Editing Cube (section)

=== 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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