Endrass Octic (160 nodes)

Endrass Octic (160 nodes) is an algebraic surface, defined by an implicit equation, immersed with singularities.

Open Endrass Octic (160 nodes) in the interactive viewer →

algebraic aperiodic def-implicit implemented 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(-3.875 \left(x^{2} + y^{2}\right)^{2} + \left(4 z^{2} + 8.824116139070417\right) \left(x^{2} + y^{2}\right) - z^{4} - 4.058504678941934 z^{2} - 5.032078619621483\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

Definition

Endrass's second D8 x Z2 octic: parameter values in his 5-parameter family (b = 4 and four derived expressions in b and sqrt(2), printed on the mirrored page) giving exactly 160 nodes. Distinct from, and found alongside, his 168-nodal record surface.

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 8.88e-16).

St. Endrass, "A Projective Surface of Degree Eight with 168 Nodes", J. Algebraic Geom. 6 (1997) 325-334 -- the family and the 112-node count of its generic member.
O. Labs, Algebraic Surface Homepage, algebraicsurface.net, octics pages (endroctconstr, stephan160, duco165, stephan120128136, vstrconstr) and advent calendar 2002 No. 6 -- the printed family, the 160-/165-nodal parameter values, the 124-nodal equation and the modified Chmutov octic. Mirrored locally under references/websites/algsurf/.
S. Breske, O. Labs and D. van Straten, "Real Line Arrangements and Surfaces with Many Real Nodes", arXiv:math/0507234 (2005) -- why real nodes are the rendering-relevant count.
V. I. Arnold, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps II, Birkhaeuser (1988), p. 419 -- Chmutov's original series.
M. Kuehnel, "A note on octic hypersurfaces with many nodes", arXiv:math/0210440 (2002) -- the bundle-theoretic 128-nodal existence result discussed (and not transcribed) above.
O. Labs, 'St. Endrass's 160-nodal D8 x Z2-symmetric Octic (with the explicit parameter values)', The Algebraic Surface Homepage, algebraicsurface.net (Wayback rescue, captures to 2017-10-01; local mirror: algsurf/book/.../ch008_stephan160.md).
O. Labs, 'Advent calendar 2002, No. 19 ('Besides constructing the octics with 168 nodes, St. Endrass found this 160-nodal octic with dihedral symmetry, which is not so well-known')', The Algebraic Surface Homepage, algebraicsurface.net (Wayback rescue, captures to 2017-10-01; local mirror: algsurf/book/.../ch037_no_19.md).
O. Labs, 'Octics (the survey of known nodal octic counts: Miyaoka bound 174, best known 168)', The Algebraic Surface Homepage, algebraicsurface.net (Wayback rescue, captures to 2017-10-01; local mirror: algsurf/book/.../ch006_octics.md).
St. Endrass, Ph.D. thesis (1996), as cited throughout the mirrored octics pages for the D8 x Z2 construction.