Barth Sextic

Barth Sextic is an algebraic surface, defined by an implicit equation, immersed with singularities, with Ih symmetry (a point group).

Open Barth Sextic in the interactive viewer →

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

Formula

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

Properties

Family: algebraic
Given by: an implicit equation
Curvature: no condition imposed
Periodicity: not periodic
Ends: 0
Embedding: immersed with singularities
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 1.42e-14).

W. Barth, 'Two projective surfaces with many nodes admitting the symmetries of the icosahedron', J. Algebraic Geometry 5 (1996).
Breske-Labs-van Straten (2005) mu(6) table and J. Garcia-Escudero (arXiv:1107.3401, 2011): Barth's 65-nodal sextic -- independent confirmations of the 65.