Rounded Dodecahedron

Rounded Dodecahedron is an algebraic surface, defined by an implicit equation, immersed, with Ih symmetry (a point group).

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algebraic aperiodic def-implicit immersed implemented tradition-classical

Formula

z^{6} - 5 \left(x^{2} + y^{2}\right) z^{4} + 5 \left(x^{2} + y^{2}\right)^{2} z^{2} - 2 \left(x^{4} - 10 x^{2} y^{2} + 5 y^{4}\right) x z + \left(x^{2} + y^{2} + z^{2}\right)^{3} - \left(x^{2} + y^{2} + z^{2}\right)^{2} + \left(x^{2} + y^{2} + z^{2}\right) - 1 = 0

Properties

Family: algebraic
Given by: an implicit equation
Curvature: no condition imposed
Periodicity: not periodic
Ends: 0
Embedding: immersed
Symmetry: Ih
Symmetry kind: point group
Fidelity: exact
Exactness: elementary

Sources

Closed form reproduced numerically against the shipped implementation (math_art/surfaces/algebraic.py) over 240 sample points: matches oracle x 1 over 240 points (worst 4.26e-14).

A. Cayley, "A Memoir on Cubic Surfaces", Phil. Trans. 159, 1869.
A. Clebsch, "Ueber die Flachen vierter Ordnung...", Journal fur die reine und angewandte Mathematik 69, 1868 -- the diagonal surface.
W. Barth, "Two projective surfaces with many nodes admitting the symmetries of the icosahedron", J. Algebraic Geometry 5, 1996.
H. Hauser, "Bildergalerie algebraischer Flaechen", Universitaet Wien -- the named gallery surfaces in the _HAUSER block below.
St. Endrass, "A Projective Surface of Degree Eight with 168 Nodes", J. Algebraic Geom. 6 (1997) 325-334 -- also the source of the five-parameter D8 x Z2 octic family; the 160- and 165-nodal parameter values are published on O. Labs's Algebraic Surface
Homepage, algebraicsurface.net (octics pages).
E. Goursat, 'Etude des surfaces qui admettent tous les plans de symetrie d'un polyedre regulier', Ann. Sci. ENS 3e serie 4 (1887).