|
Size: 3944
Comment: added cdef for speed
|
← Revision 41 as of 2019-04-06 16:11:28 ⇥
Size: 8098
Comment: py3 print
|
| Deletions are marked like this. | Additions are marked like this. |
| Line 8: | Line 8: |
| {{{ def muk_plot(m0,k): |
{{{#!sagecell def muk_plot(m0,k): |
| Line 12: | Line 12: |
| associated to m0, 1-m0, and k. | associated to m0, 1-m0, and k. |
| Line 32: | Line 32: |
| 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))): |
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)): |
| Line 37: | Line 37: |
| line(v,thickness=thickness).show(xmin=0.5, xmax=0.5, ymin=0, figsize=[8,3]) | line(v,thickness=thickness).show(xmin=0, xmax=1, ymin=0, figsize=[8,3]) |
| Line 42: | Line 42: |
| == 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])?) == {{{ |
== 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])?) {{{#!sagecell |
| Line 50: | Line 52: |
| def f(A = matrix([[1,1],[-1,1]]), D = '[[0,0],[1,0]]', k=(3..17)): print "Det = ", A.det() |
def f(A = matrix([[1,1],[-1,1]]), D = '[[0,0],[1,0]]', k=[3..17]): print("Det = {}".format(A.det())) |
| Line 59: | Line 61: |
| G = points([v.list() for v in Dn(k)]) |
G = points([v.list() for v in Dn(k)],size=50) |
| Line 63: | Line 65: |
| }}} {{attachment:1.png}} |
}}} {{attachment:1.png}} |
| Line 69: | Line 70: |
| {{{ | {{{#!sagecell |
| Line 74: | Line 75: |
| def f(A = matrix([[1,1],[-1,1]]), D = '[[0,0],[1,0]]', k=(3..17)): print "Det = ", A.det() |
def f(A = matrix([[1,1],[-1,1]]), D = '[[0,0],[1,0]]', k=[3..17]): print("Det = {}".format(A.det())) |
| Line 83: | Line 84: |
| Line 85: | Line 86: |
| Line 87: | Line 88: |
| Line 91: | Line 92: |
| == Now in 3d == {{{ |
== Now in 3D == {{{#!sagecell |
| Line 104: | Line 105: |
| @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() |
@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())) |
| Line 109: | Line 110: |
| print "D:" print D |
print("D:") print(D) |
| Line 115: | Line 116: |
| Line 128: | Line 129: |
| def mandelbrot_cython(float x0,float x1,float y0,float y1,int N=200, int L=50, float R=3): | cimport numpy as np def mandelbrot_cython(float x0,float x1,float y0,float y1, int N=200, int L=50, float R=3): |
| Line 131: | Line 135: |
| cdef complex c,z cdef int h, i, k m= np.zeros([N,N], dtype=np.int) for i in range(N): |
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): |
| Line 136: | Line 145: |
| c=complex(x0+i*(x1-x0)/N, y0+k*(y1-y0)/N) z=complex(0,0) |
c = (x0+j*deltax)+ I*(y0+k*deltay) z=0 |
| Line 139: | Line 148: |
| while (h<L) and (abs(z)<R): | while (h<L and z.real**2 + z.imag**2 < R2): |
| Line 142: | Line 152: |
| m[i,k]=h | m[j,k]=h |
| Line 146: | Line 156: |
| @interact def showme_mandelbrot(x0=-2, y0=-1.5, side=3.0,N=(100*i for i in range(1,11)), L=(20*i for i in range(1,11)) ): time m=mandelbrot_cython(x0 ,x0 + side ,y0 ,y0 + side , N, L ) time show(matrix_plot(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') |
| Line 152: | Line 196: |
== 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 {{{#!sagecell @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) }}} {{attachment:mandel-interact-02.png}} === Julia === by Harald Schilly {{{#!sagecell @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) }}} {{attachment:julia-interact-01.png}} {{{ julia_plot(-7,30,0.5,0.5,(-1.5,1.5), (-1.5,1.5)) }}} {{attachment:julia-fractal-exponent--7.png}} == Sierpiński Triangle == by Eviatar Bach {{{#!sagecell 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]) }}} {{attachment:sierpinski.png}} |
Sage Interactions - Fractal
goto interact main page
Contents
-
Sage Interactions - Fractal
- Mandelbrot's Fractal Binomial Distribution
- Fractals Generated By Digit Sets and Dilation Matrices
- Demonstrating that the Twin Dragon Matrix is likely to yield a Tiling of a Compact Interval of R^2 as k->infinity (It does!)
- Now in 3D
- Exploring Mandelbrot
- Mandelbrot & Julia Interact with variable exponent
- Sierpiński Triangle
Mandelbrot's Fractal Binomial Distribution
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])?)
Demonstrating that the Twin Dragon Matrix is likely to yield a Tiling of a Compact Interval of R^2 as k->infinity (It does!)
Now in 3D
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 mimport 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
Julia
by Harald Schilly
julia_plot(-7,30,0.5,0.5,(-1.5,1.5), (-1.5,1.5))
Sierpiński Triangle
by Eviatar Bach
