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Snub Dodecahedron


SnubDodecahedronSolidWireframeNet

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The snub dodecahedron is an Archimedean solid consisting of 92 faces (80 triangular, 12 pentagonal), 150 edges, and 60 vertices. It is sometimes called the dodecahedron simum (Kepler 1619, Weissbach and Martini 2002) or snub icosidodecahedron. It is a chiral solid, and therefore exists in two enantiomorphous forms, commonly called laevo (left-handed) and dextro (right-handed). The laevo snub dodecahedron is illustrated above together with a wireframe version and a net that can be used for its construction.

It is also the uniform polyhedron with Maeder index 29 (Maeder 1997), Wenninger index 18 (Wenninger 1989), Coxeter index 32 (Coxeter et al. 1954), and Har'El index 34 (Har'El 1993). It has Schläfli symbol s{3; 5} and Wythoff symbol |235.

SnubDodecProjections

Some symmetric projections of the snub dodecahedron are illustrated above.

It is implemented in the Wolfram Language as PolyhedronData["SnubDodecahedron"].

SnubDodecahedronMirrorImages

An attractive dual of the two enantiomers superposed on one another is illustrated above.

SnubDodecahedronAndDual

The dual polyhedron of the snub dodecahedron is the pentagonal hexecontahedron, with which it is illustrated above.

It can be constructed by snubification of a dodecahedron of unit edge length with outward offset

 d=(512000x^(12)-1920000x^(10)-460800x^8+424000x^6+53040x^4-20600x^2+961)_8
(1)

and twist angle

 theta=cos^(-1)[(64x^6+64x^5+800x^4+240x^3-800x^2-306x+59)_3].
(2)

Here, the notation (P(x))_n indicates a polynomial root.

The inradius r of the dual, midradius rho=rho_d of the solid and dual, and circumradius R of the solid for a=1 are given

r_d=(1-128x^2+6384x^4-149376x^6+1443072x^8-3900416x^(10)+856064x^(12))_8
(3)
=2.03987315...
(4)
rho=(1-40x^2+624x^4-4672x^6+16384x^8-21504x^(10)+4096x^(12))_8
(5)
=2.09705383...
(6)
R=(209-2696x^2+13872x^4-35776x^6+47104x^8-27648x^(10)+4096x^(12))_8
(7)
=2.15583737....
(8)

The surface area is given by

 S=sqrt(15[95+6sqrt(5)+8sqrt(15(5+2sqrt(5)))]),
(9)

and the volume is given by the polynomial root

 V=(187445810737515625-182124351550575000x^2+6152923794150000x^4+1030526618040000x^6+162223191936000x^8-3195335070720x^(10)+2176782336x^(12))_8.
(10)

See also

Archimedean Solid, Equilateral Zonohedron, Hexecontahedron, Snub Cube, Snub Dodecahedral Graph

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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.Geometry Technologies. "Snub Dodecahedron." http://www.scienceu.com/geometry/facts/solids/snub_dodeca.html.Har'El, Z. "Uniform Solution for Uniform Polyhedra." Geometriae Dedicata 47, 57-110, 1993.Kasahara, K. "From Regular to Semiregular Polyhedrons." Origami Omnibus: Paper-Folding for Everyone. Tokyo: Japan Publications, pp. 220-221, 1988.Kepler, J. Harmonices Mundi. 1619. Reprinted Opera Omnia, Lib. II. Frankfurt, Germany.Longuet-Higgins, M. S. "Snub Polyhedra and Organic Growth." Proc. Roy. Soc. A 465, 477-491, 2009.Maeder, R. E. "29: Snub Dodecahedron." 1997. https://www.mathconsult.ch/static/unipoly/29.html.Weissbach, B. and Martini, H. "On the Chiral Archimedean Solids." Contrib. Algebra and Geometry 43, 121-133, 2002.Wenninger, M. J. "The Snub Dodecahedron." Model 18 in Polyhedron Models. Cambridge, England: Cambridge University Press, p. 32, 1989.

Cite this as:

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

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