#Ashley Vega
#Project #5
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def fern(x,y):
r = random()
if r<.01:
return [0,.2*y]
elif r < .08:
return [.35*x-.19*y, .38*x+.44*y+2.3]
elif r < .15:
return [-.35*x+.54*y, .41*x+.36*y+.48]
else:
return [.57*x+.08*y,-.03*x+.79*y+2.6]
def fern_orbit(n):
traj = [[0,0]]
for i in range(n):
nt = fern(traj[-1][0],traj[-1][1])
traj.append(nt)
return traj
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fern (0,4)
[0.320000000000000, 5.76000000000000] [0.320000000000000, 5.76000000000000] |
show(points(fern_orbit(10000),pointsize=3),axes=False)
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#This fern looks like a maple leaf.
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# code used from http://trac.sagemath.org/sage_trac/ticket/8423
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def fern(x,y):
r = random()
if r<.01:
return [0,.16*y]
elif r < .08:
return [.2*x-.26*y, .23*x+.22*y+1.6]
elif r < .15:
return [-.15*x+.28*y, .26*x+.24*y+.44]
else:
return [.85*x+.04*y,-.04*x+.85*y+1.6]
def fern_orbit(n):
traj = [[0,0]]
for i in range(n):
nt = fern(traj[-1][0],traj[-1][1])
traj.append(nt)
return traj
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fern(1,1)
[0.890000000000000, 2.41000000000000] [0.890000000000000, 2.41000000000000] |
show(points(fern_orbit(10000),pointsize=2),axes=False)
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#This fern looks like a Christmas tree tilted to the right.
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# code used from http://trac.sagemath.org/sage_trac/ticket/8423
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def butterfly2d():
""""
EXAMPLE:
sage: butterfly2d()
"""
g = Graphics()
x1, y1 = 0, 0
from math import sin, cos, exp, pi
for theta in srange( 0, 10*pi, 0.01 ):
r = exp(cos(theta)) - 2*cos(4*theta) + sin(theta/12)^5
x = r * cos(theta) # Convert polar to rectangular coordinates
y = r * sin(theta)
xx = x*6 + 25 # Scale factors to enlarge and center the curve.
yy = y*6 + 25
if theta != 0:
l = line( [(x1, y1), (xx, yy)], rgbcolor=hue(theta/7 + 4) )
g = g + l
x1, y1 = xx, yy
g.show(dpi=100, axes=False)
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butterfly2d()
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#This is a butterfly.
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# code used from: http://wiki.sagemath.org/pics
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def butterfly2d():
g = Graphics()
x1, y1 = 3, 6
from math import sin, cos, exp, pi
for theta in srange( 0, 10*pi, 0.01 ):
r = exp(cos(theta)) - 2*cos(4*theta) + sin(theta/12)^5
x = r * cos(theta) # Convert polar to rectangular coordinates
y = r * sin(theta)
xx = x*0.523 + 7 # Scale factors to enlarge and center the curve.
yy = y*0.39 + 16
if theta != 0:
l = line( [(x1, y1), (xx, yy)], rgbcolor=hue(theta/7 + 4) )
g = g + l
x1, y1 = xx, yy
g.show(dpi=100, axes=False)
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butterfly2d()
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#This is a smaller butterfly and it is farther away.
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# code used from: http://wiki.sagemath.org/pics
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def butterfly2d():
g = Graphics()
x1, y1 = 0.65, 1.38
from math import sin, cos, exp, pi
for theta in srange( 0, 21*pi, 0.03 ):
r = exp(cos(theta)) - 2*cos(17*theta) + sin(theta/4)^2
x = r * cos(theta) # Convert polar to rectangular coordinates
y = r * sin(theta)
xx = x*3.6 + 0.83 # Scale factors to enlarge and center the curve.
yy = y*0.49 + 7
if theta != 0:
l = line( [(x1, y1), (xx, yy)], rgbcolor=hue(theta/7 + 4) )
g = g + l
x1, y1 = xx, yy
g.show(dpi=100, axes=False)
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butterfly2d()
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#This looks like a flower.
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# code used from: http://wiki.sagemath.org/pics
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%cython
import numpy as np
cimport numpy as np
def mandelbrot_cython(float x0,float x1,float y0,float y1,
int N=400, int L=25, float R=2):
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
time m=mandelbrot_cython(x0 ,x0 + side ,y0 ,y0 + side , N, L )
pylab.clf()
pylab.imshow(m, cmap = pylab.cm.gray)
time pylab.savefig('mandelbrot.png')
Click to the left again to hide and once more to show the dynamic interactive window |
# code from http://wiki.sagemath.org/interact/fractal
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import pylab
x0_default = -0.23
y0_default = -1.2
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 = 3, width=5),
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/0.65
y0 += side/-0.56
elif option == 'Upper Right':
y0 += side/0.65
elif option == 'Lower Left':
x0 += side/-0.56
if option=='Reset':
side = side_default
x0 = x0_default
y0 = y0_default
elif option != 'Stay':
side = side/-0.65
time m=mandelbrot_cython(x0 ,x0 / side ,y0 ,y0 / side , N, L )
pylab.clf()
pylab.imshow(m, cmap = pylab.cm.gray)
time pylab.savefig('mandelbrot.png')
Click to the left again to hide and once more to show the dynamic interactive window |
#This looks like a cascade of water.
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#code from http://wiki.sagemath.org/interact/fractal
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f(z) = z^5 + z - 1 + 1/z
complex_plot(f, (-2, 2), (-2, 2))
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#idea from: http://www.walkingrandomly.com/?cat=24
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f(z)=z^3-z+3+1/z
complex_plot(f, (-1,1), (-2,2))
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#idea from: http://www.walkingrandomly.com/?cat=24
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f(z)=(z^2)/(-z+3)
complex_plot(f, (-4,7), (-4,4))
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#idea from: http://www.walkingrandomly.com/?cat=24
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f(z)=(z^4)+(z^0.5)/(z^6)
complex_plot(f, (4,16), (1,9))
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#idea from: http://www.walkingrandomly.com/?cat=24
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