Quartic with 12 Cuboctahedral Nodes

Quartic with 12 Cuboctahedral Nodes is an algebraic surface, defined by an implicit equation, immersed, with Oh symmetry (a point group).

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algebraic aperiodic def-implicit immersed implemented tradition-classical

Formula

x^{4} + y^{4} + z^{4} + 0 \left(x^{2} + y^{2} + z^{2}\right)^{2} - 2 \left(x^{2} + y^{2} + z^{2}\right) + 2 = 0

Properties

Family: algebraic
Given by: an implicit equation
Curvature: no condition imposed
Periodicity: not periodic
Ends: 0
Embedding: immersed
Symmetry: Oh
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 0).

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
D. van Straten, D_d-symmetric surfaces of degree d with many real nodes, per O. Labs, algebraicsurface.net -- the 124-nodal octic is transcribed verbatim from that page; cf. S. Breske, O. Labs and D. van Straten, "Real Line Arrangements and Surfaces with Many Real Nodes", arXiv:math/0507234 (2005).
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.
E. Goursat, 'Etude des surfaces qui admettent tous les plans de symetrie d'un polyedre regulier', Ann. Sci. ENS 3e serie 4 (1887).