D036IteratedFunctionSystems.sagews

Author: Vincent J. Matsko
Date: 17 April 2016, Day036
Post: Fractals VII:  Iterated Function Systems, III

# This is the basic algorithm.
# You can adapt it many ways.  Feel free to experiment!

# n is the number of points to plot
def sierpinski(n):
    # These variables keep track of the differently colored points.
    # For more transformations, add more variables.
    ptscolor1 = []
    ptscolor2 = []
    ptscolor3 = []
    # Begin at (0,0).  You may begin anywhere you like.
    last = (0,0)
    for i in range(n):
        select = random()
        (x,y) = last
        if 0 <= select < 0.333:        # 1/3 chance for 1st transformation.
            last = (0.5*x,0.5*y)       # Note how the affine transformation is represented.
            ptscolor1.append(last)
        elif 0.333 <= select < 0.667:  # 1/3 chance for 2nd transformation.
            last = (0.5*x + 0.5,0.5*y)
            ptscolor2.append(last)
        else:                          # remaining 1/3 chance for last transformation.
            last = (0.5*x,0.5*y + 0.5)
            ptscolor3.append(last)
    # You may use any colors you wish!
    # The "aspect_ratio=1" keeps the axes at the same scale.
    show(points(ptscolor1, color='red') + points(ptscolor2, color='blue') + points(ptscolor3, color='orange'), aspect_ratio=1)

sierpinski(1000)

# The same algorithm works even if you do not use affine transformations.
# The possibilites are virtually endless!

# n is the number of points to plot
def nonaffine(n):
    # These variables keep track of the differnetly colored points.
    # For more transformations, add more variables.
    ptscolor1 = []
    ptscolor2 = []
    ptscolor3 = []
    # Begin at (0,0).  You may begin anywhere you like.
    last = (0,0)
    for i in range(n):
        select = random()
        (x,y) = last
        if 0 <= select < 0.5:                  # 1/3 chance for 1st transformation.
            last = (x**2,0.75*y)               # No longer affine!
            ptscolor1.append(last)
        elif 0.5 <= select < 0.8:              # 1/3 chance for 2nd transformation.
            last = (-0.5*x + 0.5,-y**2)        # No longer affine!
            ptscolor2.append(last)
        else:                                  # remaining 1/3 chance for last transformation.
            last = (0.5*x,-0.75*(y**2) + 0.5)  # No longer affine!
            ptscolor3.append(last)
    # You may use any colors you wish!
    # The "size=3" made the red points smaller.  You may adjust this parameter.
    # The "aspect_ratio=1" keeps the axes at the same scale.
    show(points(ptscolor1, size=3, color='red') + points(ptscolor2, color='blue') + points(ptscolor3, color='orange'), aspect_ratio=1)

nonaffine(3000)

# For the adventurous, you can create 3D fractals as well!
# The algorithm is the same, except the affine transformations are three-dimensional. 
def sierpinski3d(n):
    ptsclr1 = []
    ptsclr2 = []
    ptsclr3 = []
    ptsclr4 = []
    last = (0,0,0)
    for i in range(n):
        select = random()
        (x,y,z) = last
        if 0 <= select < 0.25:
            last = (0.5*x,0.5*y,0.5*z)
            ptsclr1.append(last)
        elif 0.25 <= select < 0.5:
            last = (0.5*x + 0.5,0.5*y,0.5*z)
            ptsclr2.append(last)
        elif 0.5 <= select < 0.75:
            last = (0.5*x,0.5*y + 0.5,0.5*z)
            ptsclr3.append(last)
        else:
            last = (0.5*x,0.5*y,0.5*z + 0.5)
            ptsclr4.append(last)
    P=point3d(ptsclr1, size=2, color='red') + point3d(ptsclr2, size=2, color='blue') + point3d(ptsclr3, size=2, color='orange') + point3d(ptsclr4, size=2, color='purple',viewpoint=(1,1,1))
    show(P)

sierpinski3d(3000)

3D rendering not yet implemented