This one has been long in the making. Ever since formulating the maths of genetic fractals, I have been wondering about describing fractal space, i.e. the space around fractals.
I have just submitted a paper to arXiv which is scheduled to become available in a few days. I will update this article with the link. In the meantime you can access it here.
The maths is not suitable for the mathophobes but there are some beautiful results in there. Let’s have a crack at this.
It all starts with the analytic tree fractals like the one below which was made with the Fractal Explorer.
The question I have been pondering for a few years is: what if we consider each branch, could we define space around it so that we can fill that space with forms?
You will see that the answer is straight forward but it took me a while to find it. When you consider that Einstein took 15 years to produce the very elegant equations for General Relativity, I don’t feel so bad.
Here’s how you do it. Let’s say you are in a modern city with a grid road system. To find any place in that city, you just go so many blocks in one direction, turn left or right and go another number of blocks in that direction. You can define coordinate system like that. If you have a point (x,y) you go ‘x’ in a straight line, turn 90 degrees left (if y is positive) and go ‘y’ along a straight line. Below is an example when (x,y) = (5,3).
Now, what if you didn’t go along a straight line, would it still work? Below is an example that shows what that would look like. You follow a curve for an amount ‘x’, turn left and follow the curve for an amount ‘y’.
At first this may look weird. With such bent axis, you could and up anywhere. But as a way of defining a geometry that follows the shape of a curve, this is perfect. When you use this on a curve and draw a coordinate system, you will see why this is the thing to do. Wherever you are on this coordinate system, the grid is defined by the curve at that point.
Since we are looking for such a geometry for fractals, we just have to calculate the coordinate grid for the other branches.
This is the equivalent of a Cartesian grid but centred around a tree fractal. Since it is a fractal tree, after each branch point, the same point (x,y) needs to be drawn on each branch. This means that if the point has an x-coordinate that would take it to the extremities of this tree fractal, it would be drawn 16 times. Welcome to the weird world of geometric fractal space.
Now that we have defined the space around a fractal, we draw objects on that space. Below is a fractal space with an ellipse drawn on it. Because of the branching, it doesn’t look like an ellipse but it is. It is a tree fractal ellipse.
The image at the top of the page is also an ellipse. This time it is projected on a Koch fractal space.
In the paper I extend this to 3 and higher dimensions. In other words, you can use this method to create fractal objects in hyperspace. Awesome, if I may say so.
The maths itself for the hyperspace case is elegant. I’m quite sure that this is original maths. Although this blog will disappear in time, the maths won’t. Whether or not the original paper will survive, it is a nice thought that I was the first to write it, forever.