Endrass Octic

Endrass Octic is an algebraic surface, defined by an implicit equation, immersed with singularities.

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

Formula

0.25 \left(x^{2} - 1\right) \left(y^{2} - 1\right) \left(\left(x + y\right)^{2} - 2\right) \left(\left(x - y\right)^{2} - 2\right) - \left(-0.6035533905932737 \left(x^{2} + y^{2}\right)^{2} + \left(x^{2} + y^{2}\right) \left(1.7071067811865475 z^{2} + 1.4874368670764582\right) - z^{4} - 0.9142135623730951 z^{2} - 1.1231601717798214\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 4.44e-16).

S. Endrass, 'A projective surface of degree eight with 168 nodes', J. Algebraic Geom. 6 (1997).
S. Endrass, 'A projective surface of degree eight with 168 nodes' (1995; converted copy in research/papers/algebraic-surfaces/) and Breske-Labs-van Straten (2005) mu(8) >= 168 -- independent confirmations of the 168.