It’s been quiet on this blog. There is something beautiful about that. Still, we must plow on.
Although I haven’t blogged at all for a while, I am steadily progressing with my research into fractals. What started as a project to “grow” organic machines and devices has become a study of mathematics. I realised that the approach of defining fractals by reference (branch) functions opens up a lot of new avenues. One that I am pursuing right now concerns fractal transforms, i.e. transform ordinary functions into a fractal space where they are defined by a fractal scaffold rather than the usual Cartesian coordinate system.
I am not sure what this transform will allow me to do but it seems an interesting avenue to pursue. One of the formulations I wrote concerns objects like the one above, i.e. tree structures whose canopy (or boundary function) exactly matches a given function (when we let it grow indefinitely).
The example above shows a paraboloid after 5 iterations of the fractal tree function. Is this a fractal? Or is it a tree? It is a tree for a given number of iterations. However, when we let the iterations go to infinity, there is a one-to-one match between that boundary function and the paraboloid for all x,y,z in R3 (within the domain). However, the derivative of the fractal surface is undefined and takes on all values between any two points no matter how close. It is thus, a fractal.
I am not certain where all this is going. I am touching graph theory, topology, number theory, complex functions, fractals and who knows what else. Let’s see.