# Compound of five cubes

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Polyhedral compound

Compound of five cubes (Animation, 3D model) Type Regular compound Coxeter symbol 2{5,3}[5{4,3}][1][2] Stellation core rhombic triacontahedron Convex hull Dodecahedron Index UC9 Polyhedra 5 cubes Faces 30 squares (visible as 360 triangles) Edges 60 Vertices 20 Dual Compound of five octahedra Symmetry group icosahedral (Ih) Subgroup restricting to one constituent pyritohedral (Th)

Model by [Max Brückner](/source/Max_Br%C3%BCckner) (1900)

Model with dodecahedron

The **[compound](/source/Polyhedral_compound) of five [cubes](/source/Cube)** is one of the five [regular polyhedral compounds](/source/Polyhedral_compound#Regular_compounds). It was first described by [Edmund Hess](/source/Edmund_Hess) in 1876.

Its vertices are those of a regular [dodecahedron](/source/Dodecahedron). Its edges form [pentagrams](/source/Pentagram#Geometry), which are the [stellations](/source/Stellation) of the pentagonal faces of the dodecahedron.

It is one of the [stellations](#As_a_stellation) of the [rhombic triacontahedron](/source/Rhombic_triacontahedron). Its dual is the [compound of five octahedra](/source/Compound_of_five_octahedra). It has [icosahedral symmetry](/source/Icosahedral_symmetry) (**I**h).

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

## Geometry

The compound is a [faceting](/source/Faceting) of the [dodecahedron](/source/Dodecahedron). Each cube represents a selection of 8 of the 20 vertices of the dodecahedron.

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](/source/Triangle)), 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](/source/Euler_characteristic) of 182 − 540 + 360 = 2.

## Edge arrangement

Its [convex hull](/source/Convex_hull) is a regular [dodecahedron](/source/Dodecahedron).[3] It additionally shares its [edge arrangement](/source/Edge_arrangement) with the [small ditrigonal icosidodecahedron](/source/Small_ditrigonal_icosidodecahedron), the [great ditrigonal icosidodecahedron](/source/Great_ditrigonal_icosidodecahedron), and the [ditrigonal dodecadodecahedron](/source/Ditrigonal_dodecadodecahedron). With these, it can form polyhedral compounds that can also be considered as degenerate [uniform star polyhedra](/source/Uniform_star_polyhedron); respectively, the [small complex rhombicosidodecahedron](https://en.wikipedia.org/w/index.php?title=Small_complex_rhombicosidodecahedron&action=edit&redlink=1), [great complex rhombicosidodecahedron](https://en.wikipedia.org/w/index.php?title=Great_complex_rhombicosidodecahedron&action=edit&redlink=1) and [complex rhombidodecadodecahedron](https://en.wikipedia.org/w/index.php?title=Complex_rhombidodecadodecahedron&action=edit&redlink=1).

Small ditrigonal icosidodecahedron Great ditrigonal icosidodecahedron Ditrigonal dodecadodecahedron Dodecahedron (convex hull) Compound of five cubes As a spherical tiling

The [compound of ten tetrahedra](/source/Compound_of_ten_tetrahedra) can be formed by taking each of these five [cubes](/source/Cube) and replacing them with the two [tetrahedra](/source/Tetrahedron) of the [stella octangula](/source/Stellated_octahedron) (which share the same vertex arrangement of a cube).

## As a stellation

This compound can be formed as a stellation of the [rhombic triacontahedron](/source/Rhombic_triacontahedron). The 30 rhombic faces exist in the planes of the 5 cubes.

Stellation facets The yellow area corresponds to one cube face. 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

Transition to compound of four cubes Compound of five octahedra Compound of three cubes Compound of four cubes Compound of six cubes Uniform polyhedron compound

## Footnotes

1. **[^](#cite_ref-FOOTNOTECoxeter1973pp._49-50_1-0)** [Coxeter 1973](#CITEREFCoxeter1973), pp. 49-50.

1. **[^](#cite_ref-FOOTNOTECoxeter1973p_98_2-0)** [Coxeter 1973](#CITEREFCoxeter1973), p 98.

1. **[^](#cite_ref-FOOTNOTECromwell1997[httpsarchiveorgdetailspolyhedra0000crompage360mode1upviewtheater_360–361]_3-0)** [Cromwell (1997)](#CITEREFCromwell1997), pp. [360–361](https://archive.org/details/polyhedra0000crom/page/360/mode/1up?view=theater).

## References

- Cromwell, Peter R. (1997), *Polyhedra*, Cambridge University Press, p. 360.

- Harman, Michael G. (c. 1974), [*Polyhedral Compounds*](http://www.georgehart.com/virtual-polyhedra/compounds-harman.html), unpublished manuscript.

- Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", *[Mathematical Proceedings of the Cambridge Philosophical Society](/source/Mathematical_Proceedings_of_the_Cambridge_Philosophical_Society)*, **79** (3): 447–457, [Bibcode](/source/Bibcode_(identifier)):[1976MPCPS..79..447S](https://ui.adsabs.harvard.edu/abs/1976MPCPS..79..447S), [doi](/source/Doi_(identifier)):[10.1017/S0305004100052440](https://doi.org/10.1017%2FS0305004100052440), [MR](/source/MR_(identifier)) [0397554](https://mathscinet.ams.org/mathscinet-getitem?mr=0397554), [S2CID](/source/S2CID_(identifier)) [123279687](https://api.semanticscholar.org/CorpusID:123279687).

- Cundy, H. and Rollett, A. "Five Cubes in a Dodecahedron." §3.10.6 in *[Mathematical Models](/source/Mathematical_Models_(Cundy_and_Rollett))*, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 135–136, 1989.

- [Coxeter, H. S. M.](/source/Harold_Scott_MacDonald_Coxeter) (1973), *[Regular Polytopes](/source/Regular_Polytopes_(book))* (3rd ed.), Dover edition, [ISBN](/source/ISBN_(identifier)) [0-486-61480-8](https://en.wikipedia.org/wiki/Special:BookSources/0-486-61480-8), 3.6 *The five regular compounds*, pp.47-50, 6.2 *Stellating the Platonic solids*, pp.96-104

- McCooey, Robert. ["Uniform Polyhedron Compounds"](http://www.polytope.net/hedrondude/regcomp3.htm). *Hedron Dude*. Retrieved 24 June 2025.

## External links

- [MathWorld: Cube 5-Compound](https://mathworld.wolfram.com/Cube5-Compound.html) - [MathWorld: Rhombic Triacontahedron Stellations](https://mathworld.wolfram.com/RhombicTriacontahedronStellations.html)

- [George Hart: Compounds of Cubes](http://www.georgehart.com/virtual-polyhedra/compound-cubes-info.html)

- [Steven Dutch: Uniform Polyhedra and Their Duals](https://web.archive.org/web/20070102152247/http://www.uwgb.edu/dutchs/SYMMETRY/polycpd.htm)

- Klitzing, Richard. ["3D compound"](https://bendwavy.org/klitzing/dimensions/../incmats/rhom.htm).

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Adapted from the Wikipedia article [Compound of five cubes](https://en.wikipedia.org/wiki/Compound_of_five_cubes) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Compound_of_five_cubes?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.