Compound of three cubes
Type	Uniform compound Uniform polyhedron compound A uniform polyhedron compound is a polyhedral compound whose constituents are identical uniform polyhedra, in an arrangement that is also uniform: the symmetry group of the compound acts transitively on the compound's vertices.The uniform polyhedron compounds were first enumerated by John Skilling...
Index	UC8
Convex hull Convex hull In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X....	Nonuniform truncated octahedron Truncated octahedron In geometry, the truncated octahedron is an Archimedean solid. It has 14 faces , 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a zonohedron....
Polyhedra	3 cubes
Faces	6+12 squares Square (geometry) In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
Edges	36
Vertices	24
Symmetry group Symmetry group The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...	octahedral Octahedral symmetry 150px|thumb|right|The [[cube]] is the most common shape with octahedral symmetryA regular octahedron has 24 rotational symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation...  (Oh)
Subgroup Subgroup In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...  restricting to one constituent	4-fold prismatic Dihedral symmetry in three dimensions This article deals with three infinite sequences of point groups in three dimensions which have a symmetry group that as abstract group is a dihedral group Dihn .See also point groups in two dimensions.Chiral:...  (D4h)

This uniform polyhedron compound

Uniform polyhedron compound

A uniform polyhedron compound is a polyhedral compound whose constituents are identical uniform polyhedra, in an arrangement that is also uniform: the symmetry group of the compound acts transitively on the compound's vertices.The uniform polyhedron compounds were first enumerated by John Skilling...

 is a symmetric arrangement of 3 cubes, considered as square prisms. It can be constructed by superimposing three identical cubes, and then rotating each by 45 degrees about a separate axis (that passes through the centres of two opposite faces).

This compound famously appears in the lithograph print Waterfall

Waterfall (M. C. Escher)

Waterfall is a lithography print by the Dutch artist M. C. Escher which was first printed in October, 1961. It shows an apparent paradox where water from the base of a waterfall appears to run downhill before reaching the top of the waterfall....

by M.C. Escher. Its dual

Dual polyhedron

In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...

, the compound of three octahedra

Compound of three octahedra

In mathematics, the compound of three octahedra or octahedron 3-compound is a polyhedral compound formed from three regular octahedra, all sharing a common center but rotated with respect to each other. Although appearing earlier in the mathematical literature, it was rediscovered and popularized...

, forms the central image in an earlier Escher woodcut

Woodcut

Woodcut—occasionally known as xylography—is a relief printing artistic technique in printmaking in which an image is carved into the surface of a block of wood, with the printing parts remaining level with the surface while the non-printing parts are removed, typically with gouges...

, Stars

Stars (M. C. Escher)

Stars is a wood engraving print by the Dutch artist M. C. Escher which was first printed in October 1948, depicting two chameleons in a polyhedral cage floating through space.-Description:...

.

Cartesian coordinates

Cartesian coordinates for the vertices of this compound are all the permutation

Permutation

In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...

s of

(±√2, 0, ±1)