Editing Cube (section)

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

=== Measurement and other metric properties ===
[[File:Cube diagonals.svg|thumb|upright=0.6|A face diagonal in red and space diagonal in blue]]
Given a cube with edge length <math> a </math>. The [[face diagonal]] of 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}}
<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}}
<math display="block"> V = a^3. </math>

[[File:Prince Ruperts cube.png|thumb|upright=0.6|[[Prince Rupert's cube]]]]
One special case is the [[unit cube]], so named for measuring a single [[unit of length]] along each edge. It follows that each face is a [[unit square]] and that the entire figure has a volume of 1 cubic unit.{{r|ball|hr-w}} [[Prince Rupert's cube]], named after [[Prince Rupert of the Rhine]], is the largest cube that can pass through a hole cut into the unit cube, despite having sides approximately 6% longer.{{r|sriraman}} A polyhedron that can pass through a copy of itself of the same size or smaller is said to have the [[Rupert property]].{{r|jwy}} A geometric problem of [[doubling the cube]]&mdash;alternatively known as the ''Delian problem''&mdash;requires the construction of a cube with a volume twice the original by using 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}}

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

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

=== Symmetry ===
The cube has [[octahedral symmetry]] <math> \mathrm{O}_\mathrm{h} </math>. 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]]
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}}
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