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.

Ideally, continuous dwell would be defined in such a way that all points on any given field line (as defined by the electric field model as given in equipotentials) have the same dwell. This is not possible with ordinary iteration, for reasons related to the fact that for most iterates Z_n, |Z_n+1| is not equal to |Z_n|². In other words, the lemniscates obtained with an escape radius of ER≥2 do not align with those when using an escape radius of ER².

However, with a fairly large escape radius the lemniscates are "close enough" to field lines so as to not be easily discerned at the resolution used for pixel images. Thus, one may choose a fairly large escape radius (such as 1000), then perform the ordinary iteration algorithm until it escapes. 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 + 1 + log₂(log₂(EscapeRadius)) - log₂(log₂(|Z_n|))

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 in the range (-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

Low Escape Radius

When a very low escape radius is used, the above formula for continuous dwell gives markedly different results. There are clearly discernable jump discontinuities along the lemniscates for the chosen escape radius, plus additional "edges" along portions of the lemniscates for the square of the chosen escape radius. Here is an example with dwell bands around R2F(2(3B1)B1), where pinch points are prominent:


Normal (integer) dwell bands	Continuous dwell

revisions: 20221202 oldest on record; 20230320 link to 3-D and ridgelines, and Wikipedia "continuous function"; 20250310 correct sign in the log(log(...)) terms, and reference equipotentials; 20250312 add low escape radius section

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