Fractals (Jeff Bezanson, Daniel Jones)

Original (second example from Daniel Jones's page on Compose): http://beowulf.csail.mit.edu/18.337/fractals.ipynb

This notebook illustrates three ways to obtain objects exhibiting fractal geometry:

Approaching a pointset with a fractal boundary, using deterministic iteration.
Explicit recursive subdivision.
Iterated function systems, using the random iteration algorithm.

I will use three different graphics libraries to do this: PyPlot (calling Python's matplotlib), Compose.jl (which is great for nested geometry), and Cairo (which can draw thousands of primitives in no time).

Julia set



In [1]:

    
using PyPlot









    



INFO: Loading help data...



In [2]:

    
# julia set
# (the familiar mandelbrot set is obtained by setting c==z initially)
function julia(z, c; maxiter=200)
    for n = 1:maxiter
        if abs2(z) > 4
            return n-1
        end
        z = z*z + c
    end
    return maxiter
end









    Out[2]:





julia (generic function with 1 method)



In [3]:

    
# varying the second argument to julia() tiny amounts results in a stunning variety of forms
@time m = [ uint8(julia(complex(r,i), complex(-.06,.67))) for i=1:-.002:-1, r=-1.5:.002:1.5 ];









    



elapsed time: 0.243365055 seconds (1502848 bytes allocated)



In [4]:

    
# the notebook is able to display ColorMaps
get_cmap("RdGy")









    Out[4]:



In [5]:

    
imshow(m, cmap="RdGy", extent=[-1.5,1.5,-1,1])









    












    Out[5]:





PyObject <matplotlib.image.AxesImage object at 0x10cbd5310>

Sierpinski triangle



In [7]:

    
using Compose

function sierpinski(n)
    if n == 0
        compose(context(), polygon([(1,1), (0,1), (1/2, 0)]))
    else
        t = sierpinski(n - 1)
        compose(context(),
                (context(1/4,   0, 1/2, 1/2), t),
                (context(  0, 1/2, 1/2, 1/2), t),
                (context(1/2, 1/2, 1/2, 1/2), t))
    end
end

img = SVG("sierpinski.svg", 4inch, 4(sqrt(3)/2)inch)
draw(img, compose(sierpinski(8), linewidth(0.1mm), fill(nothing), stroke("black")))



In [29]:

    
# get more detail by scaling up
draw(SVG(9inch,7inch), compose(sierpinski(8), fill(nothing), linewidth(0.1mm)))