Cube
A cube is a three-dimensional solid object in geometry. It 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. It is a special case of a cuboid, a parallelepiped, and a rhombohedron in which all six quadrilateral faces are squares. It is a three-dimensional hypercube, a family of polytopes that also includes the two-dimensional square and four-dimensional tesseract.
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 can be represented in many ways. One of them is by drawing a graph with vertices connected with an edge in a plane. Such a graph is called the cubical graph, a special case of the hypercube graph.
The cube is the core of many polyhedra's construction or other geometrical shapes. For example, truncating a cube's vertices results in a truncated cube. Joining one or more cubes face-to-face yields a polycube, a three-dimensional version of polyominoes. Two or more cubes can have the same centre, forming polyhedral compounds. Cubes can form a honeycomb by attaching face-to-face, filling a space without leaving a gap.
Properties
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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.[1] 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.[2][3] More generally, the cube and the rectangular cuboid are special cases of a cuboid, a polyhedron with six quadrilaterals (four-sided polygons).[4] As for all convex polyhedra, the cube has Euler characteristic of 2, according to the formula ; the three letters denote respectively the number of vertices, edges, and faces.[5]
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.[6] 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;[7] a rhombohedron, as a special case of a parallelepiped with six rhombi faces, because the interior angle of all of the faces is right;[8] and a trigonal trapezohedron, a polyhedron with congruent quadrilateral faces, since its square faces are the special cases of rhombi.[9]
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.[10]
Measurement
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