Cubic

Geometry

The cube is one of the Five solids of Plato. A cube belongs to the family of the right prisms. It has 8 tops and 12 edges. Moreover:

Two edges having a common end are orthogonal.
the opposite faces are parallel. The adjacent faces are perpendicular
All the plane angles are right.
the diagonals are intersected in a single point, the center of symmetry of the cube, the isobarycentre of the eight tops.

But by definition its edges are very equal length, say has . Its faces are thus Carré S, of the same surface, equalizes with has ². In fact:

Its surface is worth thus 6 × has ².
Its Volume is worth has ³.
the length of a diagonal is worth $a \ sqrt {3}$ .
the circumscribed sphere thus has as a ray $\ frac a2 \ sqrt {3}$ .
the tangent sphere with the edges has as a ray $\ frac a2 \ sqrt {2}$ .
the registered sphere has as a ray $\ frac a2$ .
the angle between the diagonal and each plan adjacent is worth $arctan (\ frac 1 \ sqrt {2}) \ simeq 35.26^ {^o}$

It is the expression of its volume which led to the use of the word cubic in Algèbre.

Other definitions

There exist other equivalent definitions of the cube:

the cubes are the only polyhedrons of which all the faces are square.
the cube is a Antidiamant of order 3 at regular tops and equal plane angles.

Group isométries

The cube is one of the polyhedrons offering the most symmetries. Listels them:

3 axes of rotation of order 4: axes passing by the center of two opposite faces.
6 axes of rotations of order 2: axes passing by the medium of two opposite edges
4 axes of rotation of order 3: axes passing by two opposite tops
the symmetry of center O
9 symmetry planes: 3 mediators plans of the edges, 6 plans passing by two opposite edges.

A isometry of the cube is entirely defined by the image of a top and the three edges resulting from this top (reference mark of space). This top can have as an image any of the 8 tops of the cube. The first edge has 3 possible images then, the second edge two images only and the image of the last edge is then given. This proves that the isométries leaving the cube overall invariant are 8 × 3 × 2 = 48. These isométries are divided in 24 positive isométries and 24 negative isométries. The positive isométries have all the point O like invariant: one then counts 23 rotations plus the identity.

The preceding axes of rotations then are found:

3 axes of rotation generating 3 rotations of angle not no one, either 9 rotations
6 axes of rotation generating 1 rotation of flat angle, or 6 rotations
4 axes of rotation generating 2 rotations of angle not no one, or 8 rotations.

as well as 9 symmetries compared to a plan generated by 9 rotations of flat angle made up with symmetry of center O.

What proves that the inventory was quite exhaustive.

Owners

There exist many owners of the cube, eleven different to be precise, here are examples:

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