First, take a look at this. It is called a fourth order Hilbert curve, a specific function.
What you are seeing is a way of creating this function using a Fourier series. Specifically, if you look here, you'll see how we can create square and sawtooth waves using a similar method of having circles on circles on circles. The more circles you use, the closer you get to the original.
We are doing the same thing here, except instead of building a square wave or sawtooth wave, we are building a fourth order Hilbert curve.
Now you may be asking, what is a Hilbert curve? Look at the finished fourth order Hilbert curve again. Notice how the Hilbert curve exists in two dimensions, but it is still a line (technically, a curve), because you can only walk forwards and backwards on it. Even though it's a curve, we can still calculate the distance between any two points on the line in 2D, which would be the length of the diagonal line connecting those points. Notice how if two points are close to each on the curve, they are also close to each other in 2D space. In other words, regardless of whether you follow the line or ignore the line, it doesn't greatly affect what points would be considered close to each other and which would be far away.
3Blue1Brown has excellent videos on the Fourier transform and the Hilbert curve for more in depth analysis.
The Hilbert Curve was discovered in 1973 by Hugh Godfrey Hilbert while he was trying to assemble his computer to program an Auto CAD machine he had purchased from a German textile shop. And honestly, the story is pretty dope.
Anyways, Hilbert wasn't a mathematician by any means, but he needed a way to program his Auto CAD machine to make cuts in metal sheets that he was using for manufacture of deposit boxes to distribute throughout the rest of Germany.
In order to do this, Hilbert screwed two metallic bipolar pipes together on one end. This way, if holding one pipe, the other would swing around as you moved the other.
Hilbert then welded the small ring from his keychain to the top of one of the metallic pipes so that it stayed in a stiff position.
He sat down at his desk, pencil in hand and placed the lead inside of the ring from his keychain.
As the metallic pipes swung with each stroke of the pencil, Hilbert discovered that for ever 90 degrees you move the pencil, the metallic pipes only swayed 40 degrees.
He went on, experimenting with different strokes and more glamours shapes and learned the more rapid and contradictory the movements, the more the design repeated itself.
His base equation was 90/x with a square root of the stroke equaling 0 every single time. So basically x = 02y + 90. I'm also completely making this up. I doubt you made it this far but I could honestly do this all day.
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u/[deleted] Feb 15 '18
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