Rhombic Hexecontahedron


The rhombic hexecontahedron is a 60-faced polyhedron that can be obtained by stellating the rhombic triacontahedron by placing a plane along each edge which is perpendicular to the plane of symmetry in which the edge lies, and taking the solid bounded by these planes gives a hexecontahedron (Steinhaus 1999). It is therefore a rhombic triacontahedron stellation. It appears to have been first noted and illustrated by Unkelbach (1940) as one of 20 finite equilateral polyhedra whose edges lie in planes of symmetry and whose faces are convex polygons which do not penetrate one another.

The 60 faces of the rhombic hexecontahedron are golden rhombi (Kabai 2002, p. 179).

Amazingly, the rhombic hexecontahedron is inferred to exist in nature as the central core of a quasicrystal aggregate of Al_6Li_3Cu produced by slow solidification (Guyot 1987).


The rhombic hexecontahedron is implemented in the Wolfram Language as PolyhedronData["RhombicHexecontahedron"]. It is also the logo for the Wolfram|Alpha web site (Weisstein 2009).


The rhombic hexecontahedron can be constructed by extending the long edges of each rhombic face of the rhombic triacontahedron to obtain rhombi on either side of the original that are a factor of the golden ratio phi larger that the original central rhombus (Kabai 2002, p. 181).

Rhombic hexecontahedron hulls

A tetrahedron 10-compound, octahedron 5-compound, cube 5-compound, icosahedron, dodecahedron, and icosidodecahedron can be inscribed in the vertices of a rhombic hexecontahedron, illustrated above (E. Weisstein, Dec. 24-27, 2009).

Rhombic hexecontahedron from 180 dodecahedra

20 golden rhombohedra can be combined to form a solid rhombic hexecontahedron. An addition 180 regular dodecahedra can be placed face-to-face to lie along the edges of a rhombic hexecontahedron (Kabai 2011, Fig. 40).


The skeleton of the rhombic hexecontahedron is the deltoidal hexecontahedral graph, illustrated above.

The rhombic hexecontahedron with edge length a has surface area and volume given by


and inertia tensor

 I=[1/(15)(10+3sqrt(5))Ma^2 0 0; 0 1/(15)(10+3sqrt(5))Ma^2 0; 0 0 1/(15)(10+3sqrt(5))Ma^2].

See also

Deltoidal Hexecontahedron, Golden Rhombohedron, Golden Rhombus, Pentagonal Hexecontahedron, Pentakis Dodecahedron, Rhombic Triacontahedron Stellations, Rhombohedron, Small Rhombicosidodecahedron, Snub Dodecahedron, Spikey, Triakis Icosahedron, Truncated Dodecahedron, Truncated Icosahedron

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Grünbaum, B. "A New Rhombic Hexecontahedron." Geombinatorics 6, 15-18, 1996.Grünbaum, B. "A New Rhombic Hexecontahedron--Once More." Geombinatorics, 6, 55-59, 1996.Grünbaum, B. "Still More Rhombic Hexecontahedra." Geombinatorics 6, 140-142, 1997.Grünbaum, B. "Parallelogram-Faced Isohedra with Edges in Mirror-Planes." Disc. Math. 221, 93-100, 2000.Guyot, P. "News on Five-Fold Symmetry." Nature 326, 640-641, 1987.Kabai, S. Mathematical Graphics I: Lessons in Computer Graphics Using Mathematica. Püspökladány, Hungary: Uniconstant, pp. 171, 179, and 181, 2002.Kabai, S. "Inside and Outside the Rhombic Hexecontahedron: A Study of Possible Structures with Rhombic Hexecontahedron with the Help of Physical Models and Wolfram Mathematica." In Proceedings of Bridges 2011: Mathematics, Music, Art, Architecture, Culture (Ed. R. Sarhangi and C. H. Séquin). Tessellations Publishing, pp. 387-394, 2011., S. and Bérczi, S. Rhombic Structures: Geometry and Modeling from Crystals to Space Stations. Püsspökladány, Hungary: Uniconstant, 2015.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 210, 1999.Unkelbach, H. "Die kantensymmetrischen, gleichkantigen Polyeder." Deutsche Math. 5, 306-316, 1940.Weisstein, E. W. "What's In a Name? That Which We Call a Rhombic Hexecontahedron." May 19, 2009., S. "The Story of Spikey." Dec. 28, 2018.

Cite this as:

Weisstein, Eric W. "Rhombic Hexecontahedron." From MathWorld--A Wolfram Web Resource.

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