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
Copying from a post of mine from some time ago:
If you double a line segment in size, you can cut it in 2^^**1** pieces each of which is identical to the original. If you double a filled square in each direction, you can cut it into 4 = 2^^**2** copies of the initial square. A doubled cube consists of 8 = 2^^**3** cubes of the original size. Thus it stands to reason that the **dimension** of a thing is the exponent you get there: a segment is **1**D, a square **2**D and a cube **3**D.
Now look at a fractal, lets pick the [Sierpinski triangle](https://upload.wikimedia.org/wikipedia/commons/thumb/4/45/Sierpinski_triangle.svg/1280px-Sierpinski_triangle.svg.png) as an example. If you double it in every direction, you get something that consists of _3_ copies of the original version. By the above, the dimension **d** of that thing thus must satisfy 3 = 2^^**d** . Now that is not solved by any whole number at all, instead it turns out that **d** = 1.5849625… ! Which is truly the dimension of the Sierpinski triangle, hence it is a _fractal_.
However, the meaning of “fractal” has extended nowadays. It now also includes anything that has a whole dimension if it somehow is unexpected to be so. It also includes wider variations of what structures count.
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