Cayley Nodal Cubic

Cayley Nodal Cubic is an algebraic surface, defined by an implicit equation, immersed with singularities, with Td symmetry (a point group).

Open Cayley Nodal Cubic in the interactive viewer →

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

Formula

4 \left(x^{2} + y^{2} + z^{2}\right) + 16 x y z - 1 = 0

Properties

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

A. Cayley, 'A Memoir on Cubic Surfaces', Phil. Trans. Roy. Soc. 159 (1869).
Breske-Labs-van Straten, 'Real Line Arrangements and Surfaces with Many Real Nodes' (2005), mu(3) = 4 (exact) -- independent node-count oracle.