How does the concept of a fractal differ from that of a traditional geometric shape
In: Mathematics
A fractal is a shape such that when you zoom in, it looks the same. You could zoom in or out indefinitely and see the same shape.
This does not work for a traditional shape, you can’t keep zooming in on a triangle to find you see another triangle
In geometry, we can make statements about the relationship between the lengths of different sides of a shape. Fractals’ sides are of infinite length, or they have infinitely many sides.
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You know how a _really_ big circle looks like it’s a straight line when you can only see a small piece of it? And the tighter you zoom into it, the flatter it looks until it’s virtually indistinguishable from a straight line?
Now, take a look at [this piece of land](https://www.google.com/maps/@50.2335678,-5.4961855,28521m/data=!3m1!1e3?entry=ttu), at the tip of Cornwall. As you zoom further in, instead of straight lines, you start seeing more and more details along the coast. Cuts and ridges and rocks and stuff. It doesn’t “want” to become that simple straight line you’d get from zooming in on a circle.
Fractals are like the mathematical version of that map: their distinguishing feature is that, no matter how tightly you zoom in, you never reach that “looks like a straight line” phase. There’s always more detail, more structure, more stuff happening. Where a triangle has three points where it’s not nice and smooth, fractals are not nice and smooth _anywhere at all_.
The most famous sort of fractal, and the sort everybody else is talking about, is self-similar fractals. Literally: shapes that are similar to themselves when you zoom in. [This (very trippy) gif](https://upload.wikimedia.org/wikipedia/commons/a/a4/Mandelbrot_sequence_new.gif) shows that self-similarity for the Mandelbrot set, which is perhaps the single most famous fractal.
The informal understanding of a fractal is that it’s a shape that is self-similar. That if you zoom in, you see shapes that look the same as the whole thing, and this zooming can be done infinitely.
However, a mathematical definition of a fractal is any shape that has *a fractal ‘dimension’ that exceeds its “topological dimension”* (among other definitions – there’s no unified definition of a fractal, and there are multiple different fractal dimensions), where this these dimensions are calculated by an extension of concepts like length, area or volume that we are familiar with. The definitions involved can get quite involved and deal with things relating to covering the shape so I’ll avoid detailing them here (you can look them up). It also often involves non-integer dimensions (the Sierpinski triangle has fractal dimension of about 1.585). It just so happens that many of the famous fractals do exhibit some level of self-similarity.
A traditional geometric figure encloses a finite area with a finite perimeter. It may be a complicated shape, but if you add up all the sides, it eventually settles down to a single, finite number.
A fractal that can be drawn in two dimensions has infinite length, and may or may not enclose an area. No matter how carefully you try, you can’t add up the length, because it has infinitely many sides which don’t shrink fast enough to settle down to a single result. The coastlines of islands are an example of a non-self-similar, closed fractal (it returns to where it started). The coastline of a country that doesn’t have exclusively ocean borders would be an example of a non-enclosed non-self-similar fractal.
Fractals can also exist in higher dimensions, e.g. ones made up of areas (such as the Koch snowflake) or volumes (e.g. the Menger sponge). Most fractals that have names are self-similar ones, which are technically only a tiny subset of all fractals, but this self similarity has special properties that make it interesting as an object of study for mathematicians.
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