How does the concept of a fractal differ from that of a traditional geometric shape
In: Mathematics
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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