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Comment by jrockway | Hacker News Reader

It's strange that his computer would choke on high zoom levels; computing the mandelbrot set is proportional to the number of pixels on the screen -- the zoom level is not relevant. (You just run a set of computations for the coordinates of each pixel you want to know about.) I guess if you get past the point where a double can represent the coordinates, you'll hit a slowdown, but I doubt that is at "5x zoom".

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The necessary accuracy keeps taking more and more bits, if you think about it. If you've only got, say, 200 bits of float to play with, then when your down at a resolution where things on the screen are closer together than that resolution can handle, you have to crank the bits up, or you'd just render the same color over and over. (Actually you have to crank it up before then, of course, I'm just reducing it to obvious absurdity for demonstration purposes.)

So, it's like the way adding integers is O(1), right? Sure... as long as your integers are always 32 or 64-bit and you don't mind a bit of light truncation. In general it's higher than that.

That's absolutely right, but it's not the only reason. The interesting bits of a Mandelbrot set plot are the ones that are on the boundary of the set itself. Remember that the Mandelbrot set is the set of points for which a particular iteration never diverges to infinity. So a point outside the set but very very close will diverge very very slowly: it will take a lot of iterations before you can tell it's outside the set. (The colours you see in a typical Mandelbrot set plot indicate how many iterations it takes.) So, if you zoom really close in then every point in your plot either lies outside the set but takes many iterations to determine that, or lies inside the set (and therefore takes as many iterations as you're prepared to do).

So you're doing more iterations, with higher-precision numbers, as you zoom closer in.

right, this kind of anim requires arbitrary precision arithmetic. i wonder how long it took to do. my 1080p work is about an hour per frame, eg http://www.archive.org/details/HighFidelityDemo would take >2 months on a regular computer, and HiFiDreams over 100 years.

That's not quite true – as you zoom in, you're no longer simply taking Re(c) = x pixel on the screen and Im(c) = y pixel on the screen. The more you zoom in, you're moving in on the Re/Im axes, which means you're no longer looking at integer values for the computation. You start looking at floating point values for x (Re(c)) and y (Im(c)) when computing the sequence. Therefore, the computer begins to do more and more floating point computations on, ostensibly, arbitrarily accurate fp.

EDIT: Added "pixel on the screen".

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