Cube

The cube ( throw of German because he is thrown into dice games, also regular hexahedron [ hɛksae ː dər ], from Greek hexáedron, hexahedron ', or cube, from Latin cubus, dice ') is one of the five Platonic solids, specifically a ( three-dimensional ) polyhedron ( polyhedron ) with

  • Six ( congruent ) squares as boundary surfaces
  • Twelve ( equally divided) edges and
  • Eight corners in which meet three boundary surfaces

The cube is a special ( three-dimensional ) parallelepiped ( parallelepiped ), a special (namely equilateral ) cuboid and a special straight square prism. The size of a cube is already determined by the specification of the edge length.

Symmetry

Due to its high symmetry - all corners, edges and sides are equal to each other like - the cube is a regular polytope. He has

  • Three four-fold axes of rotation ( through the center points of two opposite faces),
  • Four threefold rotational axes ( opposite diagonal corners by two )
  • Six -fold rotation axes ( opposed by the center points of two diagonal edges) and
  • Nine mirror planes ( six levels by four corners, three levels by four edge midpoints )
  • 14 rotary reflections ( six to 90 ° with the planes by four edges midpoints and 8 to 60 ° with planes through six edges centers)

And is

  • Point symmetry ( the center ).

For a fourfold rotation axis, there are three symmetry operations ( rotation by 90 °, 180 ° and 270 ° ), for a three-fold axis of rotation accordingly 2 symmetry operations. Overall, the symmetry group of the cube has 48 elements. They are designated in the notation of Schoenflies as, in the notation of Hermann / Mauguin than or commonly but somewhat inaccurately as octahedral or cube group.

Relations with other polyhedra

The cube is the dual polyhedron to an octahedron (and vice versa). Moreover, the vertices of the cube describe two point-symmetric regular tetrahedron, which together form the star tetrahedron as other regular solids.

Using cube and octahedron numerous body can be constructed which also have the cube group as a symmetry group. Thus, for example, receives

  • The Hexaederstumpf or the truncated cube with 6 octagons and 8 triangles
  • The cuboctahedron with 6 squares and 8 triangles, ie 14 pages, and 12 corners
  • The truncated octahedron or truncated octahedron with the 6 squares and 8 hexagons

As averages of a cube with an octahedron (see Archimedean bodies ) and

  • The rhombic dodecahedron with 6 8 = 14 vertices and 12 diamonds as side

As a convex hull of a union of a cube with one octahedron.

The cube is a component of the regular Würfelparkettierung.

Formulas

Generalization

The analogues of the cube are n in arbitrary dimension referred to as the (n- dimensional ) cube ( or hypercube ) and are also regular polytopes. The n-dimensional cube having sides of the limiting dimension k special cases:

  • The zero-dimensional cube ( point) has 1 corner.
  • The one-dimensional cube (path) has 2 vertices and 1 edge.
  • The two-dimensional cube ( square) has 4 corners, 4 edges and 1 face
  • The four-dimensional hypercube ( tesseract ) has 16 vertices, 32 edges, 24 squares and page 8 page cubes.
  • The n-dimensional cube has corners ( k = 0), edges ( k = 1), surfaces, (k = 2 ), volume ( k = 3) and (n-1 )-dimensional cube as k (k = n-1) -dimensional faces ( facets).

A model for the n- dimensional cube is the unit cube In in the vector space Rn. Namely, the closed unit cube

  • , The n-fold Cartesian product of the unit interval
  • The convex hull of the 2n vertices of the coordinates 0 and 1,
  • The average of the 2n half-spaces and

The unit cube is a parallel to the axis cube with an edge length of 1 and a corner at the origin. A generalization of this concept are parallelepiped in Rn, which play a role in the multivariate analysis.

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