Peano Surface

Peano Surface is an algebraic surface, defined by an implicit equation, immersed.

Open Peano Surface in the interactive viewer →

algebraic aperiodic def-implicit immersed implemented tradition-classical

Formula

z - \left(2 x^{2} - y\right) \left(y - x^{2}\right) = 0

Properties

Family: algebraic
Given by: an implicit equation
Curvature: no condition imposed
Periodicity: not periodic
Ends: 0
Embedding: immersed
Fidelity: exact
Exactness: elementary

Definition

Peano's counterexample on second-derivative tests for maxima.

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.
G. Peano's 1899 counterexample; cf. the MathWorld page's history of the criterion it defeated.
T. Nordstrand, "Weird Cube" and the tooth/pillow object, per the MathWorld pages (Nordstrand 1997).
E. W. Weisstein, MathWorld, topic 'Algebraic Surfaces' -- the enumeration this absence was measured against.