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Sorento7
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Do you know why a simple set used for construction of the Mandelbrot's fractal results in a jaw dropping shape?
Sorento7 said:Do you know why a simple set used for construction of the Mandelbrot's fractal results in a jaw dropping shape?
Good point on the impact of rounding errors on final shape. But this only changes the question without answering it, why should this simple set introduce such weird rounding errors producing a fractal?Bill Simpson said:Contrary position.
In the places where there are modest numbers of floating point calculations, and thus modest numbers of tiny floating point roundoff errors introduced before stopping, the picture is very simple.
In the places where there are vast vast numbers of floating point calculations, and thus vast vast numbers of tiny floating point roundoff errors introduced before stopping, the picture is very complicated.
Decades and decades ago an author in Byte magazine was desperately trying to get his Mandelbrot images done in time for the publishing deadline. He was using the official blessed Intel software floating point library. Because of the processor speed it was taking him hours and days to produce each image. Then someone loaned him a wildly expensive 80287 floating point chip so that he could get done in time. This was supposed to produce EXACTLY the same results. He accidentally happened to compare the results from the software and the hardware and found that in the simple places the pictures were the same, but when he looked at the more complicated parts of the pictures he saw small differences and the more complicated the more different. At that point he hit the deadline, ran out of time, mentioned all this in a few sentences in his article and that was the end of it.
Since reading that I've wondered what it would look like if someone would "subtract" the Mandelbrot done with floating point approximations from the same done with arbitrary precision exact calculations. I suspect the difference would look almost exactly like the Mandelbrot pictures that everyone displays. If so that would go a long way towards saying that the pictures we all recognize are to a substantial extent just floating point errors in color.
Years ago someone sent me a link to a journal article supposedly on exactly this subject. Unfortunately I lost the link before I was able to follow up on that.
Sorento7 said:Good point on the impact of rounding errors on final shape. But this only changes the question without answering it, why should this simple set introduce such weird rounding errors producing a fractal?
Mandelbrot's Fractal, also known as the Mandelbrot set, is a mathematical set of complex numbers that is created by iterating a simple mathematical equation. The resulting set displays an infinite amount of intricate and repeating patterns.
Mandelbrot's Fractal is created by repeatedly applying a mathematical equation to each point in a complex plane. The points are then categorized as either belonging to the set or not based on whether the resulting calculations remain bounded or not.
Mandelbrot's Fractal is mesmerizing because of its infinite complexity and the seemingly endless patterns that emerge from a simple set of rules. The fractal also displays self-similarity, meaning that smaller parts of the set look similar to the whole, adding to its mesmerizing quality.
While originally discovered purely for its mathematical significance, Mandelbrot's Fractal has found applications in various fields such as physics, engineering, and computer graphics. It has been used to analyze natural phenomena like coastlines and stock market fluctuations, as well as to create visually stunning images and animations.
While a basic understanding of complex numbers and mathematical concepts is needed to fully comprehend Mandelbrot's Fractal, anyone can appreciate and be fascinated by its beauty and complexity. There are also many accessible resources and visualizations available to help understand the concept in simpler terms.