Great Stellated Dodecahedron


The great stellated dodecahedron is one of the Kepler-Poinsot solids. It is also uniform polyhedron U_(52), Wenninger model W_(41), and is the third dodecahedron stellation (Wenninger 1989). Its dual is the great icosahedron. The great stellated dodecahedron has Schläfli symbol {5/2,3} and Wythoff symbol 3|25/2. It has 12 pentagrammic faces.

The great stellated dodecahedron was published by Wenzel Jamnitzer in 1568. It was rediscovered by Kepler (and published in his work Harmonice Mundi in 1619), and again by Poinsot in 1809.

The great stellated dodecahedron can be constructed from a unit dodecahedron by selecting the 144 sets of five coplanar vertices, then discarding sets whose edges correspond to the edges of the original dodecahedron. This gives 12 pentagrams of edge length phi^2=(3+sqrt(5))/2, where phi is the golden ratio. Rescaling to give the pentagrams unit edge lengths, the circumradius of the great stellated dodecahedron is


where phi is the golden ratio.

The skeleton of the great stellated dodecahedron is isomorphic to the dodecahedral graph.


Another way to construct a great stellated dodecahedron via augmentation is to make 20 triangular pyramids with side length phi=(1+sqrt(5))/2 (the golden ratio) times the base, as illustrated above, and attach them to the sides of an icosahedron. The height of these pyramids is then sqrt(1/6(7+3sqrt(5))).

Cumulating a unit dodecahedron to construct a great stellated dodecahedron produces a solid with edge lengths


The surface area and volume of such a great stellated dodecahedron are


The convex hull of the great stellated dodecahedron is a regular dodecahedron and the dual of the dodecahedron is the icosahedron, so the dual of the great stellated dodecahedron (i.e., the great icosahedron) is one of the icosahedron stellations (Wenninger 1983, p. 40)

See also

Dodecahedron, Dodecahedron Stellations, Great Dodecahedron, Great Icosahedron, Great Stellated Truncated Dodecahedron, Kepler-Poinsot Solid, Small Stellated Dodecahedron, Spikey, Stellation

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Cauchy, A. L. "Recherches sur les polyèdres." J. de l'École Polytechnique 9, 68-86, 1813.Cundy, H. and Rollett, A. "Great Stellated Dodecahedron. (5/2)^3." §3.6.3 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 94-95, 1989.Fischer, G. (Ed.). Plate 104 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, p. 103, 1986.Jamnitzer, W. Perspectiva Corporum Regularium. Nürnberg, Germany, 1568. Reprinted Frankfurt, 1972.Kasahara, K. Origami Omnibus: Paper-Folding for Everyone. Tokyo: Japan Publications, p. 239, 1988.Kepler, J. "Harmonice Mundi." In Opera Omnia, Vol. 5. Frankfurt, 1864.Wenninger, M. J. Dual Models. Cambridge, England: Cambridge University Press, pp. 39-40, 1983.Wenninger, M. J. Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 35 and 40, 1989.

Cite this as:

Weisstein, Eric W. "Great Stellated Dodecahedron." From MathWorld--A Wolfram Web Resource.

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