Mathematics

Cnidaria_Fractalus

All of the posts and maths below have been superseded by a paper that presents all the research to date: Derivative coordinates for analytic tree fractals and fractal engineering

The posts are perhaps more user friendly but less formal and less accurate. I have taken care in the paper to correct errors in the formulations and introduce consistent terminology and notation.

8 January 2015



Going beyond the 3rd dimension: hyper dimensional fractals using the genetic equation.

Hyper dimensional flower

Hyper dimensional flower

An analysis of the genetic equation for arbitrary function: the genetic transform.

Coral growth (genetic fractal)

Coral growth (genetic fractal)

A grand overview of a closed and analytic form of genetic fractals: Genetic fractal formulation of a “Genetic Equation”.

Universe generated by genetic fractal "creation Equation"

Universe generated by genetic fractal “Genetic Equation”

When I started analyzing genetic fractals, I was focusing on L-system fractals, I came up with an interesting approach to create smooth fractal branches, i.e. analytic from a math’s perspective.

from the usual angular L-system fractals to smooth and analytic fractals
H-fractal genetic-fractals Continuous ronded H-fractal genetic-fractals

This led to a an analysis I reported here in a few chapters:

1. Introduction

2. Fractals as sets of paths

3. Path functions

4. Fractals and binary trees

5. Reference paths

Also PDF-ed together here.

2 thoughts on “Mathematics

  1. As you foresaw, I couldn’t keep myself from popping by. I read your paper (what you’ve written by now) and found it pretty easy to follow so far. Sometimes I struggled with the notation (in some stages I wasn’t clear on whether you were using vectors or some other way to express paths, for example) but I finally understood what you were doing. I’ll look forward to seeing the rest! My impression with this part was that you were just setting the background for the actual stuff…

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    • Thanks David, Great feedback. The formulations are discrete initiale, thema’s bevond continuous and are slapping into complex functions. I will try and make that explicit.

      You are right, I had to put the foundations in place. As far as I can tell, this reference path approach is new and I needed to build it from scratch.

      I am finalizing the chapter on continuous fractals which leads to fractal spaces. All of this is new, I believe. The genetic formulation is eAsy from there on and will lead to node transfer functions.

      It takes a lot of time to write this stuff as I need to find the right formulations first, verify that these are correct and then tie them into the mathematical story. But – I’m having fun 😉

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