Fractals that have never been discovered before

Question

I have a maths competition and I've cracked till round 2. This round mainly states that I require to make a fractal that's not on the internet (yet) and explain it.

To be honest, fractals is an alien area for me and would love some help In designing new ones. My queries are:

$1.$ Can I actually make a fractal out of mathematical formula?
$2.$I read about Apollonian Gasket and Seirpinski Triangle but didn't really understand this line in it's properties:

Thus the Sierpinski triangle has Hausdorff dimension log(3) / log(2) ≈ 1.585, which follows from solving $2^d = 3 $ for $ d$.

What exactly do they mean by $\text{Hausdroff dimension}$?

$3. $ which computer software do I need to create them?

Thank you for you help!

Answer 1

1.) Yes you can. Well, not really with one, but with multiple formulas your can. The general idea is to take a set of affine functions on $\Bbb C$ with the property that for each function $f$, there exists some $\lambda_f \in [0,1)$ such that for all $z,w\in \Bbb C$, $d(f(z),f(w))\leq \lambda_f d(z,w)$, where $d$ is the usual Euclidean metric. Then this set of functions acting on the plane must have a unique compact fixed set, which is a fractal. To find this fixed set, iterate your function a bunch of times on a square in the plane and it will appear.

2.)The idea of dimension came, originally from geometry. The basic idea of dimension for shapes is that it is the unique integer such that it's volume measured in that dimension is not infinite, but in all smaller dimensions it is. For example, the 1 dimensional volume of a filled in square is infinite, it's two dimensional volume is the square of the sides, and all higher volumes are zero. This falls apart for fractals, which usually have infinite arc length but zero volume. Thus the idea of fractional dimension, hence the name fractal, was made, and we measure the new sense of dimension using the Hausdorff dimension. The basic idea is to compute this unique number where volume is meaningful. For special spaces as described in part 1, you can calculate the Hausdorff dimension by solving $$\sum \lambda_f^d = 1$$ for $d$, which is what the author did.

3.) Maple has some nice packages: IterativeMaps, Fractals

Answer 2

3Blue1Brown has a fine video explaining Hausdorff dimension.

Yes, you absolutely can make a fractal by just picking a formula and trying to iterate it. Fractals are common, almost anything you try will work.

What software should you use? Whatever you like! Fractal plotters are really, really easy to write. It should take less than twenty lines of code to draw a Dragon curve or a Julia set or whatever starting point takes your fancy. Then you can start changing things until you find something you like. Example: here's some code I knocked up to draw the Logistic map: https://jsfiddle.net/a6jcgujp/.

And while I do believe writing your own software is probably the way forward if you seek something new (you'll know exactly what it's doing, and there's nothing you can't change), I can't resist pointing you at my own explorations in this area... :)