1. 曼德勃罗集

import numpy as np

import pylab as pl

import time

from matplotlib import cm

def iter_point(c):

z = c

for i in xrange(1, 100): # 最多迭代100次

if abs(z)>2: break # 半径大于2则认为逃逸

z = z*z+c

return i # 返回迭代次数

def draw_mandelbrot(cx, cy, d):

"""

绘制点(cx, cy)附近正负d的范围的Mandelbrot

"""

x0, x1, y0, y1 = cx-d, cx+d, cy-d, cy+d

y, x = np.ogrid[y0:y1:200j, x0:x1:200j]

c = x + y*1j

start = time.clock()

mandelbrot = np.frompyfunc(iter_point,1,1)(c).astype(np.float)

print "time=",time.clock() - start

pl.imshow(mandelbrot, cmap=cm.jet, extent=[x0,x1,y0,y1])

pl.gca().set_axis_off()

x,y = 0.27322626, 0.595153338

pl.subplot(231)

draw_mandelbrot(-0.5,0,1.5)

for i in range(2,7):

pl.subplot(230+i)

draw_mandelbrot(x, y, 0.2**(i-1))

pl.subplots_adjust(0.02, 0, 0.98, 1, 0.02, 0)

pl.show()

2. 分形树叶

import numpy as np

import matplotlib.pyplot as pl

import time

# 蕨类植物叶子的迭代函数和其概率值

eq1 = np.array([[0,0,0],[0,0.16,0]])

p1 = 0.01

eq2 = np.array([[0.2,-0.26,0],[0.23,0.22,1.6]])

p2 = 0.07

eq3 = np.array([[-0.15, 0.28, 0],[0.26,0.24,0.44]])

p3 = 0.07

eq4 = np.array([[0.85, 0.04, 0],[-0.04, 0.85, 1.6]])

p4 = 0.85

def ifs(p, eq, init, n):

"""

进行函数迭代

p: 每个函数的选择概率列表

eq: 迭代函数列表

init: 迭代初始点

n: 迭代次数

返回值： 每次迭代所得的X坐标数组， Y坐标数组， 计算所用的函数下标

"""

# 迭代向量的初始化

pos = np.ones(3, dtype=np.float)

pos[:2] = init

# 通过函数概率，计算函数的选择序列

p = np.add.accumulate(p)

rands = np.random.rand(n)

select = np.ones(n, dtype=np.int)*(n-1)

for i, x in enumerate(p[::-1]):

select[rands

# 结果的初始化

result = np.zeros((n,2), dtype=np.float)

c = np.zeros(n, dtype=np.float)

for i in range(n):

eqidx = select[i] # 所选的函数下标

tmp = np.dot(eq[eqidx], pos) # 进行迭代

pos[:2] = tmp # 更新迭代向量

# 保存结果

result[i] = tmp

c[i] = eqidx

return result[:,0], result[:, 1], c

start = time.clock()

x, y, c = ifs([p1,p2,p3,p4],[eq1,eq2,eq3,eq4], [0,0], 100000)

time.clock() - start

pl.figure(figsize=(6,6))

pl.subplot(121)

pl.scatter(x, y, s=1, c="g", marker="s", linewidths=0)

pl.axis("equal")

pl.axis("off")

pl.subplot(122)

pl.scatter(x, y, s=1,c = c, marker="s", linewidths=0)

pl.axis("equal")

pl.axis("off")

pl.subplots_adjust(left=0,right=1,bottom=0,top=1,wspace=0,hspace=0)

pl.gcf().patch.set_facecolor("#D3D3D3")

pl.show()

3. 其它分形图(科赫曲线、分形龙、谢尔宾斯基三角等)

from math import sin, cos, pi

import matplotlib.pyplot as pl

from matplotlib import collections

class L_System(object):

def __init__(self, rule):

info = rule['S']

for i in range(rule['iter']):

ninfo = []

for c in info:

if c in rule:

ninfo.append(rule[c])

else:

ninfo.append(c)

info = "".join(ninfo)

self.rule = rule

self.info = info

def get_lines(self):

d = self.rule['direct']

a = self.rule['angle']

p = (0.0, 0.0)

l = 1.0

lines = []

stack = []

for c in self.info:

if c in "Ff":

r = d * pi / 180

t = p[0] + l*cos(r), p[1] + l*sin(r)

lines.append(((p[0], p[1]), (t[0], t[1])))

p = t

elif c == "+":

d += a

elif c == "-":

d -= a

elif c == "[":

stack.append((p,d))

elif c == "]":

p, d = stack[-1]

del stack[-1]

return lines

rules = [

{

"F":"F+F--F+F", "S":"F",

"direct":180,

"angle":60,

"iter":5,

"title":"Koch"

},

{

"X":"X+YF+", "Y":"-FX-Y", "S":"FX",

"direct":0,

"angle":90,

"iter":13,

"title":"Dragon"

},

{

"f":"F-f-F", "F":"f+F+f", "S":"f",

"direct":0,

"angle":60,

"iter":7,

"title":"Triangle"

},

{

"X":"F-[[X]+X]+F[+FX]-X", "F":"FF", "S":"X",

"direct":-45,

"angle":25,

"iter":6,

"title":"Plant"

},

{

"S":"X", "X":"-YF+XFX+", "Y":"+XF-YFY-FX+",

"direct":0,

"angle":90,

"iter":6,

"title":"Hilbert"

},

{

"S":"L--F--L--F", "L":"+R-F-R+", "R":"-L+F+",

"direct":0,

"angle":45,

"iter":10,

"title":"Sierpinski"

},

]

def draw(ax, rule, iter=None):

if iter!=None:

rule["iter"] = iter

lines = L_System(rule).get_lines()

linecollections = collections.LineCollection(lines)

ax.add_collection(linecollections, autolim=True)

ax.axis("equal")

ax.set_axis_off()

ax.set_xlim(ax.dataLim.xmin, ax.dataLim.xmax)

ax.invert_yaxis()

fig = pl.figure(figsize=(7,4.5))

fig.patch.set_facecolor("papayawhip")

for i in xrange(6):

ax = fig.add_subplot(231+i)

draw(ax, rules[i])

fig.subplots_adjust(left=0,right=1,bottom=0,top=1,wspace=0,hspace=0)

pl.show()

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