Smooth iteration count for the Mandelbrot set

The exterior of the Mandelbrot set is usually colored by using the number of iterations. As the number of iterations is a discrete quantity, this leads to the occurrence of banding in the coloring. In this article, I will derive a formula for a fractional iteration count. This is by no means novel; plenty of derivations can be found on the internet. However, I found most derivations to be fairly hard to follow, and hope that my explanation may be helpful to at least some people.
Comparison of the classical iteration count with the smooth iteration count

Mandelbrot formula

To check if a point is in the Mandelbrot set, we iterate

If , the set is outside the Mandelbrot set, otherwise, it is in the Mandelbrot set. In practice, we just iterate a fixed number of times, and check that stays within an escape radius . It can be shown that when for some , then . So, in a sense, is big enough. However, when we color the fractal based on the number of iterations, the escape radius matters, so in this article I’ll use a general escape radius .

I will use the following conventions: the magnitude of the iterand is denoted by . The first such that for a given is called the iteration count for the point .

While the iterands (and thus, their magnitudes) depend on the starting point in a continuous way, the iteration count ‘jumps’ to discrete values. Suppose that is a very small number. A starting point for which will have an iteration count of , while a starting point with will have iteration count . If we would have some kind of measurement for the amount by which the iterand has escaped the escape radius , we could use this to assign a fractional part of the iteration count. That is, the iterand with would have an iteration count slightly lower than , while the iterand with would have an iteration count slightly higher than .

Ideally, we would like to have a smooth function such that for , and find the smallest such that . Instead of trying to find such a function, we will try to approximate it locally.

Suppose that is the smallest number such that . We then have . Assuming that is small with respect to , we get

Likewise, we find , or, more generally, . Note that this is a perfectly fine approximation even when is real. So now we can try to solve

for . In doing so, we find . Since , we have , so for the fractional iteration count we find

where is the escape radius, , and is the classical iteration count (i.e. the smallest such that ).

Exercise: a) Suppose that we instead chose to approximate at instead of . Show that we would have end up with:

b) Show that these expressions are the same. Hint: Use and .

c) Show that taking if we would take another base instead of for the inner logarithm, we would still get a smooth coloring, but if we would take another base instead of 2 for the outer logarithm, we would get a non-smooth coloring. Hint: use that .

It is not hard to see that the smooth iteration count is continuous. Imagine we are picking a starting point in a fractal colored with the discrete iteration count. Suppose the starting point has iteration count . If we move the starting point around a little bit, but no so much that the discrete iteration count changes, the smooth iteration count is a continuous function of . We can visualize this as moving the starting point without leaving the ‘band’ that the starting point is in. When we move the starting point so much that it approaches the outer edge of its band, we see that the smooth iteration count will approach . Likewise, if we move the starting point near the inner edge of the of the band, the smooth iteration count will approach . This means that the smooth iteration count is continous, even as the starting point crosses bands, since the smooth iteration count will assume the value on the border between the th and the th band.