Cuboctaèdre

A cuboctaèdre is a Polyèdre with 14 regular faces, of which eights are equilateral triangles and six are squares. It comprises

12 tops identical, each one uniting two triangles and two opposite squares two to two;
24 stop identical, each one common to a triangle and a square.

It is thus about a quasi-regular polyhedron, i.e. a Solide of Archimedes (uniformity of the tops) with moreover, one uniformity of the edges.

Its dual Polyèdre is the rhombic Dodécaèdre.

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Other names

Heptaparallèloèdre (Buckminster Fuller) ** Fuller applied the name " Dymaxion " and Vector Equilibrium for this form.
Cubic rectified or octahedral rectified - (Norman Johnson)

And tetrahedron bevelled by a lower symmetry.

Surface and volume

The surface $A$ and $V$ volume of a cuboctaèdre on side $a$ are given by

A= (6+2 \ sqrt {3}) a^2

V= \ begin {matrix} {5 \ over3} \ end {matrix} \ sqrt {2} a^3

Geometrical relations

A cuboctaèdre can be obtained by taking a Plane section adapted of a Polytope in cross with four dimensions.

A cuboctaèdre has an octahedral symmetry. Its first Stellation is the made up of a Cube and its dual, the Octaèdre, with the localized tops of the cuboctaèdre in the middle of the edges of the other.

The cuboctaèdre is a Cube rectified and also a rectified Octaèdre.

It is also a bevelled Tétraèdre. With this construction, one gives him the symbol of Wythoff: 3 3 | 2 .

A beveling of skew of a tetrahedron produces a solid with the faces parallel with those of the cuboctaèdre, i.e. eight triangles of two sizes and six rectangles. Whereas its edges are unequal, this solid remains with uniform tops : the solid has the symmetrical Groupe complete and its tops are equivalent with this group.

The edges of a cuboctaèdres form four Hexagone S regular. If the cuboctaèdre is cut in the plan of one of these hexagons, each half is a triangular Coupole, one of the solid of Johnson; the cuboctaèdre itself can thus be called the gyrobicoupole triangular, simplest of a series (other that the Gyrobifastigium or " gyrobicoupole digonale"). If the halves are replaced whole with a torsion, then these triangles meet the triangles and the squares meet the squares, the result is another solid of Johnson, the triangular Orthobicoupole.

Both bicoupoles triangular are important in the compact Empilement. The distance starting from the center of the solid towards its tops is equal to its length of edge. Each central Sphère can have with more the twelve neighbors, and in a cubic lattice with centered faces, those take the positions of the tops of a cuboctaèdre. In a network of hexagonal compact stacking, they correspond to the corners of the triangular orthobicoupoles. In both cases, the central sphere takes the position of the center of the solid.

The cuboctaèdres seem cells in three of the convex uniform honeycombs and in nine of the uniform polychores.

The volume of the cuboctaèdre is 5/6 of the cube circumscribed and 5/8 of octahedral circumscribed; they is 5/3 √ 2 times the cube length of an edge.

Cartesian coordinates

Cartesian coordinated of the tops of a cuboctaèdre (length of edge √ 2) centered in the beginning are

(± 1, ± 1,0)

(± 1,0, ± 1)

(0, ± 1, ± 1)

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