Tag Archives: fractal

Golden Fractal Tree with Python Turtle (Source Code)

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Golden Fractal Tree is a tree based on Golden Ratio. The golden fractal tree contains the main branch and three smaller golden fractal trees: the first branch turns left by 72 degrees with ratio of main branch to the parent’s main branch = 2-golden_ratio; the second branch have the same the direction as the parent with ratio of main branch to the parent’s main branch = golden_ratio-1; the first branch turns right by 72 degrees with ratio of main branch to the parent’s main branch = 2-golden_ratio. For more details on golden fractal tree, please check out this web page.

Golden Fractal Tree

Source Code:

import turtle
import math

screen = turtle.Screen()
screen.title('Golden Fractal Tree - PythonTurtle.Academy')
screen.setup(1000,1000)
screen.setworldcoordinates(-1000,-1000,1000,1000)
turtle.speed(0)
turtle.hideturtle()
turtle.tracer(0,0)
golden_ratio = (1+5**0.5)/2

def golden_fractal_tree(x,y,direction,length):
    turtle.up()
    turtle.goto(x,y)
    turtle.seth(direction)
    turtle.pensize(math.log(length,2)/3)
    if length<10: turtle.color('forest green')
    else: turtle.color('gray')
    turtle.down()
    turtle.fd(length)
    if length < 3: return
    cx,cy = turtle.xcor(), turtle.ycor()
    golden_fractal_tree(cx,cy,direction+72,(2-golden_ratio)*length)
    golden_fractal_tree(cx,cy,direction-72,(2-golden_ratio)*length)
    golden_fractal_tree(cx,cy,direction,(golden_ratio-1)*length)

golden_fractal_tree(0,-900,90,700)
turtle.update()

Five Pointed Star Fractal with Python and Turtle (Source Code)

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Draw a five pointed star fractal that have nested five pointed star as shown.

Hint: Create a function that draws five pointed star in any location, orientation, and size.

Five Pointed Star Fractal

Source Code:

import turtle
import math

screen = turtle.Screen()
screen.title('Five Pointed Star Fractal - PythonTurtle.Academy')
screen.setup(1000,1000)
screen.setworldcoordinates(-1000,-1000,1000,1000)
turtle.speed(0)
turtle.hideturtle()
turtle.pensize(1)

def five_pointed_star(x,y,direction,r): #x,y is the center
    turtle.up()
    turtle.goto(x,y)
    turtle.seth(direction)
    turtle.fd(r)
    turtle.right(180-18)
    turtle.down()
    length = r*math.sin(math.pi*2/5)/(1+math.sin(math.pi/10))
    for _ in range(5):
        turtle.fd(length)
        turtle.left(72)
        turtle.fd(length)
        turtle.right(180-36)

def five_pointed_star_fractal(x,y,direction,r): #x,y is the center
    five_pointed_star(x,y,direction,r)
    if r < 20: return
    five_pointed_star_fractal(x,y,180+direction,r*math.sin(math.pi/10)/math.cos(math.pi/5))

five_pointed_star_fractal(0,-40,90,1000)

Pentagram Fractal with Python Turtle (Source Code)

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Draw the following fractal made up of nesting pentagrams with either recursion or iteration.

Hint: Making a function that draws pentagram in any location, radius, and direction will help!

Pentagram Fractal

Source Code:

import turtle
import math

screen = turtle.Screen()
screen.title('Pentagram Fractal - PythonTurtle.Academy')
screen.setup(1000,1000)
screen.setworldcoordinates(-1000,-1000,1000,1000)
turtle.speed(0)
turtle.hideturtle()

def star(x,y,direction,r): #x,y is the center
    turtle.up()
    turtle.goto(x,y)
    turtle.seth(direction)
    turtle.fd(r)
    turtle.right(180-18)
    turtle.down()
    length = 2*r*math.sin(math.pi*2/5)
    for _ in range(5):
        turtle.fd(length)
        turtle.right(180-36)

def star_fractal(x,y,direction,r):
    star(x,y,direction,r)
    if r < 50: return
    star_fractal(x,y,180+direction,r*math.sin(math.pi/10)/math.cos(math.pi/5))


star_fractal(0,0,90,1000)

Mandelbrot Set with Different Paramaters (Source Code)

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Using the center coordinate and zoom factor to draw the following Mandelbrot Set visualizations.

(-0.75, 0), Zoom=1
(-0.103865, -0.9584393), Zoom=167466.7.
(-0.761574, -0.0847596) Zoom=3125
(-0.761574, -0.0847596) Zoom=78125
(-0.59990625, -0.4290703125) Zoom=1024
(-1.62917, -0.0203968) Zoom=3125
(0.2613577, -0.002018128) Zoom=3354.786

Source Code:

from PIL import Image
import colorsys
import math

px, py = -0.7746806106269039, -0.1374168856037867 #Tante Renate
px, py, zoom = -0.74384935657398, -0.13170134084746293, 5788441.443619884
px, py, zoom = 2.613577e-1, -2.018128e-3, 3.354786e+3
px, py, zoom = -0.59990625, -0.4290703125, 1024
px, py, zoom = -1.038650e-1, -9.584393e-1, 1.674667e+5
px, py, zoom = -0.761574, -0.0847596, 78125
px, py, zoom = -1.62917,-0.0203968, 3125
px, py, zoom = -0.75,0,1
R = 3 
max_iteration = 512
w, h = 1250,1250
mfactor = 1

def Mandelbrot(x,y,max_iteration,minx,maxx,miny,maxy):
    zx = 0
    zy = 0
    RX1, RX2, RY1, RY2 = px-R/2, px+R/2,py-R/2,py+R/2
    cx = (x-minx)/(maxx-minx)*(RX2-RX1)+RX1
    cy = (y-miny)/(maxy-miny)*(RY2-RY1)+RY1
    i=0
    while zx**2 + zy**2 <= 4 and i < max_iteration:
        temp = zx**2 - zy**2
        zy = 2*zx*zy + cy
        zx = temp + cx
        i += 1
    return i

def gen_Mandelbrot_image():
  bitmap = Image.new("RGB", (w, h), "white")
  pix = bitmap.load()
  for x in range(w):
    for y in range(h):
      c=Mandelbrot(x,y,max_iteration,0,w-1,0,h-1)
      v = c**mfactor/max_iteration**mfactor
      hv = 0.67-v*2
      #if hv<0: hv+=1
      r,g,b = colorsys.hsv_to_rgb(hv,1,1-(v-0.1)**2/0.9**2)
      r = min(255,round(r*255))
      g = min(255,round(g*255))
      b = min(255,round(b*255))
      pix[x,y] = int(r) + (int(g) << 8) + (int(b) << 16)
  bitmap.save("Mandelbrot_"+str(px)+"_"+str(py)+"_"+str(zoom)+".jpg")
  bitmap.show()
  
R=3/zoom
gen_Mandelbrot_image()

Zooming 10^13 into Mandelbrot Set Animation with Python (Source Code)

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Write a Python program to generate 1000+ images of Mandelbrot Set with ever zooming ranges. Please note that Turtle is not used in this project for speed consideration. Then use an external software to combine the images into a video:

Zooming 10^13 into Mandelbrot Set

Source Code:

from PIL import Image
import colorsys
import math

px, py = -0.7746806106269039, -0.1374168856037867 #Tante Renate
R = 3 
max_iteration = 2500
w, h = 1024,1024
mfactor = 0.5

def Mandelbrot(x,y,max_iteration,minx,maxx,miny,maxy):
    zx = 0
    zy = 0
    RX1, RX2, RY1, RY2 = px-R/2, px+R/2,py-R/2,py+R/2
    cx = (x-minx)/(maxx-minx)*(RX2-RX1)+RX1
    cy = (y-miny)/(maxy-miny)*(RY2-RY1)+RY1
    i=0
    while zx**2 + zy**2 <= 4 and i < max_iteration:
        temp = zx**2 - zy**2
        zy = 2*zx*zy + cy
        zx = temp + cx
        i += 1
    return i

def gen_Mandelbrot_image(sequence):
  bitmap = Image.new("RGB", (w, h), "white")
  pix = bitmap.load()
  for x in range(w):
    for y in range(h):
      c=Mandelbrot(x,y,max_iteration,0,w-1,0,h-1)
      v = c**mfactor/max_iteration**mfactor
      hv = 0.67-v
      if hv<0: hv+=1
      r,g,b = colorsys.hsv_to_rgb(hv,1,1-(v-0.1)**2/0.9**2)
      r = min(255,round(r*255))
      g = min(255,round(g*255))
      b = min(255,round(b*255))
      pix[x,y] = int(r) + (int(g) << 8) + (int(b) << 16)
  bitmap.save("Mandelbrot_"+str(sequence)+".jpg")

R=3
f = 0.975
RZF = 1/1000000000000
k=1
while R>RZF:
  if k>100: break
  mfactor = 0.5 + (1/1000000000000)**0.1/R**0.1
  print(k,mfactor)
  gen_Mandelbrot_image(k)
  R *= f
  k+=1

This program generates over 1000 images and may be quite slow. You can make it faster by dividing the work to multiple programs each doing a portion of those images.

Zooming 10^13 Times into Julia Set Animation

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Write a Python program to generate hundreds of images of Julia Set with ever zooming ranges. Please note that Turtle is not used in this project for speed consideration. Then use an external software to combine the images into a video:

Zooming 10^13 Times into Julia Set

The video above zooms into the center at 0.0958598997051 + 1.1501990019993i with c = 0.355 + 0.355i. In each iteration the range shrinks by 5% until the range becomes 1/10^13 the size of the original range. You can modify the code to try different centers and c. The following is the source code:

from PIL import Image
import colorsys

cx = -0.7269
cy = 0.1889
R = (1+(1+4*(cx**2+cy**2)**0.5)**0.5)/2+0.001
max_iteration = 512
print(R)

w, h, zoom = 1500,1500,1

def julia(cx,cy,RZ,px,py,R,max_iteration,x,y,minx,maxx,miny,maxy):
    zx = (x-minx)/(maxx-minx)*2*RZ-RZ+px
    zy = (y-miny)/(maxy-miny)*2*RZ-RZ+py
    i=0
    while zx**2 + zy**2 < R**2 and i < max_iteration:
        temp = zx**2 - zy**2
        zy = 2*zx*zy + cy
        zx = temp + cx
        i += 1
    return i

def gen_julia_image(sequence,cx,cy,RZ):
  bitmap = Image.new("RGB", (w, h), "white")
  pix = bitmap.load()
  for x in range(w):
    for y in range(h):
        # smllaer: right,down
      c=julia(cx,cy,RZ,0.0958598997051,1.1501990019993,R,max_iteration,x,y,0,w-1,0,h-1)
      v = c/max_iteration
      hv = 0.67-v*0.67
      r,g,b = colorsys.hsv_to_rgb(hv,1,v/2+0.45)
      r = min(255,round(r*255))
      g = min(255,round(g*255))
      b = min(255,round(b*255))
      pix[x,y] = int(r) + (int(g) << 8) + (int(b) << 16)
  bitmap.save("julia_"+str(sequence)+".jpg")
#  bitmap.show()

f = 0.95
RZF = 1/1000000000000
RZ = 1
k=1
while RZ>RZF:
  print(k,RZ)
  gen_julia_image(k,0.355,0.355,RZ)
  RZ *= f
  k+=1

Related Projects:

Julia Set Fractal Animation

Julia Set with Different Parameters

Julia Set with Different Parameters

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In this project, we are generate several Julia Sets by varying the constant c in the normal Julia Set function. Turtle will be too slow to do this work. Instead we are going to use the PIL library. Please check out Julia Set Animation project for source code.

c = -0.618 (Max Iteration=38)
c = 0.355534-0.337292i (Max Iteration=4064)
c = 0.355+0.355i (Max Iteration=512)
c = 0.285+0.01i (Max Iteration=204)
c = -0.8+0.156i (Max Iteration=924)
c = -0.835+0.2321i (Max Iteration=104)
c = 0.7269+0.1889i (Max Iteration=1024)

Related Projects:

Zooming 10^13 Times into Julia Set Animation

Julia Set Fractal Animation