Computer Science 111

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Computer Science 111

• Fundamentals of Programming I
• Advanced Turtle Graphics
• Recursive Patterns in Art and Nature

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Recursive Patterns in Art

• The 20th century Dutch artist Piet Mondrian
  painted a series of pictures that displayed
  abstract, rectangular patterns of color
• Start with a single colored rectangle
• Subdivide the rectangle into two unequal parts
  (say, 1/3 and 2/3) and paint these in different
  colors
• Repeat this process until an aesthetically
  appropriate moment is reached

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Level 1 A Single Filled Rectangle
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Level 2 Split at the Aesthetically Appropriate
Spot
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Level 3 Continue the Same Process with Each Part
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Level 4
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Level 5
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Level 6
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Level 7
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Level 8
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Level 9
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Design a Recursive Function

• The function expects a Turtle object, the corner
  points of a rectangle, and the current level as
  arguments
• If the level is greater than 0
• Draw a filled rectangle with the given corner
  points
• Calculate the corner points of two new rectangles
  within the current one and decrement the level by
  1
• Call the function recursively to draw these two
  rectangles

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Program Structure
from turtle import Turtle import random def
drawRectangle(t, x1, y1, x2, y2) red
random.randint(0, 255) green
random.randint(0, 255) blue
random.randint(0, 255) t.pencolor(red, green,
blue) Code for drawing goes here
Definition of the recursive mondrian function
goes here def main(level 1) t Turtle()
t.speed(0) t.hideturtle() x 50 y
50 mondrian(t, -x, y, x, -y, level)
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The mondrian Function
def mondrian(t, x1, y1, x2, y2, level) if
level gt 0 drawRectangle(t, x1, y1, x2,
y2) vertical random.randint(1, 2)
if vertical 1 Vertical split
mondrian(t, x1, y1, (x2 - x1) // 3 x1,
y2, level - 1)
mondrian(t, (x2 - x1) // 3 x1, y1, x2, y2,
level - 1) else
Horizontal split
mondrian(t, x1, y1, x2, (y2 - y1) // 3 y1,
level - 1)
mondrian(t, x1, (y2 - y1) // 3 y1, x2, y2,
level - 1)
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Recursive Patterns in Nature

• A fractal is a mathematical object that exhibits
  the same pattern when it is examined in greater
  detail
• Many natural phenomena, such as coastlines and
  mountain ranges, exhibit fractal patterns

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The C-curve

• A C-curve is a fractal pattern
• A level 0 C-curve is a vertical line segment
• A level 1 C-curve is obtained by bisecting a
  level 0 C-curve and joining the sections at right
  angles
• A level N C-curve is obtained by joining two
  level N - 1 C-curves at right angles

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(No Transcript)
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Level 0 and Level 1
(50,50)
(50,50)
(0,0)
(50,-50)
(50,-50)
drawLine(50, -50, 0, 0) drawLine(0, 0, 50, 50)
drawLine(50, -50, 50, 50)
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Bisecting and Joining
(50,50)
(50,50)
(0,0)
(50,-50)
(50,-50)
0 (50 50 -50 - 50) // 2 0 (50 -50 50
- 50) // 2 drawLine(50, -50, 0, 0) drawLine(0, 0,
50, 50)
drawLine(50, -50, 50, 50)
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Generalizing
(50,50)
(50,50)
(0,0)
(50,-50)
(50,-50)
xm (x1 x2 y1 - y2) // 2 ym (x2 y1 y2
- x1) // 2 drawLine(x1, y1, xm, ym) drawLine(xm,
ym, x2, y2)
drawLine(x1, y1, x2, y2)
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Recursing
(50,50)
(50,50)
(0,0)
(50,-50)
(50,-50)
xm (x1 x2 y1 - y2) // 2 ym (x2 y1 y2
- x1) // 2 cCurve(x1, y1, xm, ym) CCurve(xm, ym,
x2, y2)
drawLine(x1, y1, x2, y2)
Base case
Recursive step
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The cCurve Function
def cCurve(t, x1, y1, x2, y2, level) if
level 0 drawLine(t, x1, y1, x2, y2)
else xm (x1 x2 y1 - y2) // 2
ym (x2 y1 y2 - x1) // 2
cCurve(t, x1, y1, xm, ym, level - 1)
cCurve(t, xm, ym, x2, y2, level - 1)
Note that recursive calls occur before any
C-curve is drawn when level gt 0
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Program Structure
from turtle import Turtle def drawLine(t, x1,
y1, x2, y2) """Draws a line segment between
the endpoints.""" t.up() t.goto(x1, y1)
t.down() t.goto(x2, y2) Definition of
the recursive cCurve function goes here def
main(level 1) t Turtle() t.speed(0)
t.hideturtle() cCurve(t, 50, -50, 50, 50,
level)
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Call Tree for ccurve(0)
A call tree diagram shows the number of calls of
a function for a given argument value
ccurve
ccurve(0) uses one call, the top-level one
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Call Tree for ccurve(1)
ccurve
ccurve
ccurve
ccurve(1) uses three calls, a top-level one and
two recursive calls
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Call Tree for ccurve(2)
ccurve(2) uses 7 calls, a top-level one and 6
recursive calls
ccurve
ccurve
ccurve
ccurve
ccurve
ccurve
ccurve
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Call Tree for ccurve(n)
ccurve(n) uses 2n1 - 1 calls, a top-level one
and 2n1 - 2 recursive calls
ccurve
ccurve
ccurve
ccurve
ccurve
ccurve
ccurve
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Call Tree for ccurve(2)
The number of line segments drawn equals the
number of calls on the frontier of the tree (2n)
ccurve
ccurve
ccurve
ccurve
ccurve
ccurve
ccurve
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Summary

• A recursive algorithm passes the buck repeatedly
  to the same function
• Recursive algorithms are well-suited for solving
  problems in domains that exhibit recursive
  patterns
• Recursive strategies can be used to simplify
  complex solutions to difficult problems

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For Next Week

• Finish Chapter 7