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I wanted to know that how can one find the area of the Mandelbrot set or any fractal for that matter ?

I.R
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    The way I'd do it is by approximating the individual stages when building the fractal from within and without (like inner measure/outer measure). If the inner and outer measures converge to the same value, then I'd associate that area to the area of the fractal. This is the most general approach. However, if you had a specific fractal in mind (e.g. the Koch snowflake), there are much better geometric ways to do it. I do not think it is guaranteed that any given fractal will be measurable (i.e. that we can associate an area to it) which is why I mentioned the inner and outer areas. – Cameron Williams Aug 06 '15 at 19:03
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    what if we use integration? – I.R Aug 06 '15 at 19:27
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    Integration is equivalent to what I mentioned. – Cameron Williams Aug 06 '15 at 19:33
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    I haven't looked through it, but this paper seems relevant. – Akiva Weinberger Aug 06 '15 at 19:53

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As mentioned in the comments there actually two orthogonal ways of thinking about the "area" of a fractal.

You could consider the area to be a measure of the amount of space the fractal encloses.

On the other hand, you could think of trying to measure the "size" of the fractal itself.

The first method is very easy to do, you just find a recursion formula for the amount of area, and/or manually count the number of squares inside the boundary. According to Wikipedia the area of the Mandelbrot Set is about $1.506...$, the site has more digits. Here's the derivation for the area of the Koch Snowflake.

The second method can be either very difficult or extremely tractable depending on what properties you'd like to investigate.

First, the hard way. We define a measure, in this case the Haussdorf measure, using this. Basically, we extend integer dimension measures, like cardinality, length, and area to fractional dimensions. The problem is that finding the Haussdorf measure of even simple shapes is an open problem. There are some estimates for the Koch Curve and a few other sets.

However, there is an easy way to work with measures. Instead of working to find a numerical value for the measure, we simply define a certain fractal to have a measure equal to unity. This might sound trivial, but if you give up the intrinsic size measuring property of the Haussdorf measure, you can measure other interesting properties. For instance, instead of focusing on the exact measure of the cantor set, you could focus on integrating a function along this measure. See an example with the cantor set here. Evaluating Integrals by Self Similarity, by Bob Strichartz, is paper that lays this method out in more detail.

In fact, there's an entire tag on this site devoted to calculus with respect to fractals. It's called Fractal Analysis and you can find it here.

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