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
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Take a line. Double all its lengths. Now it takes 2 times as much space, or 2^(1).
Take a square. Double all its lengths. Now it takes up 4 times as much space, or 2^(2).
Take a cube. Double all its lengths. Now it takes up 8 times as much space, or 2^(3).
Those are examples of 1, 2, and 3 dimensional shapes.
Now take Sierpinsky’s triangle (google it if you need). Double all its lengths. There are 3 of the original triangle inside this new doubled triangle. It takes up 3 times as much space. 2^(log2[3]). So this fractal has dimension log2[3].
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Here’s a good video on the topic:
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.
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
Actual physics educator here (retired):
A piece of paper is one dimension. But wad it up, and it can only exist that way in three dimensions. Zoom in close enough though, and the paper is still two dimensional. So, you could say the paper exists as a 2.5 dimensional surface. That’s the fractional dimension.
In more practical terms, the surface of a globe is a fractional dimension. Two dimensions that can only exist in three dimensional space. (This also begets “non-euclidean” geometry. A triangle can be drawn from the north pole, to the equator, 1/4 of the way around the equator, then back to the north pole. *Three* *90 degree angles* in one triangle. It’s a fractional dimension, non-euclidean triangle.)
A straight line on a piece of flat paper is one dimensional. A slightly curved line on that two-dimensional surface might be 1.1 dimensional (there are ways to calculate it.) A super squiggly-wiggly line on the flat paper might be 1.9 dimensional. Now wad that paper into a ball, and it gets complicated…
Time dilation due to gravity makes space a fractal dimension somewhere between three and four.
Enjoy!
I had a professor crumple a paper ball and ask if this is a 2-dimensional object. Yes, but it occupies 3-D space. If you lightly crumple it and spread it out, that wrinkled plane is slightly more than 2. If you wad it into the tightest ball you can, it’s almost 3. Similarly for lines that fill a plane. Etc
Edit: spelling
While the other responses about doubling size are all true, I think they’re all missing the fact that there are different ways to use the word “dimension” in mathematics, and that’s what really causing your problems.
The everyday usage of “dimension” that we use in relation to height, width, depth is the Euclidean dimension. Or, fancier, the topological dimension of a Euclidean space. It’s always an integer.
The dimension the others have been describing is the Hausdorff dimension. It doesn’t have to be an integer, and it doesn’t need to correspond to the topological dimension of the space it’s in. It also doesn’t need to correspond to its own TD. The TD for the Sierpinksi triangle is 1.
There are other fractional dimensions, too. The box counting dimension, for instance, is distinct from the Hausdorff dimension.
You can have infinite dimensions, in some settings.
If you applied the doubling principle to the entire 3D space, you’d have the same 3D space. So… is the dimension of a three dimensional space 0? No, of course not.
Bottom line, this whole “fractals have fractional dimension” thing has lost some of it’s rigor as it’s been translated into popular science, so that’s why you have trouble making sense of it. That isn’t to dismiss the other answers — the doubling thing is intuitive and a useful measure. But I wouldn’t say that’s the whole picture.
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