# Season’s Greetings Fractal 6

Posted by **Cleve Moler**,

I don't recall where I found this seasonal fractal. And I can't explain how it works. So please submit a comment if you can shed any light on either of these questions.

### Contents

#### Season's Greetings

I have featured a fractal at this time of the year in two previous years. I see Christmas trees or perhaps a holly decoration.

#### The Formula

All these figures are obtained by varying the parameters in a formula that generates complex numbers $z$ from the partial sums.

$$ z = \sum_n{\exp{(\phi n^p+\sigma) \pi i}} $$

A vector of points in the complex plane is produced by taking $n$ to be a vector of consecutive integers and using the MATLAB cumulative summation function `cumsum` to compute the partial sums. There are 8600 points in the figure above and 100,000 points in the figures below.

The default value of the parameter $\phi$ is my old friend the `golden ratio`.

$$ \phi = \frac{1+\sqrt{5}}{2} $$

In previous posts I've taken $\phi$ to be other rational and irrational numbers, but today I am sticking to this value.

The parameter $\sigma$ controls the angular orientation. Taking $\sigma$ near $1/8$ makes the large Christmas tree vertical.

While trying how to understand how this thing works I've varied the power $p$ from its usual value of 2 and taken hundreds of thousands of points. This produces today's pictures. Different values of $p$ produce wildly different results.

For a real variable $x$, the expression $\exp (x \pi i)$ is periodic and lies on the unit circle in the complex plane. So we are plotting the cumulative sum of values taken from around the unit circle. At first glance, this appears to be a complex valued random number generator. But it is a lousy generator because we can see Christmas trees in the output.

#### Traditional p = 2

#### p = 2/3

#### p = 5/4

#### p = 4

#### Today's code

```
type greetings_gifs
```

function greetings_gifs % Generate animated season's greeting gifs. % Generate the fractal phi = (1+sqrt(5))/2; s = 1/8; n = 100000; for gifnum = 1:4 switch gifnum case 1, p = 2; case 2, p = 2/3; case 3, p = 1.25; case 4, p = 4; end w = exp((phi*(0:n).^p+s)*pi*1i); z = cumsum(w); % Find local extrema ks = extrema(z); % Make an animated gif plotit(z,p,ks,gifnum) end % gifnum % ------------------------ function ks = extrema(z) n = length(z)-1; m = n/40; ks = []; for j = 0:m:n-m zj = z(j+1:j+m); w = zj - mean(zj); k = find(abs(w) == max(abs(w))) + j; ks = [ks k]; end end % extrema function plotit(z,p,ks,gifnum) % Make an animated gif shg plot(z,'.') axis square ax = axis; gif_frame(['greetings' int2str(gifnum) '.gif']) clf axis(ax) axis square gif_frame for j = 1:length(ks) k = ks(j); plot(z(1:k),'k.','markersize',0.5) axis(ax) axis square hold on plot(z(ks(1:j)),'g.','markersize',18) plot(z(k),'r.','markersize',24) hold off title(sprintf('p = %4.2f',p)) gif_frame end gif_frame(5) gif_frame('reset') end % plotit end % greetings_gifs

#### Postscript

*Happy New Year.*

Get the MATLAB code

Published with MATLAB® R2016a

**Category:**- Fractals,
- Fun,
- Graphics,
- Random Numbers

### Note

Comments are closed.

## 6 CommentsOldest to Newest

**1**of 6

The p=5/4 case looks like a Cornu spiral when you try to plot it when r is too small to be in the Fraunhofer regime.

**2**of 6

Thanks for the comment. Here’s the Wikipedia entry about “Cornu spiral”, aka “Euler spiral.” Euler Spiral

**3**of 6

Beautiful! Thanks for sharing, Cleve!

This actually reminds me of something (much simpler) that I created before with this code.

**4**of 6

Very nice. Thanks. — Cleve

**5**of 6

Hello, sorry about this naive question, where can I find the gif_frame function ?

**6**of 6

You can get gif_frame.m here:

gif_frame.m

— Cleve

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