Great Theoretical Ideas In Computer Science Danny Sleator Lecture 3 CS 15 251 Jan 19 2010 Unary and Binary Spring 2010 Carnegie Mellon University Oh No Homework 1 is due today at 11 59pm Give yourself sufficient time to make PDF Quiz 1 is next Thursday during lecture More on Fractals Fractals are geometric objects that are selfsimilar i e composed of infinitely many pieces all of which look the same The Koch Family of Curves Fractal Dimension We can break a line segment into N selfsimilar pieces and each of which can be magnified by a factor of N to yield the original segment We can break a square into N2 selfsimilar pieces and each of which can be magnified by a factor of N Fractal Dimension The dimension is the exponent of the number of self similar pieces with magnification factor into which the figure may be broken of self similar pieces mf dim ln of self similarpieces dim ln magnification factor Hausdorf dimension Fractal Dimension of the Plane We can break a square into N2 selfsimilar pieces and each of which can be magnified by a factor of N 2 ln N dim 2 ln N Fractal Dimension of the Koch Curve We begin with a straight line of length 1 Remove the middle third of the line and replace it with two lines that each have the same length Repeat infinitely Fractal Dimension of the Koch Curve 1 ln of self similarpieces dim ln magnification factor ln 4 dim 1 26 ln 3 The Koch Family of Curves 60o What if we increase that angle but keep all sides of the equal length Compute its dimension 72o The Koch Family of Curves 1 A 72o B C D AD 2 BC BC 2 cos 72o ln 4 dim o 1 44 ln 2 2 cos 72 The Koch Family of Curves 1 A 72o B C D Increase this angle 90o A B C D Planeln 4 filling dim 2 o curve ln 2 2 cos 90 Unary and Binary How to play the 9 stone game 1 2 3 4 5 6 9 7 8 9 stones numbered 1 9 Two players alternate moves Each move a player gets to take a new stone Any subset of 3 stones adding to 15 wins Magic Square Brought to humanity on the back of a tortoise from the river Lo in the days of Emperor Yu in ancient China 4 9 3 2 5 8 7 1 6 Magic Square Any 3 in a vertical horizontal or diagonal line add up to 15 4 9 2 3 5 7 8 1 6 Conversely any 3 that add to 15 must be on a line 4 9 2 3 5 7 8 1 6 TIC TAC TOE on a Magic Square Represents The Nine Stone Game Alternate taking squares 1 9 Get 3 in a row to win 4 9 2 3 5 7 8 1 6 Basic Idea of This Lecture Don t stick with the representation in which you encounter problems Always seek the more useful one This idea requires a lot of practice Prehistoric Unary 1 2 3 4 Consider the problem of finding a formula for the sum of the first n numbers You already used induction to verify that the answer is n n 1 A different approach 1 2 3 n 1 n S n n 1 n 2 2 1 S n 1 n 1 n 1 n 1 n 1 2S n n 1 2S n n 1 S 2 1 2 3 n 1 n S n n 1 n 2 2 1 S n n 1 2S There are n n 1 dots in the grid n n n 1 S 2 2 1 1 2 n nth Triangular Number n 1 2 3 n 1 n n n 1 2 nth Square Number n n2 n n 1 Breaking a square up in a new way 1 Breaking a square up in a new way 1 3 Breaking a square up in a new way 1 3 5 Breaking a square up in a new way 1 3 5 7 Breaking a square up in a new way 1 3 5 7 9 Breaking a square up in a new way 1 3 5 7 9 52 Breaking a square up in a new way The sum of the first n odd 2 numbers is n Pythagoras Here is an alternative dot proof of the same sum nth Square Number n n n 1 n2 nth Square Number n n n 1 n2 nth Square Number n n n 1 nth Square Number n n n 1 Sum of first n odd numbers We find a formula for the sum of the first n cubes Area of square n 2 n n n 1 2 n Area of square n 2 n 1 n 1 n n n 1 2 n Area of square n 2 n 1 n 1 n n n n 1 2 n n Area of square n 2 n 1 n 1 n n n n Area of square n 2 n 1 2 n n 1 n n n 1 2 n n 1 n n 1 2 n n2 n 1 n 1 2 n n n 1 n n n n n 1 n 1 2 n3 n n 2 n3 n 1 2 n3 n 1 3 n 2 2 n3 n 1 3 n 2 3 n2 3 n3 n 1 3 n 2 3 13 n 2 13 23 33 n3 n n 1 2 2 Can you find a formula for the sum of the first n squares Babylonians needed this sum to compute the number of blocks in their pyramids Rhind Papyrus Scribe Ahmes was Martin Gardner of his day Rhind Papyrus A man has 7 houses Each house contains 7 cats Each cat has killed 7 mice Each mouse had eaten 7 ears of spelt Each ear had 7 grains on it What is the total of all of these Sum of powers of 7 What is a closed form of the sum of powers of integers 1 X1 X2 X3 Xn 2 Xn 1 Xn 1 X 1 We ll use this fundamental sum again and again The Geometric Series Proof X 1 1 X1 X2 X3 Xn 2 Xn 1 X1 X2 X3 Xn 1 Xn 1 X1 X2 X3 Xn 2 Xn 1 Xn 1 2 1 x x x n 1 x 1 x 1 n when x 1 Geometric Series for x 2 n 1 n 1 2 4 2 2 1 Geometric Series for x 1 1 1 1 1 n 2 n 2 4 2 2 Two Case Studies Bases and Representation BASE X Representation S an 1 an 2 a1 a0 represents the number an 1 Xn 1 an 2 Xn 2 a0 X0 Base 2 Binary Notation 1 2 2 0 21 1 20 101 represents Base 7 015 represents 0 7 2 1 71 5 70 Bases In Different Cultures Sumerian Babylonian 10 …
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