Mandelbrot Mosaic

Have a funny (or intriguing) mathematical picture? Let's see it.

Mandelbrot Mosaic

I would not normally pimp my own website here, but I imagine that a significant subset of the folk here might be interested in the following image:

There is a much larger version here (it is nearly 12 MB, so think before clicking the link). It looks like a fuzzy image of the Mandelbrot set, but is actually a mosaic of 250,000 Julia sets (you can actually make out some of the detail in the larger image). A more complete explanation can be found on my website.

xander
Last edited by xander on Thu Jul 28, 2011 10:40 pm, edited 1 time in total.

xander
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Location: Sparks, NV, USA

Re: Mandelbrot Mosaic

I've been meaning to ask this question for a long time...

is the circumference of the mandelbrot shape finite or infinite?

it took me 5 minutes (and the post on your website) to realise, just how awesome that mosaic is. I first thought you just changed the colorschemes in a way that it would represent the mandelbrot set, but now I realise, that you didn't and that the mandelbrot set and the julia sets are connected with each other. amazing o_o
Q.E.D. , or not?
Zapp
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Re: Mandelbrot Mosaic

Zapp wrote:I've been meaning to ask this question for a long time...

is the circumference of the mandelbrot shape finite or infinite?

Its circumference, and every single branch of it, has infinite length, assuming an infinite iteration depth.
Kindergarten

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Re: Mandelbrot Mosaic

Zapp wrote:I've been meaning to ask this question for a long time...

is the circumference of the mandelbrot shape finite or infinite?

The perimeter of the Mandelbrot set is infinite. As you "zoom in" on the boundary of the set, you will continue to find increasing details (i.e. additional "bays" and "peninsula" along the boundary, if you assume that the Mandelbrot set itself is a kind of lake or ocean). As you measure these details with a finer and finer measuring stick, the length of the perimeter will increase without bound.

Zapp wrote:it took me 5 minutes (and the post on your website) to realise, just how awesome that mosaic is. I first thought you just changed the colorschemes in a way that it would represent the mandelbrot set, but now I realise, that you didn't and that the mandelbrot set and the julia sets are connected with each other. amazing o_o

In fact, when Mandelbrot "discovered" (invented?) the set, he was actually interested in Julia sets. The Mandelbrot set is a kind of "map" of the Julia sets. Julia sets generated with a constant from within the Mandelbrot set will be connected, while Julia sets generated with a constant from outside of the Mandelbrot set will consist of disconnected components (or can be empty, I believe---my knowledge is not that deep). If you want to get some idea of how a Julia set will behave, it is helpful to see where that set is in relation to the Mandelbrot set.

xander

xander
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Joined: Fri Feb 11, 2011 12:14 am
Location: Sparks, NV, USA

Re: Mandelbrot Mosaic

wow, learning something new everyday!
Q.E.D. , or not?
Zapp
University

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Joined: Thu Feb 10, 2011 3:49 pm

Re: Mandelbrot Mosaic

I was playing around with parameters a bit, and found this:

xander

xander
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Posts: 154
Joined: Fri Feb 11, 2011 12:14 am
Location: Sparks, NV, USA

Re: Mandelbrot Mosaic

I think this one is pretty nifty, too:

xander

xander
University

Posts: 154
Joined: Fri Feb 11, 2011 12:14 am
Location: Sparks, NV, USA