Cube 5-Compound

Cube5-CompoundPaper 5-cube model by Emily Herrstrom

A polyhedron compound consisting of the arrangement of five cubes in the polyhedron vertices of a dodecahedron (or the centers of the faces of the icosahedron). The illustration at right above shows a paper sculpture of the cube 5-compound.

Solids in which the cube 5-compound can be inscribed

The cube 5-compound can be inscribed on the vertices of an augmented dodecahedron, (first) cube 4-compound, cube-octahedron 5-compound, augmented dodecahedron, deltoidal hexecontahedron, disdyakis triacontahedron, dodecahedron, echidnahedron, great rhombic triacontahedron, great stellated dodecahedron, metabiaugmented dodecahedron, parabiaugmented dodecahedron, pentagonal hexecontahedron, pentakis dodecahedron, rhombic enneacontahedron, rhombic hexecontahedron, rhombic triacontahedron, small triambic icosahedron, spikey, triakis icosahedron, and triaugmented dodecahedron (E. Weisstein, Dec. 25-28, 2009).

The cube 5-compound is implemented in the Wolfram Language as PolyhedronData["CubeFiveCompound"].

The compound can be generated by starting with an initial cube centered at the origin and oriented along the axes, then adding four more cubes obtained from the initial cube by rotations through angles -2npi/5 about the axis (1,phi,0) for n=1, 2, 3, 4. The cube 5-compound is the dual of the octahedron 5-compound, and is one of the rhombic triacontahedron stellations (Kabai 2002, p. 185).

The cube 5-compound has the 30 facial planes of the rhombic triacontahedron (Steinhaus 1999, pp. 199 and 209; Ball and Coxeter 1987).


In the above figure, let a=1 be the length of a cube polyhedron edge. Then

theta=tan^(-1)((3-sqrt(5))/2) approx 20 degrees54^'
phi=tan^(-1)((sqrt(5)-1)/2) approx 31 degrees43^'
psi=90 degrees-phi approx 58 degrees17^'
alpha=90 degrees-theta approx 69 degrees06^'.

The compound can be constructed using pieces like those illustrated above (Cundy and Rollett 1989).


Nets are shown above for constructing the compound such that each cube can be made a different color. For cubes of unit edge lengths, the resulting edge lengths are


The surface area of the compound is

 S=165sqrt(5)-360 approx 8.95,

compared to S=6 for each of the five constituent cubes.

The circumradius for the compound composed of unit cubes is


and the surface area and volume are


Its convex hull is a dodecahedron, and its interior is a rhombic triacontahedron. The beautiful figures above show the results of starting with the interior of the compound and including successively larger portions of the space enclosed by its stellations (M. Trott, pers. comm., Feb. 10, 2006). They are therefore rhombic triacontahedron stellations.

See also

Cube, Cube 2-Compound, Cube 3-Compound, Cube 4-Compound, Cube-Octahedron 5-Compound, Cube 6-Compound, Cube 7-Compound, Cube 10-Compound, Cube 20-Compound, Dodecahedron, Octahedron 5-Compound, Polyhedron Compound, Rhombic Triacontahedron, Rhombic Triacontahedron Stellations

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Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 135 and 137, 1987.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.Hart, G. "Standard Compound of Five Cubes.", G. "Standard Compound of Five Cubes.", S. Mathematical Graphics I: Lessons in Computer Graphics Using Mathematica. Püspökladány, Hungary: Uniconstant, pp. 161 and 185, 2002.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999.Verheyen, H. F. Symmetry Orbits. Boston, MA: Birkhäuser, 2007.

Cite this as:

Weisstein, Eric W. "Cube 5-Compound." From MathWorld--A Wolfram Web Resource.

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