Sage Interactions - Fractal

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Mandelbrot's Fractal Binomial Distribution

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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

​

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

​

@interact

def _(mu0=slider(0.0001,0.999,default=0.3), k=slider([1..14],default=3), thickness=slider([0.1,0.2,..,1.0],default=1.0)):

    v = muk_plot(mu0,k)

    line(v,thickness=thickness).show(xmin=0, xmax=1, ymin=0, figsize=[8,3])

Fractals Generated By Digit Sets and Dilation Matrices

(Sage Days 9 - Avra Laarakker)

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

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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 = {}".format(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)],size=50)

​

    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!)

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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 = {}".format(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

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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=False):

    print("Det = {}".format(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([text3d(str(v),v) for v in Dn(D,A,k)])

    show(G, axes=False, frame=False)

CategoryCategory

Exploring Mandelbrot

Pablo Angulo

%cython
import numpy as np
cimport numpy as np

def mandelbrot_cython(float x0,float  x1,float  y0,float  y1,
                   int N=200, int L=50, float R=3):
    '''returns an array NxN to be plotted with matrix_plot
    '''
    cdef double complex c, z, I
    cdef float deltax, deltay, R2 = R*R
    cdef int h, j, k
    cdef np.ndarray[np.uint16_t, ndim=2] m
    m = np.zeros((N,N), dtype=np.uint16)
    I = complex(0,1)
    deltax = (x1-x0)/N
    deltay = (y1-y0)/N
    for j in range(N):
        for k in range(N):
            c = (x0+j*deltax)+ I*(y0+k*deltay)
            z=0
            h=0
            while (h<L and
                   z.real**2 + z.imag**2 < R2):
                z=z*z+c
                h+=1
            m[j,k]=h
    return m

import pylab
x0_default = -2
y0_default = -1.5
side_default = 3.0
side = side_default
x0 = x0_default
y0 = y0_default
options = ['Reset','Upper Left', 'Upper Right', 'Stay', 'Lower Left', 'Lower Right']

@interact
def show_mandelbrot(option = selector(options, nrows = 2, width=8),
                    N = slider(100, 1000,100, 300),
                    L = slider(20, 300, 20, 60),
                    plot_size = slider(2,10,1,6),
                    auto_update = False):
    global x0, y0, side
    if option == 'Lower Right':
        x0 += side/2
        y0 += side/2
    elif option == 'Upper Right':
        y0 += side/2
    elif option == 'Lower Left':
        x0 += side/2
    if option=='Reset':
        side = side_default
        x0 = x0_default
        y0 = y0_default
    elif option != 'Stay':
        side = side/2

    m=mandelbrot_cython(x0 ,x0 + side ,y0 ,y0 + side , N, L )
#    p = (matrix_plot(m) +
#             line2d([(N/2,0),(N/2,N)], color='red', zorder=2) +
#             line2d([(0,N/2),(N,N/2)], color='red', zorder=2))
#    time show(p, figsize = (plot_size, plot_size))
    pylab.clf()
    pylab.imshow(m, cmap = pylab.cm.gray)
    pylab.savefig('mandelbrot.png')

Mandelbrot & Julia Interact with variable exponent

published notebook: https://cloud.sagemath.com/projects/19575ea0-317e-402b-be57-368d04c113db/files/pub/1201-1301/1299-Mandelbrot.sagews

Mandelbrot

by Harald Schilly

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@interact

def mandel_plot(expo = slider(-10,10,0.1,2), \

      formula = ['mandel','ff'],\

      iterations=slider(1,100,1,30), \

      zoom_x = range_slider(-2,2,0.01,(-2,1)), \

      zoom_y = range_slider(-2,2,0.01,(-1.5,1.5))):

    var('z c')

    f(z,c) = z^expo + c

    ff_m = fast_callable(f, vars=[z,c], domain=CDF)

​

    # messing around with fast_callable

    for i in range(int(iterations)/3):

        f(z,c) = f(z,c)^expo+c

    ff = fast_callable(f, vars=[z,c], domain=CDF)

​

    def mandel(z):

      c = z

      for i in range(iterations):

         z = ff_m(z,c)

         if abs(z) > 2:

            return z

      return z

    print('z <- z^%s + c' % expo)

​

    # calling ff three times, otherwise it fast_callable exceeds a recursion limit

    if formula is 'ff':

     func = lambda z: ff(ff(ff(z,z),z),z)

    elif formula is 'mandel':

     func = mandel

​

    complex_plot(func, zoom_x,zoom_y, plot_points=200, dpi=150).show(frame=True, aspect_ratio=1)

Julia

by Harald Schilly

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@interact

def julia_plot(expo = slider(-10,10,0.1,2), \

      iterations=slider(1,100,1,30), \

      c_real = slider(-2,2,0.01,0.5), \

      c_imag = slider(-2,2,0.01,0.5), \

      zoom_x = range_slider(-2,2,0.01,(-1.5,1.5)), \

      zoom_y = range_slider(-2,2,0.01,(-1.5,1.5))):

    var('z')

    I = CDF.gen()

    f(z) = z^expo + c_real + c_imag*I

    ff_j = fast_callable(f, vars=[z], domain=CDF)

​

    def julia(z):

      for i in range(iterations):

         z = ff_j(z)

         if abs(z) > 2:

            return z

      return z

    print('z <- z^%s + (%s+%s*I)' % (expo, c_real, c_imag))

​

    complex_plot(julia, zoom_x,zoom_y, plot_points=200, dpi=150).show(frame=True, aspect_ratio=1)

julia_plot(-7,30,0.5,0.5,(-1.5,1.5), (-1.5,1.5))

Sierpiński Triangle

by Eviatar Bach

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def sierpinski(N):

   '''Generates the Sierpinski triangle by taking the modulo-2 of each element in Pascal's triangle'''

   return [([0] * (N // 2 - a // 2)) + [binomial(a, b) % 2 for b in range(a + 1)] + ([0] * (N // 2 - a // 2)) for a in range(0, N, 2)]

​

@interact

def _(N=slider([2 ** a for a in range(12)], label='Number of iterations', default=64), size=slider(1, 20, label='Size', step_size=1, default=9)):

    M = sierpinski(2 * N)

    matrix_plot(M, cmap='binary').show(figsize=[size, size])