Modified Chmutov Octic (144 nodes)

Modified Chmutov Octic (144 nodes) is an algebraic surface, defined by an implicit equation, immersed with singularities.

Open Modified Chmutov Octic (144 nodes) in the interactive viewer →

algebraic aperiodic def-implicit implemented singular tradition-classical

Formula

\left(\left(8 x^{2} - 8\right) x^{2} + 1\right)^{2} + \left(\left(8 y^{2} - 8\right) y^{2} + 1\right)^{2} + \left(\left(8 z^{2} - 8\right) z^{2} + 1\right)^{2} - 1 = 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

The j = 1, d = 8, n = 3 member of a series of degree-d hypersurfaces in P^n modifying Chmutov's: the sum over the coordinates of the (j+1)-th powers of the Chebyshev polynomials T_{d/2} (critical values +1/-1) set equal to 1, giving real A_j singularities.

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 5.33e-15).

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.
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.
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.
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, 'Advent calendar 2002, No. 6 (with the defining Chebyshev form)', The Algebraic Surface Homepage, algebraicsurface.net (Wayback rescue, captures to 2017-10-01; local mirror: algsurf/book/.../ch024_no_06.md).
O. Labs, 'Octics (listing it as '144: (2002?), modified Chmutov surface')', The Algebraic Surface Homepage, algebraicsurface.net (Wayback rescue, captures to 2017-10-01; local mirror: algsurf/book/.../ch006_octics.md).
V. I. Arnold et al., Singularities of Differentiable Maps, Vol. II, Birkhaeuser (1988), p. 419 (cited by the page for Chmutov's original series).