The compound of four cubes or Bakos compound is a
face-transitive polyhedron compound
In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram.
The outer vertices of a compound can be connected ...
of four
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only r ...
s with
octahedral symmetry
A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedr ...
. It is the dual of the
compound of four octahedra
The compound of four octahedra is a uniform polyhedron compound. It's composed of a symmetric arrangement of 4 octahedron, octahedra, considered as triangular antiprisms. It can be constructed by superimposing four identical octahedra, and then r ...
. Its surface area is 687/77 square lengths of the edge.
Its
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
are (±3, ±3, ±3) and the permutations of (±5, ±1, ±1).
Extension with fifth cube
The eight vertices on the 3-fold symmetry axes can be seen as the vertices of a fifth cube of the same size.
[The Wolfram pag]
Cube 5-Compound
shows a small picture of this extension under the name "first cube 4-compound". Also Grant Sanderson has used a picture of it to illustrate the term ''symmetry''.
Referring to the images below, the four old cubes are called colored, and the new one black.
Each colored cube has two opposite vertices on a 3-fold symmetry axis, which are shared with the black cube. (In the picture both 3-fold vertices of the green cube are visible.) The remaining six vertices of each colored cube correspond to the faces of the black cube. This compound shares these properties with the
compound of five cubes
The compound of five cubes is one of the five regular polyhedral compounds. It was first described by Edmund Hess in 1876.
It is one of five regular compounds, and dual to the compound of five octahedra. It can be seen as a faceting of a regula ...
(related to the
dodecahedron), into which it can be transformed by rotating the colored cubes on their 3-fold axes.
See also
*
Compound of three octahedra
*
Compound of five octahedra
*
Compound of ten octahedra
*
Compound of twenty octahedra
*
Compound of three cubes
In geometry, the compound of three cubes is a uniform polyhedron compound formed from three cubes arranged with octahedral symmetry. It has been depicted in works by Max Brückner and M.C. Escher.
History
This compound appears in Max Brückner's ...
*
Compound of five cubes
The compound of five cubes is one of the five regular polyhedral compounds. It was first described by Edmund Hess in 1876.
It is one of five regular compounds, and dual to the compound of five octahedra. It can be seen as a faceting of a regula ...
*
Compound of six cubes
*
Uniform polyhedron compound
In geometry, a uniform polyhedron compound is a polyhedral compound whose constituents are identical (although possibly enantiomorphous) uniform polyhedra, in an arrangement that is also uniform, i.e. the symmetry group of the compound acts tran ...
References
Polyhedral compounds
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