Van Straten Octic (165 nodes)

Van Straten Octic (165 nodes) is an algebraic surface, defined by an implicit equation, immersed with singularities.

Open Van Straten Octic (165 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(-0.125 \left(x^{2} + y^{2}\right)^{2} + \left(z^{2} + 0.4142135623730952\right) \left(x^{2} + y^{2}\right) - z^{4} + 0.5857864376269052 z^{2} - 0.9999999999999996\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

Member of Endrass's 5-parameter D8 x Z2 octic family (see endrass-octic-family) found by D. van Straten: the parameters are those of Endrass's 160-nodal octic except b = 1. The site notes its earlier publication venue as 'the printout on his office door'.

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).

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.
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/.
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, 'Van Straten's D8 x Z2-symmetric Octic with 165 Nodes', The Algebraic Surface Homepage, algebraicsurface.net (Wayback rescue, captures to 2017-10-01; local mirror: algsurf/book/.../ch004_duco165.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.