The cube can be represented in many ways, one of which is the graph known as thecubical graph. It can be constructed by using theCartesian product of graphs. The cube is the three-dimensionalhypercube, a family ofpolytopes also including the two-dimensional square and four-dimensionaltesseract. A cube withunit side length is the canonical unit ofvolume in three-dimensional space, relative to which other solid objects are measured. Other related figures involve the construction of polyhedra,space-filling andhoneycombs,polycubes, as well as cube in compounds, spherical, and topological space.
The cube was discovered in antiquity, associated with the nature ofearth byPlato, the founder of Platonic solid. It was used as a part of theSolar System, proposed byJohannes Kepler. It can be derived differently to create more polyhedrons, and it has applications to construct a newpolyhedron by attaching others. Other applications include popular culture of toys and games, arts, optical illusions, architectural buildings, as well as the natural science and technology.
A cube is a special case ofrectangular cuboid in which the edges are equal in length.[1] Like other cuboids, every face of a cube has four vertices, each of which connects with three congruent lines. These edges formsquare faces, making thedihedral angle of a cube between every two adjacent squares being theinterior angle of a square, 90°. Hence, the cube has six faces, twelve edges, and eight vertices.[2] Because of such properties, it is categorized as one of the fivePlatonic solids, apolyhedron in which all theregular polygons arecongruent and the same number of faces meet at each vertex.[3] Every three square faces surrounding a vertex isorthogonal each other, so the cube is classified asorthogonal polyhedron.[4] The cube may also be considered as theparallelepiped in which all of its edges are equal edges.[5]
Given a cube with edge length. Theface diagonal of a cube is thediagonal of a square, and thespace diagonal of a cube is a line connecting two vertices that is not in the same face, formulated as. Both formulas can be determined by usingPythagorean theorem. The surface area of a cube is six times the area of a square:[6]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 termcubic to mean raising any number to the third power:[7][6]
One special case is theunit cube, so named for measuring a singleunit of length along each edge. It follows that each face is aunit square and that the entire figure has a volume of 1 cubic unit.[8][9]Prince Rupert's cube, named afterPrince 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.[10] A polyhedron that can pass through a copy of itself of the same size or smaller is said to have theRupert property.[11] A geometric problem ofdoubling the cube—alternatively known as theDelian problem—requires the construction of a cube with a volume twice the original by using acompass and straightedge solely. Ancient mathematicians could not solve this old problem until French mathematicianPierre Wantzel in 1837 proved it was impossible.[12]
With edge length, theinscribed sphere of a cube is the sphere tangent to the faces of a cube at their centroids, with radius. Themidsphere of a cube is the sphere tangent to the edges of a cube, with radius. Thecircumscribed sphere of a cube is the sphere tangent to the vertices of a cube, with radius.[13]
For a cube whose circumscribed sphere has radius, and for a given point in its three-dimensional space with distances from the cube's eight vertices, it is:[14]
The cube hasoctahedral symmetry. It is composed ofreflection 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 ofrotational symmetry, a symmetry by rotating it around the axis, from which the appearance is interchangeable. It has octahedral rotation symmetry: 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°).[15][16][17]
The dual polyhedron of a cube is the regular octahedron
Thedual polyhedron can be obtained from each of the polyhedron's vertices tangent to a plane by the process known aspolar reciprocation.[18] One property of dual polyhedrons generally is that the polyhedron and its dual share theirthree-dimensional symmetry point group. In this case, the dual polyhedron of a cube is theregular octahedron, and both of these polyhedron has the same symmetry, the octahedral symmetry.[19]
The cube isface-transitive, meaning its two squares are alike and can be mapped by rotation and reflection.[20] It isvertex-transitive, meaning all of its vertices are equivalent and can be mappedisometrically under its symmetry.[21] It is alsoedge-transitive, meaning the same kind of faces surround each of its vertices in the same or reverse order, all two adjacent faces have the samedihedral angle. Therefore, the cube isregular polyhedron because it requires those properties.[22] Each vertex is surrounded by three squares, so the cube is byvertex configuration or inSchläfli symbol.[23]
ThePlatonic solid is a set of polyhedrons known since antiquity. It was named afterPlato in hisTimaeus dialogue, who attributed these solids to nature. One of them, the cube, represented theclassical element ofearth because of its stability.[43]Euclid'sElements defined the Platonic solids, including the cube, and using these solids with the problem involving to find the ratio of the circumscribed sphere's diameter to the edge length.[44] Following its attribution with nature by Plato,Johannes Kepler in hisHarmonices Mundi sketched each of the Platonic solids, one of them being a cube in which Kepler decorated a tree on it.[43] In hisMysterium Cosmographicum, Kepler also proposed theSolar System by using the Platonic solids setting into another one and separating them with six spheres resembling the six planets. The ordered solids started from the innermost to the outermost:regular octahedron,regular icosahedron,regular dodecahedron,regular tetrahedron, and cube.[45]
An elementary way to construct is using itsnet, 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.[46]
Inanalytic geometry, a cube may be constructed using theCartesian coordinate systems. For a cube centered at the origin, with edges parallel to the axes and with an edge length of 2, theCartesian coordinates of the vertices are.[47] Its interior consists of all points with for all. A cube's surface with center and edge length of is thelocus of all points such that
The cube isHanner polytope, because it can be constructed by usingCartesian product of three line segments. Its dual polyhedron, the regular octahedron, is constructed bydirect sum of three line segments.[48]
The cube may be regarded as two tetrahedra attached onto the bases of atriangular antiprism.[49]
According toSteinitz's theorem, thegraph can be represented as theskeleton of a polyhedron; roughly speaking, a framework of a polyhedron. Such a graph has two properties:planar (the edges of a graph are connected to every vertex without crossing other edges), and3-connected (whenever a graph with more than three vertices, and two of the vertices are removed, the edges remain connected).[50][51] The skeleton of a cube can be represented as the graph, and it is called thecubical graph, aPlatonic graph. It has the same number of vertices and edges as the cube, twelve vertices and eight edges.[52] The cubical graph is also classified as aprism graph, resembling the skeleton of a cuboid.[53]
The cubical graph is a special case ofhypercube graph or-cube—denoted as—because it can be constructed by using the operation known as theCartesian product of graphs: it involves two graphs connecting the pair of vertices with an edge to form a new graph.[54] In the case of the cubical graph, it is the product of two; 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.[55] As a part of the hypercube graph, it is also an example of aunit distance graph.[56]
An object illuminated by parallel rays of light casts a shadow on a plane perpendicular to those rays, called anorthogonal projection. A polyhedron is consideredequiprojective 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 aregular hexagon.[60]
The cube can be represented asconfiguration matrix. A configuration matrix is amatrix in which the rows and columns correspond to the elements of a polyhedron as in the vertices, edges, and faces. Thediagonal 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:[61]
The cube can appear in the construction of a polyhedron, and some of its types can be derived differently in the following:
Whenfaceting a cube, meaning removing part of the polygonal faces without creating new vertices of a cube, the resulting polyhedron is thestellated octahedron.[62]
The cube isnon-composite polyhedron, meaning it is a convex polyhedron that cannot be separated into two or more regular polyhedrons. The cube can be applied to construct a new convex polyhedron by attaching another.[63] Attaching asquare pyramid to each square face of a cube produces itsKleetope, a polyhedron known as thetetrakis hexahedron.[64] Suppose one and two equilateral square pyramids are attached to their square faces. In that case, they are the construction of anelongated square pyramid andelongated square bipyramid respectively, theJohnson solid's examples.[65]
Each of the cube's vertices can betruncated, and the resulting polyhedron is theArchimedean solid, thetruncated cube.[66] When its edges are truncated, it is arhombicuboctahedron.[67] 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.[68]
The corner region of a cube can also be truncated by a plane (e.g., spanned by the three neighboring vertices), resulting in atrirectangular tetrahedron.[69]
Thesnub 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 assnub.[70]
The cube can be constructed with sixsquare pyramids, tiling space by attaching their apices. In some cases, this produces therhombic dodecahedron circumscribing a cube.[71][72]
Polycube is a polyhedron in which the faces of many cubes are attached. Analogously, it can be interpreted as thepolyominoes in three-dimensional space.[73] When four cubes are stacked vertically, and the other four are attached to the second-from-top cube of the stack, the resulting polycube isDali cross, afterSalvador Dali. In addition to popular cultures, the Dali cross is a tile space polyhedron,[74][75] which can be represented as the net of atesseract. A tesseract is a cube analogous'four-dimensional space bounded by twenty-four squares and eight cubes.[76]
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 was, 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 inventionDehn invariant, answering that not all polyhedra can be reassembled into a cube.[77] It showed that two equal volume polyhedra should have the same Dehn invariant, except for the two tetrahedra whose Dehn invariants were different.[78]
The cube has a Dehn invariant of zero. This indicates the cube is applied forhoneycomb. More strongly, the cube is aspace-filling tile in three-dimensional space in which the construction begins by attaching a polyhedron onto its faces without leaving a gap.[79] The cube is aplesiohedron, a special kind of space-filling polyhedron that can be defined as theVoronoi cell of a symmetricDelone set.[80] The plesiohedra include theparallelohedrons, which can betranslated 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.[81] Every three-dimensional parallelohedron iszonohedron, acentrally symmetric polyhedron whose faces arecentrally symmetric polygons.[82] In the case of cube, it can be represented as thecell. Some honeycombs have cubes as the only cells; one example iscubic honeycomb, the only regular honeycomb in Euclidean three-dimensional space, having four cubes around each edge.[83][84]
Enumeration according toSkilling (1976): compound of six cubes with rotational freedom,three cubes, and five cubes
Compound of cubes is thepolyhedral compounds in which the cubes are sharing the same centre. They belong to theuniform polyhedron compound, meaning they are polyhedral compounds whose constituents are identical (although possiblyenantiomorphous)uniform polyhedra, in an arrangement that is also uniform. The list of compounds enumerated bySkilling (1976) in seventh to ninth uniform compound for the compound of six cubes with rotational freedom,three cubes, and five cubes respectively.[85] Two compounds, consisting oftwo and three cubes were found inEscher'swood engraving printStars andMax Brückner's bookVielecke und Vielflache.[86]
The spherical cube represents thespherical polyhedron, consisting of six spherical squares with 120° interior angle on each vertex. It hasvector equilibrium, meaning that the distance from the centroid and each vertex is the same as the distance from that and each edge.[87][88] Its dual is thespherical octahedron. The spherical cube can be modeled by thearc ofgreat circles, creating bounds as the edges of aspherical square.[89]
The topological objectthree-dimensional torus is a topological space defined to behomeomorphic to the Cartesian product of three circles. It can be represented as a three-dimensional model of the cube shape.[90]
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