Escape-Iterations

Robert P. Munafo, 2025 Mar 10

"Escape-Iterations" is the name of a function often used to apply an easily-computable viewing representation to the Mandelbrot Set. It is a simgle-valued integer function that computes the iterates to using the simple algorithm shown below.

It is the easiest to compute, but perhaps the least informative: It serves adequately at showing the mu-atoms, but only hints at the presence of the filaments by providing contours that generally tend to be spaced more closely near the filaments.

function escape-iterations param("Z"): complex param("max_iterations"): integer result: integer begin function declare C : complex declare Z : complex declare iterations : integer declare still_iterating : boolean let still_iterating = true let iterations = 0 let C = Z while (still_iterating) do if (magnitude(Z) > escape_limit) then let still_iterating = false else if (iterations >= max_iterations) then let still_iterating = false else let Z = Z * Z + C let iterations = iterations + 1 end if end while let result = iterations end function

The magnitude function computes the absolute value of the complex number Z.

There are other functions that can be used instead of escape iterations. The distance estimator function gives the best picture of what the Mandelbrot set "really" looks like.

revisions: 20080317 oldest on record; 20250310 update code to give a dwell of 0 for "already-escaped" values of C, to agree with the standard definition of dwell

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This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2025 Mar 10.

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