If you step across into the weirder side of maths, you might come across pictures of odd blobby shapes with lace-like edges. These are rather beautiful visual representations of Fractals.
If you haven't, see the Mandelbrot set and connected fractals below or do a quick google.
But what is a Fractal?
Visually, they are shapes with infinite detail at every scale. No matter how much you 'zoom' there will always be more detail. Additionally, the coolest of them are self reflecting, which means there are the same structures repeated at different scales.
They are also defined as shapes with a fractal dimension (I'll give me best go at explaining but you are probably better off watching a cool video or reading from a book).
Unlike Euclidean dimension, fractal dimension is expressed by a non-integer.
Take, for example, Helge von Koch's snowflake curve. It is a shape with a six-fold symmetry, much like a snowflake. It is created by getting an equilateral triangle, and then dividing each side into 3 and sticking an equilateral triangle on the middle part, and repeating the same process infinitely.
It is self-similar: it has three identical parts, which are each comprised of four parts that are scaled-down versions of the whole. Each of these four parts itself consists of four parts that are-scaled down versions of the whole.
The reason this produces such a powerful shape is the difference between the scaling factor and the number of parts it is divided into at each stage. Unlike a circle or an arc, who's scaling factor is equal to the parts it is divided into at every stage, the scaling factor of our snowflake is three, while the shape is divided into four identical subsections at every 'zoom'. The fractal dimension, D, is the power 3 is raised to to produce 4 (i.e.) D = log 4/log 3, or 1.26 (ish).
But... why should we care about odd little shapes like these?
You might have already seen from the snowflake example, or perhaps what you have observed yourself, that we can see fractals everywhere in our natural world. Mandelbrot, the creator of the most famous fractal out there (which I might dedicate a WHOLE POST to its so cool), was the first to suggest that fractals could be used to provide insight into modelling and describing these phenomena.
Some of the coolest of these applications include:
Medical: Cells grow in a fractal pattern, while cancerous cells tends to grow randomly. Mathematical analysis of tissue can help identify patterns that point towards disease.
Modelling wildfires.
Modelling the growth of cities as they expand and absorb smaller towns and communities.
Modelling the financial market