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Great Dodecahedron


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The great dodecahedron is the Kepler-Poinsot polyhedron whose dual is the small stellated dodecahedron. It is also uniform polyhedron with Maeder index 35 (Maeder 1997), Wenninger index 21 (Wenninger 1989), Coxeter index 44 (Coxeter et al. 1954), and Har'El index 40 (Har'El 1993). Its Schläfli symbol is {5,5/2} and its Wythoff symbol is 5/2|25. It consists of 12 intersecting pentagonal faces (12{5}).

The great dodecahedron is implemented in the Wolfram Language as UniformPolyhedron[21], UniformPolyhedron["GreatDodecahedron"], UniformPolyhedron[{"Coxeter", 44}], UniformPolyhedron[{"Kaleido", 40}], UniformPolyhedron[{"Uniform", 35}], or UniformPolyhedron[{"Wenninger", 21}]. It is also implemented in the Wolfram Language as PolyhedronData["GreatDodecahedron"].

Schläfli (1901, p. 134) did not recognize the great dodecahedron as a regular solid because it violates the polyhedral formula, instead satisfying

 N_0-N_1+N_2=12-30+12=-6,
(1)

where N_0 is the number of vertices, N_1 the number of edges, and N_2 the number of faces (Coxeter 1973, p. 172).

The skeleton of the great dodecahedron is isomorphic to the icosahedral graph.

The 12 pentagonal faces can be constructing from an icosahedron by finding the 12 sets five vertices that are coplanar and connecting each set to form a pentagon. A version split along face intersections can be constructed by augmentation of a unit edge-length icosahedron by a pyramid with height -sqrt(1/6(7-3sqrt(5))). This gives side of lengths

s_1=1/2(sqrt(5)-1)
(2)
=phi-1
(3)
s_2=1,
(4)

where phi is the golden ratio.

Its circumradius for unit edge length is

R=1/25^(1/4)phi^(1/2)
(5)
=1/45^(1/4)sqrt(2(1+sqrt(5))),
(6)

where phi is the golden ratio.

The resulting solid has surface area and volume

S=15sqrt(5-2sqrt(5))
(7)
V=5/4(sqrt(5)-1).
(8)
GreatDodecahedronHull

The convex hull of the great dodecahedron is a regular icosahedron and the dual of the icosahedron is the dodecahedron, so the dual of the great dodecahedron (the small stellated dodecahedron) is one of the dodecahedron stellations (Wenninger 1983, pp. 35 and 40)


See also

Dodecahedron, Great Icosahedron, Great Stellated Dodecahedron, Kepler-Poinsot Polyhedron, Small Stellated Dodecahedron, Stellation

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References

Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller, J. C. P. "Uniform Polyhedra." Phil. Trans. Roy. Soc. London Ser. A 246, 401-450, 1954.Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973.Cundy, H. and Rollett, A. "The Great Dodecahedron. 5^(5/2)." §3.6.2 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 92-93, 1989.Fischer, G. (Ed.). Plate 105 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, p. 104, 1986.Har'El, Z. "Uniform Solution for Uniform Polyhedra." Geometriae Dedicata 47, 57-110, 1993.Schläfli, L. "Theorie der vielfachen Kontinuität." Denkschriften der Schweizerischen naturforschenden Gessel. 38, 1-237, 1901.Maeder, R. E. "35: Great Dodecahedron." 1997. https://www.mathconsult.ch/static/unipoly/35.html.Wenninger, M. J. Dual Models. Cambridge, England: Cambridge University Press, p. 39, 1983.Wenninger, M. J. "Great Dodecahedron." Model 21 in Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 35 and 39, 1989.

Cite this as:

Weisstein, Eric W. "Great Dodecahedron." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GreatDodecahedron.html

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