# Littlewood Fractals

18 thoughts
last posted Dec. 22, 2012, 4:54 a.m.
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Via the science fiction author Greg Egan I've found out about the wonderful fractals discovered by Dan Christensen and Sam Derbyshire based on the Littlewood polynomials.

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A Littlewood polynomial is one where the coefficients are all either -1 or 1.

If you take all the complex roots of an n-th degree Littlewood polynomial and plot them on the complex plane you get a wonderful fractal.

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The above is an 18th-degree Littlewood fractal I generated using some Python code I'll soon put up on Github.

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With itertools.product it's easy to generate all the Littlewood coefficients:

for coefficients in product(*([[-1, 1]] * DEGREE)):


and then with numpy.roots it's easy to generate all the roots for that polynomial:

    for root in roots((1,) + coefficients):

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My current approach to visualization is to take the number of roots in the region represented by a pixel and take the ratio of the log of that number to the log of the maximum number of roots any pixel has:

value_at_pixel = log(num_roots_at_pixel) / log(max_roots)


I then generate an RGB value from an HSV of

(value_at_pixel / 4, 1 - value_at_pixel, 0.5 + value_at_pixel)


with colorsys.hsv_to_rgb.

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A 4000 x 2828 rendering of the 22-degree fractal is available here (NOTE: 4.3MB)

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Given that it looks like a cross between the One Ring and the Eye of Sauron perhaps the fractal should have a Tolkien-inspired name :-)

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A close up of a dragon-curve-like fractal from the inside bottom right:

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I wonder if I could speed up the PNG generation by memoizing the HSV to RGB (or more specifically, the root count to RGB) code.

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Caching root count to RGB helps a fair amount (around 25% time reduction on my initial tests)

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Running with pypy would speed things up, but numpypy doesn't support roots from what I can tell.

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The root generation and the heatmap generation can be split into two completely separate processes so I'm working on that now.

This means the roots could be calculated once for a particular degree without having to do it over and over again for different image resolutions.

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Dan Christensen, in a comment on John Baez's blog post about the Littlewood fractals says:

Using python and scipy, and carefully taking into account the 8-fold symmetry, I can generate the degree 24 roots in about 3 hours, and plot them in about 10 minutes. I store about 55 million roots (again using symmetry).

As a point of comparison, my root generation for degree 24 on my iMac a year later is currently just under 2 hours.

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I'm taking advantage of 4-fold symmetry as I wasn't aware of an additional symmetry but I've just plotted the top quadrant using polar coordinates and now it's obvious:

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So in root generation not only need I only consider Re(z) >=0 and Im(z) >=0 but also |z| <= 1.

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The Beauty of Roots by John Baez, Dan Christensen, Sam Derbyshire and Greg Egan: easy version / hard version

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Pushed separate root-generation and heatmap-rendering code (including polar version) to Github.

Run roots.py then heatmap.py.