The great stellated dodecahedron is one of the Kepler-Poinsot polyhedra. It is also the uniform polyhedron
with Maeder index 52 (Maeder 1997), Wenninger index 22 (Wenninger 1989), Coxeter
index 68 (Coxeter et al. 1954), and Har'El index 57 (Har'El 1993). It is the
third dodecahedron stellation (Wenninger
1989). The great stellated dodecahedron has Schläfli
symbol
and Wythoff symbol . 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 is implemented in the Wolfram Language as UniformPolyhedron["GreatStellatedDodecahedron"].
Precomputed properties are available as PolyhedronData["GreatStellatedDodecahedron",
prop].

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 ,
where
is the golden ratio. Rescaling to give the pentagrams
unit edge lengths, the circumradius of the great
stellated dodecahedron is

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
(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 .

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

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Polyhedra." Phil. Trans. Roy. Soc. London Ser. A246, 401-450,
1954.Cundy, H. and Rollett, A. "Great Stellated Dodecahedron. ."
§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.Har'El, Z. "Uniform Solution
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Great Stellated Dodecahedron." 1997. https://www.mathconsult.ch/static/unipoly/52.html.Wenninger,
M. J. Dual
Models. Cambridge, England: Cambridge University Press, pp. 39-40, 1983.Wenninger,
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