Continuous Dwell

Robert P. Munafo, 2023 Mar 20.

"Continuous" dwell refers to any method of calculating dwell that gives a continuous value, i.e. a value that varies smoothly rather than jumping between integer values. This allows for more variety in coloring algorithms, and is essential for making a 3-D shape of the Mandelbrot set, or for visualisation of such things as ridgelines.

The simplest method is to choose a fairly large escape radius (such as 1000), then perform the ordinary iteration algorithm. Let Z_n be the value of the iterate that exceeds the escape radius. The normal dwell value would be n (as shown in the pseudo-code in the escape-iterations article). The "continuous dwell" value is given by

D = n + log₂(log₂(|Z_n|)) - log₂(log₂(EscapeRadius))

Variation

The formula above will give something that is devoid of any dwell band boundaries, because the part with the two log₂(log₂()) expressions will be a fractional value from 0.0 to 1.0, and each band will blend smoothly into the next. You may wish to create distinct color transitions at the integer dwell bands by multiplying the log₂(log₂()) section by a constant K so that it goes from 0.0 to K.

Comparison


Normal (integer) dwell bands	Continuous dwell

The entire Mandelbrot set


Normal (integer) dwell bands	Continuous dwell

An area in seahorse valley, -0.74546 +0.11669i @ 0.01276

revisions: 20221202 oldest on record; 20230320 link to 3-D and ridgelines, and Wikipedia "continuous function"

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