Several days ago we went on an adventure exploring the Koch Snowflake. During our exploration, I told you to trust me when I wrote that
A few days later, we learned that if , we could write
And since in our example , my original equation was correct.
Of course, I also stated that the snowflake somehow existed in a space that wasn’t quite 1 dimensional, but it also wasn’t quite 2 dimensional. That is, it existed in a non-integer dimension. What the what? That’s crazy talk, you say. Probably. But let’s explore this further.
Most of you have probably heard of a Fractal. The basic idea here is that you have an object that has a property known as self-similarity. In a nutshell, if I were to zoom in on any part of the object, it would look like the original object. If you think about it, one could easily get lost in a fractal because every region looks like every other region under some magnification. It’s almost trippy if you think about it too long.
Regardless, the important point is that fractals are self-similar. Can we say the same about other shapes? And if so, what might this tell us about dimension?
Consider something in a 1 dimensional world: a line of length 1. If we magnify the line 2 times, clearly we would end up with 2 copies of the original line. That is, a 2x magnification gives us 2 copies of the original object. If we magnified the line 3 times, we’d have 3 copies. In fact, if we magnified the line times, we’d have copies.
What about something in 2 dimensions such as a square with side length of 1? If we magnify the square by a factor of 2, you can imagine that we would end up with 4 copies of the original square. In this case, a 2 times magnification gives us 4 copies of the original object. A 3 times magnification would give us 9 copies, and in general, an times magnification would give us copies.
In a similar manner, a regular object in 3 dimensions would give us 8 copies if the original were magnified 2 times. And in general, an times magnification would give us copies.
- Scaling up something by a factor of in 1 dimension gives copies of the original,
- Scaling up something by a factor of in 2 dimensions gives copies of the original, and
- Scaling up something by a factor of in 3 dimensions gives copies of the original.
In general, if we scale something up by a factor of in an -dimensional space, we will end up with copies. Cool! That is, our scaling value raised to the dimension gives us the number of copies . In mathy terms
We’re going to rearrange this in order to determine the dimension of a fractal. That is, we need to solve for . Taking the of both sides and isolating we get
So, the dimension of an object is simply the ratio of the of the number of copies one gets by scaling the object by a factor of , to the of the scaling factor . Sweet freaking awesome.
In the case of the Koch Snowflake, remember that we started with a line of length 1. We then removed the middle third and inserted 2 lines of equal length (both being one-third the length of the original line). If we were to magnify the line by a factor of 3 so that each of the line segments of length one-third looked like the original line segment of length 1, you’d see that we’d have 4 copies of the original line. That is, while . This gives us a dimension of
Cripes on a cracker - an integer dimension. That is, the Koch Snowflake isn’t quite 1 dimensional, and it isn’t quite 2 dimensional either. Wild stuff!
For those that think this is all bunk, consider this: the Koch Snowflake starts with a line. We then remove the middle third and add 2 line segments measuring one-third the original. The result is a bent line. If we continue this forever, we still end up with a bent line. More specifically, an infinitely bent line. So no matter how much we zoom in, it’s always going to look fuzzy (due to the infinite bending). We won’t ever see crisp lines – things that exist in the 1 dimensional world. But by the same token, we can’t say that the fuzziness that we see represents something that exists in the 2-dimensional world either. Technically the lines have only length, but no width. So clearly this object exists in some sort of nexus between these dimensions. How mind-blowing-ly awesome is that? Personally, I find the idea of a non-integer dimension completely satisfying.
Happy Mathematics Awareness Month all y’all!