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Compound of three cubes

{{Short description|Polyhedral compound}}

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!bgcolor=#e7dcc3 colspan=2|Compound of three cubes

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bgcolor=#e7dcc3|TypeUniform compound
bgcolor=#e7dcc3|IndexUC8
bgcolor=#e7dcc3|Convex hullNonuniform truncated octahedron
bgcolor=#e7dcc3|Polyhedra3 cubes
bgcolor=#e7dcc3|Faces6+12 squares
bgcolor=#e7dcc3|Edges36
bgcolor=#e7dcc3|Vertices24
bgcolor=#e7dcc3|Symmetry groupoctahedral (Oh)
bgcolor=#e7dcc3|Subgroup restricting to one constituent4-fold prismatic (D4h)

In geometry, the compound of three cubes is a uniform polyhedron compound formed from three cubes arranged with octahedral symmetry.{{r|verheyen}} 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),{{r|bruckner}} and in the lithograph print Waterfall (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, Stars.{{r|h19}}

In the 15th-century manuscript De quinque corporibus regularibus, Piero della Francesca 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.{{r|h98}}

{{multiple image|total_width=540|align=center|image1=Piero della Francesca - Libellus de quinque corporibus regularibus - p41b (cropped).jpg|caption1=Cube in an octahedron from De quinque corporibus regularibus|image2=Piero Libellus - Cube dans octaèdre.jpg|caption2=Another view of the cube in an octahedron}}

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).{{r|h19}}

Cartesian coordinates for the vertices of this compound can be chosen as all the permutations of (0,\pm 1,\pm\sqrt{2}).

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|at=[https://archive.org/details/vieleckeundvielf00bruoft/page/n259 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 particular [https://books.google.com/books?id=exzSBwAAQBAJ&pg=PA136 p. 136].

}}

Category:Polyhedral compounds