Desmic Quartic (12 nodes, (1,2) member)

Desmic Quartic (12 nodes, (1,2) member) is an algebraic surface, defined by an implicit equation, immersed.

Open Desmic Quartic (12 nodes, (1,2) member) in the interactive viewer →

algebraic aperiodic def-implicit immersed implemented tradition-classical

Formula

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

Associated with a desmic system of three tetrahedra: Delta_1 = (x^2-w^2)(y^2-z^2) and its two cyclic mates satisfy Delta_1 + Delta_2 + Delta_3 = 0, and the desmic surfaces are the pencil a Delta_1 + b Delta_2. THE SHIPPED ROW IS THE (a, b) = (1, 2) MEMBER, chart w = 1; its 12 nodes are exact rational points (the vertices of the three tetrahedra: the coordinate points and (1:+-1:+-1:+-1)), each verified exactly by the self-test.

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.
E. W. Weisstein, MathWorld, topic 'Algebraic Surfaces' -- the enumeration this absence was measured against.