite 3-orbit polyhedra in ordinary space, II
+  | 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
@@ Line 324: / Line 461: @@
   | last2 = Rollett | first2 = A.P.
   | edition = 2nd
- | location = Oxford
   | mr = 0124167
   | publisher = Clarendon Press
-  | title = Mathematical models
+  | title = Mathematical Models
   | title-link = Mathematical Models (Cundy and Rollett)
   | year = 1961
@@ Line 341: / Line 477: @@
   | url = https://archive.org/details/polyhedra0000crom/page/309
   | page = 309
-  | isbn = 978-0-521-55432-9
+  | isbn = 9780521554329
-}}</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>
@@ Line 362: / Line 488: @@
   | url = https://books.google.com/books?id=zA_CAgAAQBAJ&pg=PA25
   | page = 25
-| isbn = 978-0-486-29730-9
+ | isbn = 9780486297309
-  }}</ref>
+}}</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
@@ Line 372: / Line 531: @@
   | mr = 1703597
   | pages = 527–549
-  | title = Zonotopes, dicings, and Voronoi's conjecture on parallelohedra
+  | 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
+  }}. 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
@@ Line 396: / Line 555: @@
   | url = https://books.google.com/books?id=LgeP62-ZxikC&pg=PA62
   | page = 62
-  | isbn = 978-1-61444-509-8
+  | isbn = 9781614445098
 }}</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=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
@@ Line 416: / Line 585: @@
   | 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
+  | url = https://web.archive.org/web/20180304110822id_/http://www.ams.org:80/distribution/mmj/vol7-2-2007/fuchs.pdf
-| doi-broken-date = 9 June 2025
+}} See Figure 11, p. 273, for showing three types of cube's geodesics.</ref>
- }}</ref>
 <ref name=fuller>{{cite book
@@ Line 427: / Line 595: @@
   | url = https://books.google.com/books?id=AKDgDQAAQBAJ&pg=PA173
   | page = 173
-| isbn = 978-0-02-065320-2
+ | isbn = 9780020653202
-  }}</ref>
+}}</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 | s2cid = 11301955 }}</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
@@ Line 436: / Line 623: @@
   | 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
+  | isbn = 9783540711322
   | mr = 2335496
   | pages = 280–291
@@ Line 457: / Line 644: @@
 <ref name=grunbaum-2003>{{cite book
   | last = Grünbaum | first = Branko | author-link = Branko Grünbaum
-  | contribution = 13.1 Steinitz's theorem
+  | contribution = 13.1 Steinitz's Theorem
   | edition = 2nd
-  | isbn = 0-387-40409-0
+  | isbn = 0387404090
   | pages = 235–244
   | publisher = Springer-Verlag
@@ Line 467: / Line 654: @@
   | 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>
@@ Line 480: / Line 678: @@
   | year = 1893
   | jstor = 2369565
-  | title = The projection of fourfold figures on a three-flat
+  | 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>
+ | doi = 10.2307/2369565
+}}</ref>
 <ref name=hart>{{cite conference
@@ Line 496: / Line 695: @@
   | contribution = Max Brücknerʼs Wunderkammer of Paper Polyhedra
   | contribution-url = https://archive.bridgesmathart.org/2019/bridges2019-59.pdf
-  | isbn =  978-1-938664-30-4
+  | isbn =  9781938664304
   | publisher = Tessellations Publishing, [[Phoenix, Arizona]]
 }}</ref>
@@ Line 510: / Line 709: @@
 }}</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=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
+  | title = A Survey of the Theory of Hypercube Graphs
   | journal = Computers & Mathematics with Applications
   | volume = 15
@@ Line 546: / Line 753: @@
   | publisher = Springer
   | doi = 10.1007/978-3-030-35110-6
-  | isbn = 978-3-030-35110-6
+  | isbn = 9783030351106
 }}</ref>
@@ Line 555: / Line 762: @@
   | publisher = [[Springer Science+Business Media|Springer]]
   | url = https://books.google.com/books?id=zXwQGo8jyHUC
-  | isbn = 978-3-642-14441-7
+  | 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>
@@ Line 567: / Line 785: @@
   | mr = 2610282
   | pages = 1783–1792
-  | title = Products of unit distance graphs
+  | title = Products of Unit Distance Graphs
   | volume = 310
   | year = 2010
@@ Line 580: / Line 798: @@
 }}</ref>
-<ref name=hs>{{cite book
+<ref name=husain>{{cite journal
-  | last1 = Herrmann | first1 = Diane L.
+  | last = Husain | first = Sakhlaq
-  | last2 = Sally | first2 = Paul J.
+  | journal = Annals of Communications in Mathematics
-  | year = 2013
+  | volume = 8 | issue = 2 | year = 2025 | pages = 293&ndash;298
- | title = Number, Shape, & Symmetry: An Introduction to Number Theory, Geometry, and Group Theory
+  | doi = 10.62072/acm.2025.080211
- | publisher = Taylor & Francis
+  | issn = 2582-0818
- | isbn = 978-1-4665-5464-1
+  | title = On Jerusalem Square Fractal, Its Construction and Properties
-  | url = https://books.google.com/books?id=b2fjR81h6yEC&pg=PA252
+| doi-access = free}}</ref>
- | 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
@@ Line 617: / Line 824: @@
   | mr = 0226494
   | pages = 90–96
-  | title = Orthogonal icosahedra
+  | title = Orthogonal Icosahedra
   | volume = 15
   | year = 1967
@@ Line 635: / Line 842: @@
   | last = Johnson | first = Norman W. | authorlink = Norman W. Johnson
   | year = 1966
-  | title = Convex polyhedra with regular faces
+  | title = Convex Polyhedra with Regular Faces
   | journal = [[Canadian Journal of Mathematics]]
   | volume = 18
@@ Line 652: / Line 859: @@
   | page = 76
   | edition = 2nd
-  | isbn = 978-0-8018-9012-3
+  | isbn = 9780801890123
   | publisher = The Johns Hopkins University Press
 }}</ref>
-<ref name=jwy>{{cite journal
+<ref name=jw>{{cite journal
   | last1 = Jerrard | first1 = Richard P.
   | last2 = Wetzel | first2 = John E.
-  | last3 = Yuan | first3 = Liping
+  | doi = 10.2307/4145012
-  | title = Platonic passages
+  | issue = 1
-  | journal = [[Mathematics Magazine]]
+  | journal = The American Mathematical Monthly
-  | date = April 2017
+  | mr = 2026310
-  | volume = 90 | issue = 2 | pages = 87–98
+  | pages = 22–31
-  | publisher = [[Mathematical Association of America]]
+  | title = Prince Rupert's rectangles
-  | location = Washington, DC
+  | volume = 111
-  | doi = 10.4169/math.mag.90.2.87
+  | year = 2004| jstor = 4145012
- | s2cid = 218542147
 }}</ref>
@@ Line 676: / Line 882: @@
   | url = https://books.google.com/books?id=KmL1uuiMyFUC&pg=PA16
   | page = 16
-| publisher = Springer
+ | publisher = Springer
-  | isbn = 978-0-387-98979-2
+  | isbn = 9780387989792
- }}</ref>
+}}</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=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
@@ Line 690: / Line 905: @@
   | publisher = [[Pearson Education]]
   | edition = 2nd
-  | isbn = 978-81-317-1682-3
+  | isbn = 9788131716823
 }}</ref>
@@ Line 701: / Line 916: @@
   | year = 2015
   | doi = 10.1007/978-3-319-25859-1
-  | isbn = 978-3-319-25859-1
+  | isbn = 9783319258591
 }}</ref>
@@ Line 722: / Line 937: @@
 <ref name=kozachok>{{cite conference
   | last = Kozachok | first = Marina
-  | contribution = Perfect prismatoids and the conjecture concerning with face numbers of centrally symmetric polytopes
+  | 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)
+  | 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>
@@ Line 740: / Line 979: @@
   | url = https://books.google.com/books?id=_4C7oid1kQQC&pg=RA7-PA41
   | page = 41
-| isbn = 978-0-08-096529-1
+| isbn = 9780080965291
   }}</ref>
@@ Line 751: / Line 990: @@
   | year = 2003
   | publisher = [[Random House|Broadway Books]]
- | location = New York City
+  | isbn = 9780767908160
-  | isbn = 978-0-7679-0816-0
   | page = 147
 }}</ref>
@@ Line 764: / Line 1,002: @@
   | mr = 1318797
   | pages = 573–583
-  | title = Polytopes that fill <math>\mathbb{R}^n</math> and scissors congruence
+  | title = Polytopes that Fill <math>\mathbb{R}^n</math> and Scissors Congruence
   | volume = 13
-  | year = 1995| doi-access = free
+  | year = 1995
+ | doi-access = free
 }}</ref>
@@ Line 772: / Line 1,011: @@
   | last1 = Langer | first1 = Joel C.
   | last2 = Singer | first2 = David A.
-  | journal = [[Milan Journal of Mathematics]]
+  | 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
@@ Line 788: / Line 1,027: @@
   | year = 1972
   | pages = 101–108
-  | isbn = 978-1-48325-512-5
+  | isbn = 9781483255125
   | url = https://books.google.com/books?id=ja7iBQAAQBAJ&pg=PA101
 }}</ref>
@@ Line 801: / Line 1,040: @@
   | volume = 52 | issue = 1 | pages = 4–37
   | doi = 10.1111/j.1600-0498.2009.00160.x
-}}</ref>
+| url-access = subscription
+ }}</ref>
 <ref name=ly>{{cite book
@@ Line 811: / Line 1,051: @@
   | page = 273
   | publisher = Taylor & Francis
-| isbn = 978-1-58488-505-4
+ | isbn = 9781584885054
- }}</ref>
+}}</ref>
 <ref name=marar>{{cite book
@@ Line 822: / Line 1,062: @@
   | publisher = Springer
   | doi = 10.1007/978-3-031-07442-4
-  | isbn = 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
+  | title = Renaissance Mathematics and Architectural Proportion in Alberti's De re aedificatoria
   |journal=Architectural Research Quarterly
   |date=1996
@@ Line 847: / Line 1,087: @@
   | last = McLean | first = K. Robin
   | year = 1990
-  | title = Dungeons, dragons, and dice
+  | title = Dungeons, Dragons, and Dice
   | journal = [[The Mathematical Gazette]]
   | volume = 74 | issue = 469 | pages = 243–256
@@ Line 854: / Line 1,094: @@
   | 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
@@ Line 872: / Line 1,122: @@
   | publisher = Heinemann
   | url = https://books.google.com/books?id=dvFfTAR6XwEC&pg=PA16
-  | isbn = 978-0-435-02474-1
+  | isbn = 9780435024741
   | page = 16
 }}</ref>
@@ Line 879: / Line 1,129: @@
   | last1 = Nelson | first1 = Roice
   | last2 = Segerman | first2 = Henry
-  | title = Visualizing hyperbolic honeycombs
+  | 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
+ | arxiv = 1511.02851
- }}</ref>
+}}</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=padmanabhan>{{cite book
+ | last = Padmanabhan | first = Thanu
-<ref name=poo-sung>{{cite journal
+ | title = Sleeping Beauties in Theoretical Physics
-  | last = Poo-Sung
+ | chapter = The Grand Cube of Theoretical Physics
- | first = Park, Poo-Sung
+ | publisher = Springer
- | year = 2016
+ | year = 2015 | pages = 1–8
- | title = Regular polytope distances
+  | isbn = 9783319134420
- | journal = [[Forum Geometricorum]]
+}}</ref>
- | 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>
@@ Line 914: / Line 1,156: @@
   | page = 21
   | publisher = Springer
-  | isbn = 978-0-8176-8363-4
+  | isbn = 9780817683634
   | doi = 10.1007/978-0-8176-8364-1
 }}</ref>
 <ref name="pucc">{{cite conference
-  | contribution = Polycube unfoldings satisfying Conway's criterion
+  | contribution = Polycube Unfoldings Satisfying Conway's Criterion
   | last1 = Langerman | first1 = Stefan | author1-link = Stefan Langerman
   | last2 = Winslow | first2 = Andrew
@@ Line 925: / Line 1,167: @@
   | 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>
@@ Line 937: / Line 1,199: @@
   | publisher = Hindustan Book Agency
   | page = 84&ndash;89
-  | isbn = 978-93-86279-06-4
+  | 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
@@ Line 948: / Line 1,229: @@
   | page = 25
   | publisher = [[Cambridge University Press]]
-  | doi = 10.1007/9781316466919
+  | doi = 10.1017/9781316466919
- | doi-broken-date = 9 June 2025
   | isbn = 9781316466919
 }}</ref>
@@ Line 960: / Line 1,240: @@
   | url = https://books.google.com/books?id=xgOtV4Nynd0C&pg=PA77
   | page = 77
-  | isbn = 978-0-8135-3890-7
+  | isbn = 9780813538907
   | publisher = Rutgers University Press
 }}</ref>
@@ Line 981: / Line 1,261: @@
   | mr = 365332
   | pages = 261–269
-  | title = Space-filling zonotopes
+  | title = Space-filling Zonotopes
   | volume = 21
   | year = 1974| issue = 2
@@ Line 995: / Line 1,275: @@
   | page = 392
   | publisher = [[John Wiley & Sons]]
-| isbn = 978-1-118-03103-2
+| isbn = 9781118031032
   }}</ref>
@@ Line 1,010: / Line 1,290: @@
   | 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
@@ Line 1,044: / Line 1,308: @@
   | page = 45
   | publisher = [[ASM International (society)|ASM International]]
- | location = [[Materials Park, Ohio]]
+  | isbn = 9781615032419
-  | isbn = 978-1-61503-241-9
 }}</ref>
-<ref name=trudeau>{{cite book
+<ref name="turney">{{cite journal
-  | last = Trudeau | first = Richard J.
+  | last = Turney | first = Peter D.
-  | title = Introduction to Graph Theory
+  | year = 1984–1985
-  | year = 1976
+  | title = Unfolding the Tesseract
-  | isbn = 978-0-486-67870-2
+  | journal = [[Journal of Recreational Mathematics]]
-  | publisher = Dover Publications
+  | volume = 17 | issue = 1
-  | url = https://books.google.com/books?id=eRLEAgAAQBAJ&pg=PA122
+  | url = https://unfolding.apperceptual.com/
- | page = 122
 }}</ref>
@@ Line 1,065: / Line 1,327: @@
   | page = 167
   | publisher = [[Dover Publications]]
-  | isbn = 978-0-486-40919-1
+  | isbn = 9780486409191
 }} 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=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
@@ Line 1,084: / Line 1,355: @@
   | doi = 10.1007/978-3-319-95588-9
   | page = 1123
-| isbn = 978-3-319-95587-2
+| isbn = 9783319955872
   }} See Fig. 6.</ref>
@@ Line 1,096: / Line 1,367: @@
   | url = https://books.google.com/books?id=nVx-tu596twC&pg=PA50
   | page = 50
-  | isbn = 978-3-642-01898-5
+  | isbn = 9783642018985
   | doi = 10.1007/978-3-642-01899-2
 }}</ref>
@@ Line 1,110: / Line 1,381: @@
   | contribution-url = https://archive.bridgesmathart.org/2009/bridges2009-123.pdf
   | pages = 123&ndash;130
-  | isbn = 978-0-96652-020-0
+  | isbn = 9780966520200
 }}</ref>
@@ Line 1,128: / Line 1,399: @@
   | last = Ziegler | first = Günter M. | author-link = Günter M. Ziegler
   | contribution = Chapter 4: Steinitz' Theorem for 3-Polytopes
-  | isbn = 0-387-94365-X
+  | isbn = 038794365X
   | pages = 103–126
   | publisher = Springer-Verlag
@@ Line 1,136: / Line 1,407: @@
   | year = 1995
 }}</ref>
+</references>
-}}
 ==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}}
+{{Convex polyhedron navigator}}
 {{Authority control}}

Cube: Difference between revisions

Latest revision as of 13:10, 18 November 2025

Properties

Measurement

Insphere, midsphere, circumsphere

Unit cube

Symmetry

Appearances

In popular cultures

In nature and science

In antiquity

Constructions

Representation

As a graph

In orthogonal projection

As a configuration matrix

Related topics

Construction of polyhedra

Polycubes

Space-filling

Miscellanea

See also

References

External links

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