The compound of four cubes or Bakos compound is a
face-transitive
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its Face (geometry), faces are the same. More specifically, all faces must be not ...
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 connecte ...
of four
cube
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
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
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
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.
Its vertices are those of a regular dodecahedron. Its edges form pentagrams, which are the stellations of the pentag ...
(related to the
dodecahedron
In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Po ...
), into which it can be transformed by rotating the colored cubes on their 3-fold axes.
See also
*
Compound of three octahedra
In mathematics, the compound of three octahedra or octahedron 3-compound is a polyhedral compound formed from three octahedron, regular octahedra, all sharing a common center but rotated with respect to each other. Although appearing earlier in th ...
*
Compound of five octahedra
The compound of five octahedra is one of the five regular polyhedron compounds, and can also be seen as a stellation. It was first described by Edmund Hess in 1876. It is unique among the regular compounds for not having a regular convex hull.
...
*
Compound of ten octahedra
*
Compound of twenty octahedra
*
Compound of three cubes
*
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.
Its vertices are those of a regular dodecahedron. Its edges form pentagrams, which are the stellations of the pentag ...
*
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 t ...
References
Polyhedral compounds
{{polyhedron-stub