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# Rhombic Triacontahedron Stellations

Ede (1958) enumerates 13 basic series of stellations of the rhombic triacontahedron, the total number of which is extremely large. Pawley (1973) gave a set of restrictions upon which a complete enumeration of stellations can be achieved (Wenninger 1983, p. 36). Messer (1995) describes 227 stellations (including the original solid in the count as usual), some of which are illustrated above.

The Great Stella stellation software reproduces Messer's 227 fully supported stellations. Using Miller's rules gives 358833098 stellations, 84959 of them reflexible and 358748139 of them chiral.

The convex hull of the dodecadodecahedron is an icosidodecahedron and the dual of the icosidodecahedron is the rhombic triacontahedron, so the dual of the dodecadodecahedron (the medial rhombic triacontahedron) is one of the rhombic triacontahedron stellations (Wenninger 1983, p. 41). Others include the great rhombic triacontahedron, cube 5-compound, and rhombic hexecontahedron (Kabai 2002, p. 185).

The beautiful figures above show the results of starting with the interior of the cube 5-compound and including successively larger portions of the space enclosed by its stellations (M. Trott, pers. comm., Feb. 10, 2006).

Archimedean Dual Stellations, Cube 5-Compound, Fully Supported Stellation, Great Rhombic Triacontahedron, Medial Rhombic Triacontahedron, Miller's Rules, Rhombic Hexecontahedron, Rhombic Triacontahedron, Stellation

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## References

Ede, J. D. "Rhombic Triacontahedra." Math. Gaz. 42, 98-100, 1958.Kabai, S. Mathematical Graphics I: Lessons in Computer Graphics Using Mathematica. Püspökladány, Hungary: Uniconstant, p. 185, 2002.Messer, P. W. "Stellations of the Rhombic Triacontahedron and Beyond." Structural Topology 21, 25-46, 1995.Pawley, G. S. "The 227 Triacontahedra." Geom. Dedicata 4, 221-232, 1975.Webb, R. "Enumeration of Stellations." http://www.software3d.com/Enumerate.php.Webb, R. "Stellation of Rhombic Triacontahedron." http://www.software3d.com/RTC_Hollow.html.Wenninger, M. J. Dual Models. Cambridge, England: Cambridge University Press, p. 36, 1983.

## Referenced on Wolfram|Alpha

Rhombic Triacontahedron Stellations

## Cite this as:

Weisstein, Eric W. "Rhombic Triacontahedron Stellations." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RhombicTriacontahedronStellations.html