I grasp how fractals can be self-similar and have other weird properties. But I don’t quite get how they can have fractional dimensionality, even though that’s the property they’re named after.
How can a shape have a dimensionality *between*, say, two and three?
In: Mathematics
Imagine you’re drawing a line. Easy, right? That’s one dimension, like a road stretching out forever in front of you. Now, picture a square. That’s two dimensions, like a piece of paper you can walk around on.
But here’s where things get wild: fractals are like shapes that live in between these dimensions. They’re like a mixtape of dimensions, if you will! They have this crazy ability to wiggle and squiggle in all sorts of ways, repeating patterns on smaller and smaller scales.
So, when we talk about fractal dimensionality being fractional, we’re saying that these shapes fill up space in a really special way. They might look like they’re two-dimensional, but when you zoom in, you’ll find they’re actually more complex, like a never-ending maze of twists and turns.
It’s like trying to measure the coastline of a super wiggly island. The more you zoom in with your measuring stick, the longer the coastline gets because you’re counting all those tiny bends and curves. That’s why fractals can have dimensions that fall between whole numbers, like 2 and 3.
So, next time you see a fractal, remember: it’s not just a shape, it’s a dimensional enigma waiting to blow your mind
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