What would the perimeter of the mandelbrot set be? Would it be infinite? Using logical reasoning, I feel that it should be some finite number. I came to this conclusion because it could be roughly approximated with low detail then more detail could be added to get a more accurate number. However, the added detail becomes less and less significant as it is added. Therefore, it must approach some finite number.
Is this way of thinking correct? I don't believe I have the mathematical skills to calculate this number(I'm not even in university yet). would there be a way to calculate the mandelbrot set's perimeter? If so, how would you go about it? Thanks to anyone who can help :)
In this picture some equipotentials around the Mandelbrot set are plotted, labelled by approximate length. As you can see, at first the lengths decrease as you get closer, but it reaches a minimum at about $14.9$ before increasing rapidly because the curves have to wriggle a lot (the shape of the Mandelbrot set is complicated).
As per the comments on the question, the boundary of the Mandelbrot set has a Hausdorff dimension strictly greater than $1$, that is, it looks wriggly no matter how far you zoom in. In other words, counter to your intuition, the added details do not become less significant. This means as the $1$-dimensional equipotential curves more closely approach it they increase in $1$-dimensional length without bound, $\to\infty$. See https://en.wikipedia.org/wiki/Coastline_paradox
On the other hand, your intuition does hold for the $2$-dimensional area: the area enclosed by each curve in the picture does reduce, so you can bound the area of the Mandelbrot set (which is not known exactly) by the sequence of areas of the curves. It doesn't converge at all quickly, though.
In fact, as the Hausdorff dimension of the boundary of the Mandelbrot set is $2$, it could have a $2$-dimensional area (but this is not known either)
These are still open research questions that dozens (or hundreds?) of professional mathematicians haven't been able to solve yet, see https://mathoverflow.net/questions/37229/area-of-the-boundary-of-the-mandelbrot-set for some references.
Some other fractals are easier to work with than the Mandelbrot set. For example, the Koch snowflake curve can be formed by adding smaller triangles to the edges of the existing triangles, forever. It's not too hard to get explicit formulas for the perimeter and area of the $n$th-level construction, and taking limits as $n \to \infty$ see that the area is finite and get its exact value, but that the perimeter goes to infinity. The details of the calculations can be found at https://en.wikipedia.org/wiki/Koch_snowflake.