# 2D genetic fractals: a Makers Guide

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Having plastered pretty pictures all over the internet and even having a genetic fractal in an art gallery (yes…!), the time must have come to out genetic fractals beyond sharing beautiful maths equations. So here it is, a series on constructing genetic fractals in 2 dimensions. “Why only two?”, I hear you groan. Because 3 dimensions are HARD. Not too hard but we’d lose sight of how these fractals work. So, 2D first and if courage remains, we’ll do 3D.

This will be series of posts which I will also paste together in an article. Or a booklet. Hell, let’s go crazy! But as all good stories, in the beginning…

### The Genetic Equation

You may have seen this before on this site and that is not surprising. The creation equation is a mathematical formulation of a genetic fractal. Any genetic fractal.Here it is again.

$C =\int_s D_{r} e^{i\int_s D _{\varphi }}$

Or in trigonometric form:

$C=\left\{\begin{matrix}\int_s D_{r}sin(\int_s D_{\varphi}ds)ds\\ \int_s D_{r}cos(\int_sD_{\varphi}ds)ds\end{matrix}\right..$

In today’s installment, we’re just going to focus on geometry inside this equation. No fancy branches and fractal patterns, just the simple lines (y against x) that this equation will generate.

To try out this equation, we need the driver functions $D_{r}$ and $D_{\varphi}$. We can make them as simple or complex as we want. Since for this first test we can use Excel, we can make these driver functions arbitrary and play around with them. But just to reassure you that there is no slight of hand, if $D_{r}=1$ and $D_{\varphi}=0.3141$ then the creation equation becomes

$C =\int_s e^{i\int_s 0.3141}$

Below is an Excel implementation of this. Source here.  (Go on, click and play)

These simple driver functions generate a circle. No excitement yet. But now you have an Excel model and you can mess around with the drivers. To get you going, I’ve added a few extra sheets in the excel so you get the idea. Have fun – just mess around with the Dr and Dphi columns. (Better not change the other columns). If you make the data list longer in order to get more ” graph”, just change the data selection of the graphs or create new x-y graphs. You know how.

Below are screenshots of the Excel output: “spiral”, “random Dphi”, “specs”, “hexagon”, “hive”.

To be continued: in the next installment we’ll add branches (and make real fractals!!!)

Enjoy.

Next article: plotting fractals

## 7 thoughts on “2D genetic fractals: a Makers Guide”

1. Whooooosh is the sound this makes flying over my head 🙂

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2. Phil Boswell says:

“Or in trigonometric form”, formula does not parse.

I can see the raw code in a tooltip but it makes little sense to me, sorry!

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• Thanks Phil – somewhere along the line the “\” in latex got stripped. Fixed now. In the meantime I have adopted a better notation. Have a look at the end of this paper: https://arxiv.org/pdf/1703.06307.pdf
Thanks for letting me know. I’ve fixed the code.
Henk

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