Labs Septic (99 nodes)

Labs Septic (99 nodes) is an algebraic surface, defined by an implicit equation, immersed with singularities.

Open Labs Septic (99 nodes) in the interactive viewer →

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

Formula

x \left(x^{6} - 21 x^{4} y^{2} + 35 x^{2} y^{4} - 7 y^{6}\right) + 7 z \left(\left(x^{2} + y^{2}\right)^{3} - 8 z^{2} \left(x^{2} + y^{2}\right)^{2} + 16 z^{4} \left(x^{2} + y^{2}\right)\right) - 64 z^{7} - \left(z - 0.01925201509012657\right) \left(-0.07852820149698514 z^{3} - 4.158360592288018 z^{2} - 4.147143488965512 z - 1.1881659380714786 + \left(z + 1\right) \left(x^{2} + y^{2}\right)\right)^{2} = 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

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 3.55e-15).

O. Labs, 'A Septic with 99 real Nodes', Rend. Sem. Mat. Univ. Padova 116 (2006).
O. Labs, 'A septic with 99 real nodes' (2004; converted copy in research/papers/algebraic-surfaces/) and Breske-Labs-van Straten (2005) mu(7) >= 99 -- independent confirmations of the 99.