Schur Quartic (64 lines)
Schur Quartic (64 lines) is an algebraic surface, defined by an implicit equation, embedded.
Open Schur Quartic (64 lines) in the interactive viewer →
algebraic aperiodic def-implicit embedded implemented tradition-classical
Formula
Properties
- Family
- algebraic
- Given by
- an implicit equation
- Curvature
- no condition imposed
- Periodicity
- not periodic
- Ends
- 0
- Embedding
- embedded
- Fidelity
- exact
- Exactness
- elementary
Definition
B. Segre proved a quartic surface can contain at most 64 lines and gave this example attaining the bound; the distinction is the line count, not singularities (the surface is smooth), and most of the 64 lines are complex, so they are invisible in the real picture.
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).
- E. W. Weisstein, "Peano Surface", "Chair", "Crossed Trough", "Handkerchief Surface", "Hunt's Surface", "Kiss Surface", "Menn's Surface", "Miter Surface", "Nordstrand's Weird Surface", "Tooth Surface", MathWorld -- A Wolfram Web Resource, mathworld.wolfram.com.
- T. Nordstrand, "Weird Cube" and the tooth/pillow object, per the MathWorld pages (Nordstrand 1997).
- G. Peano's 1899 counterexample; cf. the MathWorld page's history of the criterion it defeated.
- O. Labs, 'Advent calendar 2002, No. 24', The Algebraic Surface Homepage, algebraicsurface.net (Wayback rescue, captures to 2017-10-01; local mirror: algsurf/book/.../ch042_no_24.md).