Kummer Quartic

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

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

Formula

\left(x^{2} + y^{2} + z^{2} - 1.6900000000000002\right)^{2} - 3.106870229007634 \left(1 - z - 1.4142135623730951 x\right) \left(1 - z + 1.4142135623730951 x\right) \left(1 + z + 1.4142135623730951 y\right) \left(1 + z - 1.4142135623730951 y\right) = 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).

E. E. Kummer, 'Uber die Flachen vierten Grades mit sechzehn singularen Punkten', Monatsber. Akad. Wiss. Berlin (1864).
mathcurve.com, Kummer surface (mirror ch1294): 16 ordinary singular points at the tetrahedral vertices; Breske-Labs-van Straten (2005) mu(4) = 16 (exact) -- independent node-count oracle.