Togliatti Quintic

Togliatti Quintic is an algebraic surface, defined by an implicit equation, immersed with singularities.

Open Togliatti Quintic in the interactive viewer →

algebraic aperiodic def-implicit implemented record-holder singular tradition-classical

Formula

64 \left(x - 1\right) \left(x^{4} - 4 x^{3} - 10 x^{2} y^{2} - 4 x^{2} + 16 x - 20 x y^{2} + 5 y^{4} + 16 - 20 y^{2}\right) - 8.312538755549069 \left(2 z - 1.6625077511098136\right) \left(4 \left(x^{2} + y^{2} - z^{2}\right) + 7.708203932499369\right)^{2} = 0

Properties

Family: algebraic
Given by: an implicit equation
Curvature: no condition imposed
Periodicity: not periodic
Ends: 0
Embedding: immersed with singularities
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 2.73e-12).

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).