Pictures of Julia and Mandelbrot Sets/Postscript

Julia and Mandlebrot sets broke new ground in 1980s

The Julia and Mandelbrot sets were the first fractals that really amazed the world when they appeared in the 1980s. These pictures displayed patterns that differed entirely from anything previously known and beyond anyone's imagination. They could furthermore possess real beauty. The hitherto known fractal patterns were sets that had more or less interesting mathematical properties, and when they could be visualized, they could exhibit thought-provoking shapes, but they were not as impressive as the new types.

Popularity of Fractal Art

Because of the Julia and Mandelbrot sets - not least the simplicity of the software for drawing them - the concept of fractal became very popular. This concept has since become extremely widespread leading on to many varieties of fractal art. This development has revealed an extreme variation in possible patterns. Pictures and animations have been made of great artistic value or displaying an elaborate piece of mathematical work. However, all the serious work has been on other things than improving the basic presentation of the Julia and Mandelbrot sets. Pictures of fractal sets in their "pure" form, without artistic elaboration, do not seem to appeal to people as much today as they did in the 1980's and work on improving the basics has slid into the background.

Lack of Awareness of flaws in 'standard' Renderings

Until early 2000 there was little or no progress as regards the basic drawing technique. Until 2003 not a single picture had been made where the boundary was drawn, apart from some pictures of the usual Mandelbrot set and its Julia sets. Also the colouring based on the real iteration number only really works when the cycle is a fixed point, and even there leads to avoidable 'banding' effects, yet this colouring is in almost universal use, even when the cycle is not a fixed point. Furthermore, until early 2000, not a single Mandelbrot set had been constructed from two finite and different critical points.

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  Mandelbrot set constructed from one point that is not a critical point; rendered without boundary and with jumps in the colouring
• 
  Mandelbrot set constructed using two different critical points; shows boundary in the upper half part and continuous colouring

The two images above show the opportunity for a more mathematically correct and more aesthetically pleasing way to render fractals than the method commonly used. The right hand image shows the boundary correctly and it does not exhibit the banding effects. Can this opportunity for improvement really be true? Can it be true, that twenty years after the Julia and Mandelbrot sets first saw the light of day, only now are technically perfect pictures beginning to turn up?

A possible explanation

The most influential book about Mandelbrot and Julia sets is probably Peitgen & Richter: The Beauty of Fractals from 1986. In this book you can see the first real pictures of the Mandelbrot set, because they were drawn by distance estimation and in black-and-white. Mandelbrot saw exciting patterns when he zoomed down into his set - stellate patterns of logarithmic spirals and uncanny and "hieroglyphical" reduced-size versions of the Julia sets - but he only see all this, because his maximum iteration number was low. The right hand picture below, with boundary and maximum iteration number 10,000 shows how the Mandlebrot set should look. The left picture is less mathematically precise, using 1,000 iterations, and is not calculating the boundary. If the left picture were drawn with the same maximum iteration number of 10,000, but still without the boundary calculation, we would only have seen the mini-mandelbrot visible in the centre of the right image. The other detail would be 'too fine' to see.

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  Without boundary calculation and with max iteration number 1,000
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  With boundary calculation and with max iteration number 10,000

Misconceptions from a Flawed Example Program

The Beauty of Fractals gives an almost correct computer program for the distance estimation shown in the right image. A possible reason that that method did not gain ground is that the procedure in this program is seriously flawed: The calculation of $z_{k}$ is performed (and completed) before the calculation of $z'_{k}$, and not after as it ought to be ($z'_{k+1}$ uses $z_{k}$, not $z_{k+1}$). For the successive calculation of $z'_{k}$, we must know $f'(z_{k})$ (which in this case is 2$z_{k}$). In order to avoid the calculation of $z_{k}$ (k = 0, 1, 2, ...) again, this sequence is saved in an array. Using this array, $z'_{k+1}=2z_{k}z'_{k}+1$ is calculated up to the last iteration number, and it is stated that overflow can occur. If overflow occurs the point is regarded as belonging to the boundary (the bail-out condition). If overflow does not occur, the calculation of the distance can be performed. Apart from it being untrue that overflow can occur, the method makes use of an unnecessary storing and repetition of the iteration, making it unnecessarily slower and less attractive. The following remark in the book is nor inviting either: "It turns out that the images depend very sensitively on the various choices" (bail-out radius, maximum iteration number, overflow, thickness of the boundary and blow-up factor). Is it this nonsense that has got people to lose all desire for using and generalizing the method?