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

Note

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6 CommentsOldest to Newest

Adriaan Visser replied on : 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.