Hyde Sextic

Hyde Sextic is an algebraic surface, defined by an implicit equation, immersed with singularities.

Open Hyde Sextic in the interactive viewer →

algebraic aperiodic def-implicit implemented singular tradition-classical

Formula

27 \left(-1.5 + x^{2} + y^{2} + z^{2}\right) \left(-1 - x^{2} + 0.5 y^{2} + 2 z^{2}\right) - 13.5 \left(-1 - x^{2} + 0.5 y^{2} + 2 z^{2}\right) + 2.25 \left(-1.5 + x^{2} + y^{2} + z^{2}\right)^{2} - 4 \left(-1.5 + x^{2} + y^{2} + z^{2}\right)^{3} - 27 \left(-1 - x^{2} + 0.5 y^{2} + 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 with singularities
Fidelity: exact
Exactness: elementary

Definition

Hyde's 1901 sextic, touched by the axes of all screws reciprocal to three given screws (classical screw theory); singular along a cuspidal double curve and at 6 points.

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).

O. Labs, "A Sextic with 35 Cusps", arXiv:math/0502520 (2005) -- the D5 family (1) and Theorem 3, the real member with 30 cusps and 10 nodes shipped here.
E. W. Hyde, "On a Surface of the Sixth Order Which Is Touched by the Axes of All Screws Reciprocal to Three Given Screws", Ann. of Math. (2) 2 (1901) 179-188 -- the surface the HYDE row rebuilds from its classical definition.
O. Labs, Algebraic Surface Homepage, algebraicsurface.net, advent calendar 2002 Nos. 1 and 17 -- the descriptions the dodecic and Hyde rows are verified against. Mirrored locally under references/websites/algsurf/.
O. Labs, "Dessins d'Enfants and Hypersurfaces with Many A_j-Singularities", J. London Math. Soc. (2) 74 (2006) 607-622; arXiv:math/0505022 -- construction (4), the folding-polynomial critical counts, Theorem 7 and the 132 = 12*11 row of Table 1.
S. V. Chmutov, "Examples of Projective Surfaces with Many Singularities", J. Algebraic Geom. 1 (1992) 191-196 -- the nodal construction the A_j series modifies.
R. S. Ball, "A Treatise on the Theory of Screws", Cambridge University Press (1900) -- the pitch quadric and the canonical form of a three-system used in the derivation.
O. Labs, 'Advent calendar 2002, No. 17', The Algebraic Surface Homepage, algebraicsurface.net (Wayback rescue, captures to 2017-10-01; local mirror: algsurf/book/.../ch035_no_17.md).
Hyde, Ann. of Math., 2nd series, vol. 2 (1901), and Ann. of Math. 4 (1888) (the page's references for the screw-theory construction); description and equation communicated to O. Labs by W. Barth.