Cube

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.

Cube
Type	Hanner polytope, orthogonal polyhedron, parallelohedron, Platonic solid, plesiohedron, regular polyhedron, zonohedron
Faces	6 square
Edges	12
Vertices	8
Euler char.	2
Vertex configuration	${\displaystyle 8\times (4^{3})}$
Schläfli symbol	${\displaystyle \{4,3\}}$
Symmetry group	octahedral symmetry ${\displaystyle \mathrm {O} _{\mathrm {h} }}$
Volume	side^3
Dihedral angle (degrees)	90°
Dual polyhedron	regular octahedron
Properties	convex, edge-transitive, face-transitive, non-composite, orthogonal faces, Rupert property: can pass through a hole with its copy, vertex-transitive

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

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. 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. More generally, the cube and the rectangular cuboid are special cases of a cuboid, a polyhedron with six quadrilaterals (four-sided polygons). As for all convex polyhedra, the cube has Euler characteristic of 2, according to the formula ${\displaystyle V-E+F=2}$; the three letters denote respectively the number of vertices, edges, and faces.

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. 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; a rhombohedron, as a special case of a parallelepiped with six rhombi faces, because the interior angle of all of the faces is right; and a trigonal trapezohedron, a polyhedron with congruent quadrilateral faces, since its square faces are the special cases of rhombi.

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.

Measurement

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

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 ${\displaystyle 3{\sqrt {2}}}$ times that of an edge. The third type is a non-planar hexagon.

Insphere, midsphere, circumsphere

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

Unit 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. 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. One way to obtain this result is by using the Pythagorean theorem or the formula for Euclidean distance in three-dimensional space.

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, ${\displaystyle {\sqrt[{3}]{2}}}$—is not constructible. However, this problem was solved with folding an origami paper by Messer (1986).

Symmetry

The cube has octahedral symmetry ${\displaystyle \mathrm {O} _{\mathrm {h} }}$ 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°).

The dual polyhedron can be obtained from each of the polyhedra's vertices tangent to a plane by a process known as polar reciprocation. 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.

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

Appearances

In popular cultures

A six-sided die

A completed Skewb

The Alamo sculpture