Van Straten D_d Nodal Series
Van Straten D_d Nodal Series is an algebraic surface, defined by an implicit equation, immersed with singularities.
Open Van Straten D_d Nodal Series in the interactive viewer →
algebraic aperiodic def-implicit implemented singular tradition-classical
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
Polynomial not stored: no closed form is stored for this surface, and its implementation could not be read back as one expression (call to isinstance()). The shipped implementation in math_art/surfaces/algebraic.py is authoritative; an unverified transcription would silently define a different surface. Van Straten's modification of Chmutov's nodal series: replace the folding polynomial by the regular d-gon polynomial R_d(x,y), giving the D_d-symmetric surface R_d(x,y) + lambda*(T_d(z)+1) = 0 with lambda the critical value of R_d and T_d the Chebyshev polynomial with critical values 0 and 1. Trades some of Chmutov's (partly complex) nodes for fewer, ALL REAL ones.
Sources
Derived from the Math Art generator registries and cross-referenced to the literature; not transcribed from any gallery's compilation.
- D. van Straten, D_d-symmetric surfaces of degree d with many real nodes, per O. Labs, algebraicsurface.net -- the 124-nodal octic is transcribed verbatim from that page; cf. S. Breske, O. Labs and D. van Straten, "Real Line Arrangements and Surfaces with Many Real Nodes", arXiv:math/0507234 (2005).
- A. Cayley, "A Memoir on Cubic Surfaces", Phil. Trans. 159, 1869.
- A. Clebsch, "Ueber die Flachen vierter Ordnung...", Journal fur die reine und angewandte Mathematik 69, 1868 -- the diagonal surface.
- W. Barth, "Two projective surfaces with many nodes admitting the symmetries of the icosahedron", J. Algebraic Geometry 5, 1996.
- H. Hauser, "Bildergalerie algebraischer Flaechen", Universitaet Wien -- the named gallery surfaces in the _HAUSER block below.
- St. Endrass, "A Projective Surface of Degree Eight with 168 Nodes", J. Algebraic Geom. 6 (1997) 325-334 -- also the source of the five-parameter D8 x Z2 octic family; the 160- and 165-nodal parameter values are published on O. Labs's Algebraic Surface
- O. Labs, 'Van Straten's Construction of D_d-symmetric Surfaces of Degree d with many Nodes', The Algebraic Surface Homepage, algebraicsurface.net (Wayback rescue, captures to 2017-10-01; local mirror: algsurf/book/.../ch009_vstrconstr.md).
- O. Labs, 'Advent calendar 2002, No. 7 (the septic member)', The Algebraic Surface Homepage, algebraicsurface.net (Wayback rescue, captures to 2017-10-01; local mirror: algsurf/book/.../ch025_no_07.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).
- S. Chmutov, J. Algebraic Geom. 1 (1992) 191-196 (the construction being modified; cited by the mirrored pages).