geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the compound of three cubes is a

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 ...

formed from three

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 arranged 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 has been depicted in works by Max Brückner and M.C. Escher.

History

This compound appears in Max Brückner's book ''Vielecke und Vielflache'' (1900), and in the

lithograph Lithography () is a planographic method of printing originally based on the miscibility, immiscibility of oil and water. The printing is from a stone (lithographic limestone) or a metal plate with a smooth surface. It was invented in 1796 by ...

print ''

Waterfall A waterfall is any point in a river or stream where water flows over a vertical drop or a series of steep drops. Waterfalls also occur where meltwater drops over the edge of a tabular iceberg or ice shelf. Waterfalls can be formed in seve ...

'' (1961) by M.C. Escher, who learned of it from Brückner's book. Its dual, the compound of three octahedra, forms the central image in an earlier Escher

woodcut Woodcut is a relief printing technique in printmaking. An artist carves an image into the surface of a block of wood—typically with gouges—leaving the printing parts level with the surface while removing the non-printing parts. Areas that ...

, ''

Stars A star is a luminous spheroid of plasma held together by self-gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night; their immense distances from Earth make them appear as fixed points of ...

''. In the 15th-century manuscript '' De quinque corporibus regularibus'',

Piero della Francesca Piero della Francesca ( , ; ; ; – 12 October 1492) was an Italian Renaissance painter, Italian painter, mathematician and List of geometers, geometer of the Early Renaissance, nowadays chiefly appreciated for his art. His painting is charact ...

includes a drawing of an octahedron circumscribed around a cube, with eight of the cube edges lying in the octahedron's eight faces. Three cubes inscribed in this way within a single octahedron would form the compound of three cubes, but della Francesca does not depict the compound.

Construction and coordinates

This compound 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).

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 ...

for the vertices of this compound can be chosen as all the

permutation In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first mean ...

s of

(0,\pm 1,\pm\sqrt)

References

{{reflist, refs= {{citation, first=Max, last=Brückner, author-link=Max Brückner, title=Vielecke und Vielflache, Theorie und Geschichte, location=Leipzig, publisher=B.G. Teubner, year=1900, a
Plate 23
} {{citation, url=http://www.georgehart.com/virtual-polyhedra/piero.html, last=Hart, first=George W., authorlink=George W. Hart, contribution=Piero della Francesca's Polyhedra, title=Virtual Polyhedra, year=1998. {{citation, contribution=Max Brücknerʼs Wunderkammer of Paper Polyhedra, last=Hart, first=George W., authorlink=George W. Hart, title=Bridges 2019 Conference Proceedings, url=http://archive.bridgesmathart.org/2019/bridges2019-59.pdf, pages=59–66 {{citation, last=Verheyen, first=Hugo F., contribution=Chapter 4: Classification of the finite compounds of cubes, doi=10.1007/978-1-4612-4074-7_5, isbn=0-8176-3661-7, location=Boston, mr=1363715, pages=95–159, publisher=Birkhäuser, series=Design Science Collection, title=Symmetry Orbits, year=1996; see in particula
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Polyhedral compounds