I stumbled upon a curious mathematical beast: a fractal with infinite dimensions. I will be producing 3 dimensional renderings to get an idea what this ‘looks’ like, but let’s start with the maths. Imagine a trunk of a 1-dimensional tree, i.e. a fat line. After a while it splits into two branches in a 2-dimensional…
Published on arXiv, my new paper on analytic tree fractals (a.k.a. as genetic fractals). Title: Definition of geometric space around analytic fractal trees using derivative coordinate funtions Author: Henk Mulder Abstract: The concept of derivative coordinate functions proved useful in the formulation of analytic fractal functions to represent smooth symmetric binary fractal trees [1]. In…
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.…
Recalling derivative coordinate functions When I developed the equation for genetic fractals (paper) , I first defined a “path function”, i.e. an equation that could represent arbitrary curves. This is given as: The notation is slightly different with respect to the paper. is a derivative of the coordinate function and is a derivative of the…
As part of the search for the answer to the ultimate question that philosophers i.e. ourselves under a starry night, we end up asking the paradoxical question: what happened before the first thing happened? Or in the present day context, what happened before the Big Bang.
Sometimes I pursue an idea that doesn’t immediately lead anywhere. This is one of those. I wondered what the canopies of tree fractals would look like if you plotted them next to one and other whilst varying
The Koch curve, one of Mandelbrot’s “monsters” becomes a little more friendly when plotted using smooth (analytic) tree fractals. For details on this curve, refer to this paper.
This has been in the making for a while and I’ve finally taken, borrowed and stolen the time to treat genetic fractals scientifically. A paper entitled: “Derivative coordinates for analytic tree fractals and fractal engineering”. Abstract below. I’ve submitted this to Arxiv (reference: 1501.01675).
In this blog I have referred to artificial DNA or aDNA without really explaining what I mean by this. Initially I started using this term in the context of genetic fractals which are a mathematical model of natural structures and even organisms.
Update: please refer to this post for a detailed article We can extend the Genetic Equation to higher dimensions. Remember that the two-dimensional exponential form of the Genetic Equation is: Or in trigonometric form:
