The compound of five

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 is one of the five regular polyhedral compounds. It was first described by

Edmund Hess Edmund Hess (17 February 1843 – 24 December 1903) was a German mathematician who discovered several regular polytopes. See also * Schläfli–Hess polychoron * Hess polytope References * ''Regular Polytopes In mathematics, a regu ...

in 1876. It is one of five regular compounds, and dual to the compound of five octahedra. It can be seen as a

faceting Stella octangula as a faceting of the cube In geometry, faceting (also spelled facetting) is the process of removing parts of a polygon, polyhedron or polytope, without creating any new Vertex (geometry), vertices. New edges of a faceted pol ...

of a regular dodecahedron. It is one of the stellations of the rhombic triacontahedron. It has icosahedral symmetry (I_h).

Geometry

The compound is a faceting of a dodecahedron (where pentagrams can be seen correlating to the pentagonal faces). Each cube represents a selection of 8 of the 20 vertices of the dodecahedron. If the shape is considered as a union of five cubes yielding a simple nonconvex solid without self-intersecting surfaces, then it has 360 faces (all triangles), 182 vertices (60 with degree 3, 30 with degree 4, 12 with degree 5, 60 with degree 8, and 20 with degree 12), and 540 edges, yielding an

Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...

of 182 − 540 + 360 = 2.

Edge arrangement

Its

convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...

is a regular dodecahedron. It additionally shares its edge arrangement with the

small ditrigonal icosidodecahedron In geometry, the small ditrigonal icosidodecahedron (or small ditrigonary icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U30. It has 32 faces (20 triangles and 12 pentagrams), 60 edges, and 20 vertices. It has extended Schläfl ...

, the

great ditrigonal icosidodecahedron In geometry, the great ditrigonal icosidodecahedron (or great ditrigonary icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U47. It has 32 faces (20 triangles and 12 pentagons), 60 edges, and 20 vertices. It has 4 Schwarz triangle ...

, and the

ditrigonal dodecadodecahedron In geometry, the ditrigonal dodecadodecahedron (or ditrigonary dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U41. It has 24 faces (12 pentagons and 12 pentagrams), 60 edges, and 20 vertices. It has extended Schläfli symbol ...

. With these, it can form polyhedral compounds that can also be considered as degenerate uniform star polyhedra; the

small complex rhombicosidodecahedron In geometry, the small complex rhombicosidodecahedron (also known as the small complex ditrigonal rhombicosidodecahedron) is a degenerate uniform star polyhedron. It has 62 faces (20 triangles, 12 pentagrams and 30 squares), 120 (doubled) edge ...

great complex rhombicosidodecahedron In geometry, the small complex rhombicosidodecahedron (also known as the small complex ditrigonal rhombicosidodecahedron) is a degenerate uniform star polyhedron. It has 62 faces (20 triangles, 12 pentagrams and 30 squares), 120 (doubled) edge ...

and

complex rhombidodecadodecahedron Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with eac ...

. The

compound of ten tetrahedra The compound of ten tetrahedra is one of the five regular polyhedral compounds. This polyhedron can be seen as either a stellation of the icosahedron or a compound. This compound was first described by Edmund Hess in 1876. It can be seen as a f ...

can be formed by taking each of these five

s and replacing them with the two

tetrahedra In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...

of the stella octangula (which share the same vertex arrangement of a cube).

As a stellation

References

*. p 360 *. *. * Cundy, H. and Rollett, A. "Five Cubes in a Dodecahedron." §3.10.6 in ''

Mathematical Models A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, b ...

'', 3rd ed. Stradbroke, England: Tarquin Pub., pp. 135–136, 1989. *

H.S.M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...

, '' Regular Polytopes'', (3rd edition, 1973), Dover edition, , 3.6 ''The five regular compounds'', pp.47-50, 6.2 ''Stellating the Platonic solids'', pp.96-104

External links

MathWorld: Cube 5-Compound
*

* VRML model

* Polyhedral stellation Polyhedral compounds {{polyhedron-stub