Burkhardt Quartic Section (y0 = sum/4 slice)

Burkhardt Quartic Section (y0 = sum/4 slice) is an algebraic surface, defined by an implicit equation, immersed.

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algebraic aperiodic def-implicit immersed implemented tradition-classical

Formula

\left(0.25 \left(x + y + z + 1\right)\right)^{4} - 0.25 \left(x + y + z + 1\right) \left(x^{3} + y^{3} + z^{3} + 1\right) + 3 x y z = 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

A quartic with 45 nodes; Heinrich Burkhardt, 1891. The 45-nodal object is a THREEFOLD in P^4 (y0^4 - y0(y1^3+..+y4^3) + 3 y1 y2 y3 y4 = 0), so any 3D picture is a section: THE SHIPPED ROW IS THE y0 = (y1+y2+y3+y4)/4 HYPERPLANE SECTION, chosen because it contains all seven of the threefold's real rational nodes; four are affine in the y4 = 1 chart and are 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.