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CS 121 Today
Fractals and Turtles!
The Koch Curve
how random…
When good programs go bad…
def power(b, p): """ Returns b**p for p >= 0 """ if p == 0: return 1 else: return b*power(b, p)
print: Making programs talk to you!
Debugging had to be discovered. I can remember the exact instant when I realized that a large part of my life from then on was going to be spent in
finding mistakes in my own programs.- Maurice Wilkes
Programming: the art of debugging an empty file.
- The Jargon File
When good programs go bad…
def power(b, p): """ Returns b**p for p >= 0 """ print ("p is", p, "; b is", b) if p == 0: return 1 else: return b*power(b, p)
Careful! print != return
def power(b, p): """ Returns b**p for p >= 0 """ if p == 0: return 1 else: return b*power(b, p-1)
def powerPrint(b, p): """ Returns(?) b**p for p >= 0 """ if p == 0: print (1) else: print (b*powerPrint(b, p-1))
sum, range
def sum(L):
""" input: a list of numbers, L
output: L's sum
"""
sum, range
def sum(L):
""" input: a list of numbers, L
output: L's sum
"""
if len(L) == 0:
return 0.0
else:
return L[0] + sum(L[1:])
Base Caseif the input
has no elements, its sum is zero
Recursive Case
if L does have an element, add
that element's value to the sum of the REST of the
list…
This input to the recursive call must be "smaller" somehow…
sum, range
def range(low,hi):
""" input: two ints, low and hi
output: int list from low up to hi
"""
excluding hi
sum, range
def range(low,hi):
""" input: two ints, low and hi
output: int list from low up to hi
"""
if hi <= low:
return []
else:
return
excluding hi
sum, range
def range(low,hi):
""" input: two ints, low and hi
output: int list from low up to hi
"""
if hi <= low:
return []
else:
return [low] + range(low+1,hi)
excluding hi
A random aside…
import random
random.choice( L )
random.uniform(low,hi)
random.choice( ['north', 'case', 'west'] )
random.uniform(41.9,42.1)
chooses 1 element from the list L
(with uniform probability)
chooses a random float from low to hi
for more explanation, try dir(random) or help(random)
How likely is this to return 42 ?
How would you get a random int from 0 to 9?
Randomness vs. Determinism
Are there random numbers?
Output
RNG
Can a computer generate them?
A “black box” model of a random number generator.
Randomness vs. Determinism
Are there random numbers?
Can a computer generate them?
The RNG revealed.
Output
Yes Not without help!
http://en.wikipedia.org/wiki/Mersenne_twister
Periodic!
p = 219937-1
Some random history…
True randomness is valuable!
http://www.rand.org/pubs/monograph_reports/MR1418/index.html
but not that valuable!
A random function…
print the guesses ?
return the number of guesses ?
from random import *
def guess( hidden ): """ guesses the user's hidden # """ compguess = choice( range(100) )
if compguess == hidden: # at last! print ('I got it!’)
else: guess( hidden )
This is a bit suspicious…
slow down…
The final version
from random import *import time
def guessFinal( hidden ): """ guesses the user's hidden # """ compguess = choice( range(100) )
print ('I choose', compguess) time.sleep(0.05)
if compguess == hidden: # at last! print ('I got it!’)
return 0 else: return 1 + guessFinal( hidden )
The two Monte Carlos
Monte Carlo casino, Monaco
Making random numbers work
for you!
Monte Carlo methods, Math/CS
Monte Carlo in action
Suppose you roll two dice.What are the chances that you roll doubles?
def countDoubles( N ): """ inputs a # of dice rolls outputs the # of doubles """ if N == 0: return 0 # zero rolls, zero doubles… else: d1 = choice( [1,2,3,4,5,6] ) d2 = choice( range(1,7) ) if d1 != d2: return countDoubles( N-1 ) # not doubles else: return # doubles!
one roll of the dice
where is the doubles check?
input is the total number of rolls
what should the last line be?
Monty Hall
Let’s make a deal ’63-’86
inspiring the “Monty Hall paradox”
Monte Carlo Monty Hall
Suppose you always switch to the other door...What are the chances that you will win the car ?
Run it (randomly) 1000 times and see!
Monte Carlo Monty Hall
def MCMH( init, sors, N ): """ plays the same "Let's make a deal" game, N times returns the number of times you win the car """ if N == 0: return 0 # don't play, can't win carDoor = choice([1,2,3]) # where is the car?
if init == carDoor and sors == 'stay': result = 'Car!' elif init == carDoor and sors == 'switch': result = 'Spam.' elif init != carDoor and sors == 'switch': result = 'Car!' else: result = 'Spam.'
print ('You get the', result) if result == 'Car!': return 1 + MCMH( init, sors, N-1 ) else: return 0 + MCMH( init, sors, N-1 )
Your initial choice! 'switch' or 'stay'number of times to play
Python's Etch-a-Sketch
A new human-computer interface?
from turtle import *
reset()
left(90)
forward(50)
right(90)
backward(50)
down() or up()
color('green')
tracer(1) or tracer(0)
width(5)
done()
and lots more!
degrees!
states if the pen draws or not
states if the pen animates
or not
Recursive Graphics
def tri(): """ draws a polygon """ forward(100) left(120) forward(100) left(120) forward(100) left(120)
there is no tri … Could we tri this with recursion?(1)
Could we create any regular n-gon?(2)
def chai(size): """ mystery! """ forward(size) left(90) forward(size/2.0) right(90) right(90) forward(size) left(90) left(90) forward(size/2.0) right(90) backward(size)
What does chai draw?(1)
How could you add more to each end?
Why are there two identical commands in a row?
Spiral
100
90
81
72.9
spiral( initLength, angle, multiplier )
close-up of innermost part of the spiral…
spiral( 100, 90, 0.9 )
svTree
svTree( trunkLength, levels )
svTree( 100, 4 )
and more! (if you want)
Help! My turtle window froze!
Your turtle window becomes unresponsive after your programruns. Type:
>>> done()
to unlock it (but then you have to close it)
The Koch curve
snowflake( 100, 0 )snowflake( 100, 1 ) snowflake( 100, 2 )
snowflake( 100, 3 ) snowflake( 100, 4 ) snowflake( 100, 5 )
Have fun!
fill(1) color("blue")
