Compound of five cubes
{{Short description|Polyhedral compound}}
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!bgcolor=#e7dcc3 colspan=2|Compound of five cubes | |
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| bgcolor=#e7dcc3|Type | Regular compound |
| bgcolor=#e7dcc3|Coxeter symbol | 2{5,3}[5{4,3}]{{sfn|Coxeter|1973|loc=pp. 49-50}}{{sfn|Coxeter|1973|loc=p 98}} |
| bgcolor=#e7dcc3|Stellation core | rhombic triacontahedron |
| bgcolor=#e7dcc3|Convex hull | Dodecahedron |
| bgcolor=#e7dcc3|Index | UC9 |
| bgcolor=#e7dcc3|Polyhedra | 5 cubes |
| bgcolor=#e7dcc3|Faces | 30 squares (visible as 360 triangles) |
| bgcolor=#e7dcc3|Edges | 60 |
| bgcolor=#e7dcc3|Vertices | 20 |
| bgcolor=#e7dcc3|Dual | Compound of five octahedra |
| bgcolor=#e7dcc3|Symmetry group | icosahedral (Ih) |
| bgcolor=#e7dcc3|Subgroup restricting to one constituent | pyritohedral (Th) |
File:Brückner 1900 compund of five cubes.png (1900)]]
File:Simetria rotacional 04.jpg
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 pentagonal faces of the dodecahedron.
It is one of the stellations of the rhombic triacontahedron. Its dual is the compound of five octahedra. It has icosahedral symmetry (Ih).
Geometry
The compound is a faceting of the dodecahedron. Each cube represents a selection of 8 of the 20 vertices of the dodecahedron.
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| colspan="3"| Views from 2-fold, 5-fold and 3-fold symmetry axis |
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 of 182 − 540 + 360 = 2.
Edge arrangement
Its convex hull is a regular dodecahedron. It additionally shares its edge arrangement with the small ditrigonal icosidodecahedron, the great ditrigonal icosidodecahedron, and the ditrigonal dodecadodecahedron. With these, it can form polyhedral compounds that can also be considered as degenerate uniform star polyhedra; the small complex rhombicosidodecahedron, great complex rhombicosidodecahedron and complex rhombidodecadodecahedron.
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The compound of ten tetrahedra can be formed by taking each of these five cubes and replacing them with the two tetrahedra of the stella octangula (which share the same vertex arrangement of a cube).
As a stellation
This compound can be formed as a stellation of the rhombic triacontahedron.
The 30 rhombic faces exist in the planes of the 5 cubes.
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| File:Stellation of rhombic triacontahedron 5 cubes facets.png | File:Hess polyhedra 1876 a.jpg (1876) |
See also
Footnotes
{{reflist}}
References
- {{citation|first=Peter R.|last=Cromwell|title=Polyhedra|location=Cambridge|year=1997|page=360}}.
- {{citation|first=Michael G.|last=Harman|title=Polyhedral Compounds|publisher=unpublished manuscript|year=c. 1974|url=http://www.georgehart.com/virtual-polyhedra/compounds-harman.html}}.
- {{citation|first=John|last=Skilling|title=Uniform Compounds of Uniform Polyhedra|journal=Mathematical Proceedings of the Cambridge Philosophical Society|volume=79|pages=447–457|year=1976|issue=3 |doi=10.1017/S0305004100052440|bibcode=1976MPCPS..79..447S |mr=0397554|s2cid=123279687 }}.
- Cundy, H. and Rollett, A. "Five Cubes in a Dodecahedron." §3.10.6 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 135–136, 1989.
- {{citation
| last1=Coxeter | first1= H. S. M. | authorlink1=Harold Scott MacDonald Coxeter
| title=Regular Polytopes
| edition=3rd
| date=1973
| publisher=Dover edition
| isbn=0-486-61480-8}}, 3.6 The five regular compounds, pp.47-50, 6.2 Stellating the Platonic solids, pp.96-104
External links
- [http://mathworld.wolfram.com/Cube5-Compound.html MathWorld: Cube 5-Compound]
- [http://mathworld.wolfram.com/RhombicTriacontahedronStellations.html MathWorld: Rhombic Triacontahedron Stellations]
- [http://www.georgehart.com/virtual-polyhedra/compound-cubes-info.html George Hart: Compounds of Cubes]
- [https://web.archive.org/web/20070102152247/http://www.uwgb.edu/dutchs/SYMMETRY/polycpd.htm Steven Dutch: Uniform Polyhedra and Their Duals]
- {{KlitzingPolytopes|../incmats/rhom.htm|3D compound|}}
Category:Polyhedral stellation
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