Fractals 3 ways

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using PyPlot


    

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# 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


    

    
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julia (generic function with 1 method)

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# 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 ];


    

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


    

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


    

    

















    
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PyObject <matplotlib.image.AxesImage object at 0x7f53e91a2510>

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using Compose


    

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# drawing an SVG shape with Compose
draw(SVG(1inch,(sqrt(3)/2)inch), compose(canvas(), polygon((1,1),(0,1),(1/2,0))))


    

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# directly from dcjones' Compose README
function sierpinski(n)
    if n == 0
        compose(canvas(), polygon((1,1), (0,1), (1/2, 0)))
    else
        t = sierpinski(n - 1)
        compose(canvas(),
                (canvas(1/4,   0, 1/2, 1/2), t),
                (canvas(  0, 1/2, 1/2, 1/2), t),
                (canvas(1/2, 1/2, 1/2, 1/2), t))
    end
end


    

    
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sierpinski (generic function with 1 method)

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


    

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# draw from a discrete CDF
pick(p) = searchsortedfirst(p, rand())


    

    
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pick (generic function with 1 method)

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# the iterated function system (IFS) "code" for Barnsley's fern.
# consists of four matrices and a probability for each.
fern_M = {[ 0    0    0; 0    .16 0;    0 0 1],
          [ 0.85  .04 0; -.04 .85 1.6;  0 0 1],
          [ 0.2  -.26 0;  .23 .22 1.6;  0 0 1],
          [-0.15  .28 0;  .26 .24  .44; 0 0 1]}
fern_P = cumsum([.01, .85, .07, .07]);


    

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# first, an implementation using Compose.
# This is a huge point cloud, which Compose is definitely not designed for, and runs quite slowly.

point(x,y) = polygon((x,y),(x+.001,y+.001))

function ifs(M, p, niter)
    pt = [0.5,0.5,1]     # start at an arbitrary point
    c = compose(canvas(), linewidth(0.3mm))
    for i = 1:niter+10
        pt = M[pick(p)]*pt
        if i > 10        # wait 10 iterations to make sure we approach the attractor
            c = compose(c, point((pt[1]+4)/14,1-pt[2]/10.2))
        end
    end
    c
end


    

    
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ifs (generic function with 1 method)

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@time draw(PNG(4.25inch,2.5inch), ifs(fern_M, fern_P, 16000))


    

    













    




elapsed time: 25.186542174 seconds (7203514612 bytes allocated)

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using Cairo
using Color
using Base.Graphics


    

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# next an implementation using Cairo directly to draw small circles.
# this is drastically faster (at least 100x)
function ifs(GC, M, p, niter, scale=0.01)
    pt = [0.5,0.5,1]
    for i = 1:niter+10
        pt = M[pick(p)]*pt
        if i > 10
            circle(GC, pt[1], pt[2], scale)
            stroke(GC)
        end
    end
end


    

    
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ifs (generic function with 3 methods)

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# a convenient utility for setting up Cairo drawings with an initial fill color and coordinate system
function drawing(w,h; fill=nothing, coords=nothing, left=0, right=w, top=0, bottom=h)
    s = CairoRGBSurface(w, h)
    gc = creategc(s)
    if coords != nothing
        left, right, top, bottom = coords
    end
    Graphics.set_coords(gc, 0, 0, w, h, left, right, top, bottom)
    if fill != nothing
        set_source(gc, fill)
        paint(gc)
    end
    set_source(gc, RGB(0,0,0))
    gc
end


    

    
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drawing (generic function with 1 method)