21–27 of 27 posts
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Bingo. There are a finite number of atoms in a coastline therefore there is a finite length. I suspect the calculus to derive it isn't even all that hard.
"There are a finite number of atoms in a coastline" And atoms do not consist of smaller parts?
Earlier quoted context omitted.
I never understood how the algorithm could be so simple, using just IEEE floats ( https://en.wikipedia.org/wiki/Plotting_algorithms_for_the_Ma... ) These programs do thousands of iterations. How does one prove that the images show the Mandelbrot set and not some artifact of issues with numerical stability?
One can check this fairly easily. The iterative step is z -> z^2 + u, or in components a + bi -> (a^2 - b^2 + c) + (2ab + d)i Multiplying and adding real numbers between -1 and 1 are fundamentally stable operations when done with floats. The only possible source of instability is if a and b are very close, that a^2 - b^2 will magnify some errors from the limits of numerical accuracy. However, this is mitigated by two…
I've wondered, is the Mandelbrot set one continuous shape, or is broken into many smaller shapes? Put crudely, if it was imported into a paint program, and you performed a fill action, would the whole shape change to a single color?
Not sure why the existence of this isomorphism implies it is connected, but there you go
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You can run it with BigDecimals or whatever unlimited number type your language has just fine, and it will give a very similar looking image.
That doesn’t help in understanding why this thing is numerically quite stable. It only gives some confidence that it is.
I’m sure an error limit can be given, as it uses a quite basic arithmetics only.
Earlier quoted context omitted.
Bingo. There are a finite number of atoms in a coastline therefore there is a finite length. I suspect the calculus to derive it isn't even all that hard.
"There are a finite number of atoms in a coastline" And atoms do not consist of smaller parts?