Sextic with 9 Triple Points (EPS (4,4,4) member)

Sextic with 9 Triple Points (EPS (4,4,4) member) is an algebraic surface, defined by an implicit equation, immersed with singularities.

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algebraic aperiodic def-implicit implemented singular tradition-classical

Formula

\left(z^{2} - y\right) \left(x^{2} - z\right) \left(y^{2} - x\right) + \left(-1 - \left(x + y + z\right) - \left(x y + y z + z x\right)\right)^{3} = 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

A sextic in P^3 carrying nine ordinary triple points. The paper's search began by looking for a sextic with ELEVEN triple points, which would have been very interesting; eleven turns out to be a priori impossible, and nine is what exists.

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. W. Weisstein, "Peano Surface", "Chair", "Crossed Trough", "Handkerchief Surface", "Hunt's Surface", "Kiss Surface", "Menn's Surface", "Miter Surface", "Nordstrand's Weird Surface", "Tooth Surface", MathWorld -- A Wolfram Web Resource, mathworld.wolfram.com.
T. Nordstrand, "Weird Cube" and the tooth/pillow object, per the MathWorld pages (Nordstrand 1997).
G. Peano's 1899 counterexample; cf. the MathWorld page's history of the criterion it defeated.
S. Endrass, U. Persson and J. Stevens, 'Surfaces with triple points', arXiv:math/0010163v1 [math.AG] (2000), 37 pp.