Compound of five cubes

{{Short description|Polyhedral compound}} {| class=wikitable align=right width="250" !bgcolor=#e7dcc3 colspan=2|Compound of five cubes |- |align=center colspan=2|[[File:Compound of five cubes, perspective.png|230px]] ([[:File:Compound of five cubes, gray and rgby.gif|Animation]], [[:File:Compound of five cubes.stl|3D model]]) |- |bgcolor=#e7dcc3|Type||[[Regular polyhedral compound|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 [[cube]]s |- |bgcolor=#e7dcc3|Faces||30 [[Square (geometry)|squares]] (visible as 360 [[Triangle|triangles]]) |- |bgcolor=#e7dcc3|Edges||60 |- |bgcolor=#e7dcc3|Vertices||20 |- |bgcolor=#e7dcc3|Dual||[[Compound of five octahedra]] |- |bgcolor=#e7dcc3|[[Symmetry group]]||[[Icosahedral symmetry|icosahedral]] (''I''h) |- |bgcolor=#e7dcc3|[[Subgroup]] restricting to one constituent||[[Tetrahedral symmetry|pyritohedral]] (''T''h) |} [[File:Brückner 1900 compund of five cubes.png|thumb|Model by [[Max Brückner]] (1900)]] [[File:Simetria rotacional 04.jpg|thumb|Model with dodecahedron]] The '''[[polyhedral compound|compound]] of five [[cube]]s''' is one of the five [[Polyhedral compound#Regular compounds|regular polyhedral compounds]]. It was first described by [[Edmund Hess]] in 1876.

Its vertices are those of a regular [[dodecahedron]]. Its edges form [[Pentagram#Geometry|pentagrams]], which are the [[stellation]]s of the pentagonal faces of the dodecahedron.

It is one of the [[#As a stellation|stellations]] of the [[rhombic triacontahedron]]. Its dual is the [[compound of five octahedra]]. It has [[icosahedral symmetry]] ('''I'''h).

The compound of five cubes can also be known as a rhombihedron.

== Geometry ==

The compound is a [[faceting]] of the [[dodecahedron]]. Each cube represents a selection of 8 of the 20 vertices of the dodecahedron.

{| class="wikitable" style="text-align: center;" |rowspan="2"| [[File:Black cube in white dodecahedron.png|160px]] | [[File:Compound of five cubes, 2-fold.png|130px]] | [[File:Compound of five cubes, 5-fold.png|130px]] | [[File:Compound of five cubes, 3-fold.png|130px]] |- |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 [[Triangle|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]].{{sfnp|Cromwell| 1997|pp=[https://archive.org/details/polyhedra0000crom/page/360/mode/1up?view=theater 360–361]}} 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 polyhedron|uniform star polyhedra]]; respectively, the [[small complex rhombicosidodecahedron]], [[great complex rhombicosidodecahedron]] and [[complex rhombidodecadodecahedron]].

{| class="wikitable" width="400" style="vertical-align:top;text-align:center" |[[Image:Small ditrigonal icosidodecahedron.png|100px]] [[Small ditrigonal icosidodecahedron]] |[[Image:Great ditrigonal icosidodecahedron.png|100px]] [[Great ditrigonal icosidodecahedron]] |[[Image:Ditrigonal dodecadodecahedron.png|100px]] [[Ditrigonal dodecadodecahedron]] |- align=center |[[Image:Dodecahedron.png|100px]] [[Dodecahedron]] ([[convex hull]]) |[[Image:Compound of five cubes.png|100px]] '''Compound of five cubes''' |[[Image:Spherical compound of five cubes.png|100px]] As a [[spherical tiling]] |}

The [[compound of ten tetrahedra]] can be formed by taking each of these five [[cube]]s and replacing them with the two [[Tetrahedron|tetrahedra]] of the [[Stellated octahedron|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.

{| |- style="vertical-align: top;" | [[File:Stellation of rhombic triacontahedron 5 cubes facets.png|thumb|x300px|left|Stellation facets The yellow area corresponds to one cube face.]] | [[File:Hess polyhedra 1876 a.jpg|thumb|x300px|left|Illustrations by [[Edmund Hess]] (1876) In the top right the same figure as on the left. In the bottom right a stellation diagram of the [[compound of five octahedra]].]] |}

==See also == {| |[[File:Icosahedral to octahedral compound of cubes.gif|thumb|left|Transition to [[compound of four cubes]]]] |style="vertical-align: top;"| *[[Compound of five octahedra]] *[[Compound of three cubes]] *[[Compound of four cubes]] *[[Compound of six cubes]] *[[Uniform polyhedron compound]] |}

== Footnotes == {{reflist}}

== References == *{{citation|first=Peter R.|last=Cromwell|title=Polyhedra|publisher=Cambridge University Press |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 (Cundy and Rollett)|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 (book)|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 * {{cite web |last=McCooey |first=Robert |title=Uniform Polyhedron Compounds |url=http://www.polytope.net/hedrondude/regcomp3.htm |website=Hedron Dude |access-date=24 June 2025}}

== External links == * [https://mathworld.wolfram.com/Cube5-Compound.html MathWorld: Cube 5-Compound] ** [https://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]] [[Category:Polyhedral compounds]]