I never really understood, please bring examples as well.
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In the broadest terms, fractals are any shapes that have detail no matter how much you zoom in. If you zoom in on a [sine wave](https://ars.els-cdn.com/content/image/3-s2.0-B9780080252490501154-f110-01-9780080252490.jpg), eventually you’ll be too close to see the bumps, and after that you’ll be too close to even see the curve – most everyday mathematical functions become straight lines if you zoom in far enough, or perhaps have a few well-defined kinks. If you zoom in on the [Weierstrass function](https://en.wikipedia.org/wiki/Weierstrass_function), it doesn’t matter how close you get – it will always look just as bumpy as it did from the highest level.
There two properties that most fractals we look at have – fractional Hausdorff dimension, and self similarity (these are connected).
The Hausdorff dimension of a shape is how much the amount of that shape changes as you scale it up. For example, if you double the side of a square, its perimeter doubles – the curve making up the perimeter has Hausdorff dimension 1 (one doubling of the size leads to one doubling of the amount of the shape). However, the area of the square will become 4 times as large, meaning that segment of the plane has Hausdorff dimension 2 (one doubling of the size leads to two doublings of the area). If the square was the base of a cube, that cube’s volume will get 8 times as large, corresponding to Hausdorff dimension 3. You can see that for many everyday objects, the Hausdorff dimension is the same as the intuitive spatial dimension (1D, 2D, 3D).
Many fractals have an interesting property that their Hausdorff dimension is larger than it apparently should be. For example, if you double the scale of the x-axis that the Weierstrass function is sitting on, the amount of the curve will more than double. ‘Amount of curve’ is kind of tricky to define (it has infinite length), so I’ll skip the formality, but suffice to say that many fractals are so jagged, they act like they’re partway to being a higher-dimensional object – with respect to zooming, there’s so much curve it behaves a little bit like it’s an area.
A related property is that the shapes have some kind of self-similarity. This is often portrayed as literal self-similarity, like in the [Koch curve](https://en.wikipedia.org/wiki/Self-similarity#/media/File:KochSnowGif16_800x500_2.gif) – if you zoom in you literally see smaller identical versions of the shape repeat. However, fractals have a broader self similarity, in which the level of detail at each zoom level is the same. This is well exhibited by a [Mandelbrot zoom](https://www.youtube.com/watch?v=b005iHf8Z3g) – the amount of visible objects of various sizes is similar no matter how far you zoom in, even though you never get perfect repetition. In fact if this phrasing seems a little bit familiar, it is this self-similarity of level of detail that lets us define the Hausdorff dimension, which is about how much more of the shape comes into view as we scale it up.
However, like I said up top, while these properties are shared by many fractals, the truly defining characteristic is the roughness at every scale. If a line is infinitely rough, even if it perfectly behaves like a line, it will be a fractal. The point of fractal geometry was to study shapes that are not susceptible to calculus, which concerns itself with shapes that are smooth up close. Weierstrass, Hausdorff, and Mandelbrot had to invent entire new methods of analyzing shapes, because the tools of calculus are so utterly useless in this domain.
Apart from looking mesmerising, what practical, real-world applications do fractals have/serve?
A fractal is something that looks like itself when zoomed in. For instance a coastline on a map looks squiggly, but if you would (theoretically) zoom in, you would see more and more detail appear without it starting to look “different”, still squiggly as the line stars describing the edge of the water between rocks, between pebbles, beetween grains of sand etc…
Fractals are shapes that are “rough,” which means they look detailed no matter how far you zoom in, and they are all around us! Plants, rocks, mountains, coastlines, snowflakes, etc.
For the longest time, we’ve only done geometry with very simple, ideal shapes like squares, triangles, and circles because those are easy to understand and model. However, the real world is always rough, and some things like trees are just too detailed to be represented well by just those simple shapes.
Fractal geometry is a relatively recent invention (a few decades old) and is great because it finally gives us the tools to model and do math with shapes that are rough, which has allowed us to do cool things like make nature and terrain in video games look more realistic.
They are also quite pretty in and of themselves, probably because of their similarity to nature. We have been painting them since long before we understood them in a mathematical sense. Just look up “The Great Wave”
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