# Field Lines

Robert P. Munafo, 2023 Feb 16.

The "field lines" of Kenelm Philip are (curved) lines which are everywhere perpendicular to the equipotential lines. They are described in his 1992 paper "Field lines in the Mandelbrot Set".

A technique for displaying color images of field lines surrounding the Mandelbrot Set using angle slicing decomposition and a "contrast color lookup table," is described in [this paper]. A modification of this method, which compensates for the spatial-frequency doubling inherent in angle-slicing decomposition, can produce black and white images that even more accurately portray the field lines. It would be interesting to have a simple way to display the "field lines" of hypothetical electric field, for the Mandelbrot set or for other images involving the level set method. A first approximation to these field lines may be obtained through binary decomposition.

"angle-slicing decomposition" refers to binary decomposition and the "electric field" is the same as that for the equipotential lines.

Colloquially, the equipotentials are "loops" that go around the Mandelbrot set without touching it, while the field lines touch the Mandelbrot set at one end and go off to infinity at the other end.

See Connectedness Proof.

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2023.

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