Sage Interactions - Fractal

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TableOfContents

Mandelbrot's Fractal Binomial Distribution

def muk_plot(m0,k):  
    """
    Return a plot of the binomial fractal measure mu_k
    associated to m0, 1-m0, and k.   
    """
    k = int(k)
    m0 = float(m0)
    m1 = float(1 - m0)
    assert m0 > 0 and m1 > 0, "both must be positive"
    v = [(0,0)]
    t = 0
    two = int(2)
    delta = float(1/2^k)
    multiplier = float(2^k)
    for i in [0..2^k-1]:
        t = i * delta
        phi1 = i.str(two).count("1")
        phi0 = k - phi1
        y = m0^(phi0)*m1^(phi1)*multiplier
        v.append((t,y))
        v.append((t+delta,y))
    return v

html("<h1>Mandelbrot's Fractal Binomial Measure</h1>")

@interact
def _(mu0=(0.3,(0.0001,0.999)), k=(3,(1..14)), thickness=(1.0,(0.1,0.2,..,1.0))):
    v = muk_plot(mu0,k)
    line(v,thickness=thickness).show(xmin=0.5, xmax=0.5, ymin=0, figsize=[8,3])

attachment:binomial.png

Fractals Generated By Digit Sets and Dilation Matrices

Attempt at Generating all integer vectors with Digits D and Matrix A (How about vector([0,-1])?)

A = matrix([[1,1],[-1,1]])
D = [vector([0,0]), vector([1,0])]

@interact
def f(A = matrix([[1,1],[-1,1]]), D = '[[0,0],[1,0]]', k=(3..17)):
    print "Det = ", A.det()
    D = matrix(eval(D)).rows()
    def Dn(k):
        ans = []
        for d in Tuples(D, k):
            s = sum(A^n*d[n] for n in range(k))
            ans.append(s)
        return ans
    
    G = points([v.list() for v in Dn(k)])
   
    show(G, frame=True, axes=False)

Demonstrating that the Twin Dragon Matrix is likely to yield a Tiling of a Compact Interval of R^2 as k->infinity (It does!)

A = matrix([[1,1],[-1,1]])
D = [vector([0,0]), vector([1,0])]

@interact
def f(A = matrix([[1,1],[-1,1]]), D = '[[0,0],[1,0]]', k=(3..17)):
    print "Det = ", A.det()
    D = matrix(eval(D)).rows()
    def Dn(k):
        ans = []
        for d in Tuples(D, k):
            s = sum(A^(-n)*d[n] for n in range(k))
            ans.append(s)
        return ans
    
    G = points([v.list() for v in Dn(k)])
   
    show(G, frame=True, axes=False)

Now in 3d

A = matrix([[0,0,2],[1,0,1],[0,1,-1]])
D = '[[0,0,0],[1,0,0]]'

def Dn(D,A,k):
    ans = []
    for d in Tuples(D, k):
        s = sum(A^n*d[n] for n in range(k))
        ans.append(s)
    return ans
    
@interact
def f(A = matrix([[0,0,2],[1,0,1],[0,1,-1]]), D = '[[0,0,0],[1,0,0]]', k=(3..15), labels=True):
    print "Det = ", A.det()
    D = matrix(eval(D)).rows()
    print "D:"
    print D
    G = point3d([v.list() for v in Dn(D,A,k)], size=8)#, opacity=.85)
    if labels:
        G += sum([text(str(v),v) for v in Dn(D,A,k)])
    show(G, axes=False, frame=False)