Cube
Construction

An elementary way to construct a cube is using its 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.

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

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.
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