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November 13

Does the Mandelbrot set zoom have any limits?

Dear Ladies and Gentlemen

My question is about the Mandelbrot set zoom which has become popular on internet video websites such as YouTube. Do these zooms have any limits or are they similar like a visual representation of a basically limitless potential? Are there theoretical limits as far as dimensions go? I beg your utmost pardon for my crude English, but foreign languages have never been my strength.

With kind regards--2A02:120B:C3E7:E650:18C8:AAB0:523A:9250 (talk) 01:17, 13 November 2019 (UTC)[reply]

The scale of an accurate zoom is limited by the arithmetic precision of the computer you're using; but if you mean whether there is a scale beyond which the M-set's boundary does not show fractal detail, then no. —Tamfang (talk) 01:44, 13 November 2019 (UTC)[reply]

Your English is better than many native speakers. :-) SinisterLefty (talk) 01:48, 13 November 2019 (UTC)[reply]

Slight amendment to Tamfang's answer, for the really deep zooms you see on YouTube you need Multiple-precision arithmetic to get the hundreds of digits of accuracy used. For normal exploration though, the arithmetic precision of a typical modern computer is fine. --RDBury (talk) 01:43, 14 November 2019 (UTC)[reply]

Question, can you do like Google Maps and recalculate at each new zoom level ? Or doesn't that approach work with fractals ? SinisterLefty (talk) 03:10, 14 November 2019 (UTC)[reply]

There are many ways to implement the basic process, but I think what you're getting at is something like XaoS which zooms interactively. There are other Mandelbrot viewers that do that now but XaoS was the first. --RDBury (talk) 12:46, 14 November 2019 (UTC)[reply]

I'm not sure if you got the full implication of my Q. If you zoom in 1000x, a million X, a billion X, a trillion X, etc., and find a scene that is exactly identical to the original configuration, couldn't you then just reset to the original configuration, and zoom in from there, instead of from the 1000x point, etc. ? This would theoretically allow you to simulate zooming in infinitely, with no concern over hitting computational limits. SinisterLefty (talk) 19:27, 15 November 2019 (UTC)[reply]

One of the interesting things about the Mandelbrot set is that no two configurations are the same. You may find "copies" of the set within the original, but there will be differences. For one thing, the copy will be attached to the main body by a thin strand while the main set has no such strand. Also, the copy will have decorations attached to it and the end of each antenna tip, and these decorations are different on each copy. If you look at a copy within a copy then you get two layers of decorations. With many fractals, specifically self-similar ones, you can do the infinite zoom thing you're describing, and if you look at the article for one of them then there's a good chance someone has done that and turned it into an animated GIF. (See e.g. Koch snowflake third picture down on the right.) The thing is, while the Mandelbrot set is often called a fractal, it's not actually a fractal according to the mathematical definition. While fractals keep the same level of complexity no matter how far you zoom in, the Mandelbrot set actually gets more complicated the deeper you zoom in. This isn't usually apparent in really deep zooms because the level of detail needed to see it is smaller that a typical screen resolution, or the human eye for that matter. I hope that answers your question; you mentioned Google Maps and that was confusing me. Last time I checked Google maps doesn't do infinite zooms. --RDBury (talk) 23:58, 16 November 2019 (UTC)[reply]

@RDBury: "Not actually a fractal according to the mathematical definition." The only precise mathematical definition I know for "fractal" is Mandelbrot's original one, namely that the Hausdorff dimension exceeds the topological dimension, and I believe that does hold for the boundary of the Mandelbrot set (it doesn't hold for the set itself, which has Hausdorff dimension and topological dimension both equal to 2).

Our fractal article says that Mandelbrot originally came to find that definition too restrictive, but does not offer any precise alternative definition. In any case, if the boundary satisfies the original definition, then it obviously satisfies any less "restrictive" one.

That said, looking into this led me to the multifractal article, which is interesting, so thanks for that. --Trovatore (talk) 01:26, 18 November 2019 (UTC)[reply]

Thanks for the explanation. The point I was trying to make with the comparison with Google Maps is that it doesn't just zoom in, you actually get different maps at different zoom levels. If they attempted to render the entire planet at the finest level of detail, it would no doubt take out their servers, but by only displaying the level of detail appropriate for each zoom level, they achieve acceptable performance while zooming. So, even if not self-similar, perhaps that part does apply to the Mandelbrot set. SinisterLefty (talk) 03:11, 17 November 2019 (UTC)[reply]

For maps, I don't think it would "take out the servers". It only takes a certain amount of data to generate the maps. Now, if you could zoom in that much in Google Earth, then you're right - that would take an enormous amount of data. But for the Mandelbrot set, you don't have to store much data - just do a lot of calculations. This is sort of a space–time tradeoff tradeoff. So zooming in Google Earth would ultimately be limited by the amount of data stored. The entire Mandlebrot set isn't stored on the computers, but it is calculated as needed. Bubba73 ^{You talkin' to me?} 03:35, 17 November 2019 (UTC)[reply]

Let me try again to specify what I mean. If you had a model of the universe, with every atom recorded with it's position, we would neither be able to store that nor display it all at once. Now let's say that this model of the universe has a variable level of detail. It may have the position of every galaxy, but only has the individual stars noted for the Milky Way, and only the description of the planets, moons, asteroids, comets, etc., noted for our solar system. Then, only Earth would have all the land forms noted, let's say just the US has the cities stored, and only Los Angeles has street level details. Within that only UCLA has individual buildings described, and the biology lab has the equipment listed. A particular piece of equipment has each Petri Dish listed, and for one, each bacterial colony is described in full detail, with one particular bacterium described down to the atom. So, we could now zoom in from the universe to an atom of that bacterium. So, then, could we do something similar with the Mandelbrot set, to discard all calculation outside the border we are interested in, and also at lower and higher zoom levels than we are interested in, and use the full computing power of the computer solely to render than particular screen's worth ? SinisterLefty (talk) 04:12, 17 November 2019 (UTC)[reply]

Oh, yes. When doing the Mandelbrot set, you only have to do the calculations for the area you want to display. Map each pixel in the display to the corresponding complex number, and do the calculations for that number. You can ignore everything else. Bubba73 ^{You talkin' to me?} 04:48, 17 November 2019 (UTC)[reply]

I mentioned storing the data for the Mandelbrot set. In a sense, the equation encodes everything there is about the set - you just have to do calculations to get it out. Bubba73 ^{You talkin' to me?} 23:23, 17 November 2019 (UTC)[reply]

There are no limits on the Mandelbot set - it goes on forever. Computers are limited in what they can compute, however. Bubba73 ^{You talkin' to me?} 02:28, 14 November 2019 (UTC)[reply]