SOME FIBONACCI SURPRISES The Power of Visualization James Tanton MAA Mathematician-at-Large Curriculum Inspirations:Mathematical Stuff:Mathematical Courses: Visualization in the curriculum * Visual or Visualization appears 34 times in the ninety-three pages of the U.S. Common Cores State Standards - 22 times in reference to grade 2-6 students using visual models for fractions - 1 time in grade 2 re comparing shapes - 5 times re representing data in statistics and modeling - 4 times re graphing functions and interpreting features of graphs - 2 times in geometry re visualizing relationships between two- and three-dimensional objects. * Alberta curriculum: Recognised HS core mathematical process: [V] Visualization involves thinking in pictures and images, and the ability to perceive, transform and recreate different aspects of the world (Armstrong, 1993, p. 10). The use of visualization in the study of mathematics provides students with opportunities to understand mathematical concepts and make connections among them. The Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, arise in a myriad of contexts. For example, count the number of sequences of Os and Xs of a given length, avoiding two consecutive Xs. Go the other extreme: Insist that Xs come in pairs! Count the number of ordered partitions of a given integer that avoid 1: Or count the number of ordered partitions that use two different types of 1! Count the number of ways to arrange non-nested parentheses around a string of objects: Count the number of ways to stack (two-dimensional) cannon balls so that each row is contiguous. OR The language of ABEEBA uses only three letters of the alphabet: A, B, and E. * No word begins with an E. * No word has the letter E immediately following an A. * All other combinations of letters are words. Actually, words that begin with an E are allowed. They are swear words. Lets count the swear words in the language of ABEEBA. And so on! There is one visual model that explains all these examples - and so much more! 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, The Fibonacci numbers arise from a classic honeycomb path-counting puzzle: We see that the number of paths from the left cell to the Nth cell of the honeycomb is the Nth Fibonacci number F. N Actually the number of paths between any two cells N cells apart is F. N LETS NOW HAVE SOME FUN! Heres a path between two cells on the top row. The path touches down one groups of dots in the bottom row. This matches placing non-nested parentheses around those dots. Each step in a path either follows a diagonal step or skips over two diagonal steps. We see a partition of a number into 1s and 2s. N diagonals corresponds to a path to the N+1 th cell. Indeed every second Fibonacci number. 2 = = = = partitions 3 = = = = = = = = = = = = partitions 1 = 1 2 partitions Count the ordered partitions of a number with two different types of 1. Each diagonal step in a path is a break between dots. There are an odd number of dots between breaks = 12 EXAMPLE: 5 = = = = There are five odd partitions of five. Missed circles define the path. Focus on the dots that a path misses. The first dot is never circled, the last dot is never circled, no two consecutive dots are ever circled. Ignoring end dots, draw Os on the dots hit and Xs on the dots missed. Get sequence of Os and Xs along the zigzag avoiding two consecutive Xs. EXAMPLE: {1,2,3} has { }, {1}, {2}, {3}, {1,3} as five such subsets. OXOOXOXOOX No two consecutive dots circled No section just 1 segment long. Each dot a path misses breaks the zigzag line of segments: Draw extra line segments at the beginning and end of the zigzag. So we have a one-free partition of the number of the zig-zag steps. EXAMPLE: 5 = 2+3 = 3+2 There are three one-free partitions of five. The language of ABEEBA. Consider paths between cells on the top row. Cannonball stacks: Each stack gives a sequence of diagonal and horizontal steps which gives a path between two cells on the top row. Another approach to cannonballs: There are three ways to make a stack with an extra row: * add a ball to the left of a previously made stack * add a ball to the right of a previously made stack * place a previously made stack on top of a next row But there is double counting. But we saw today We have the identity: (Inspired by a conversation with Sam Vandervelde) PRODUCTS OF PARTITIONS ALWAYS FIBONACCI? Take all partitions on N, multiply terms, and add. Consider paths that end on a lower cell. Consider all possible locations of the UP steps. Single DOWN in each section. The sum of all such products counts all paths. Answer must be. F 24 A partition of 12 with terms multiplied together. 12 dots on top row 24 dots in all There 2x4x1x2x3 ways to place the DOWNs. Consider every second Fibonacci number: 1, 2, 5, 13, 34, 89, SUM ALWAYS ONE LESS THAN A FIBONACCI NUMBER? TRIANGULAR SUMS Consider paths that end on a particular top dot, 2N + 1. (N+1 dots on top row, N dots on bottom row.) 1 path N paths touch just 1 lower dot This accounts for all paths:. (N-k) places for span of k dots. F paths using those dots. 2k+1 (N-k) x F paths touch a span of k dots. FIBONACCI IDENTITIES Take your favourite Fibonacci identity and try to prove via paths. EXAMPLE: PROOF: Here is my favourite identity: Ive always wondered Is there a formula for the quotient? EXAMPLE: 12 is divisible by 2, 3, 4 and 6, and F = 144 is divisible by F = 1, F = 2, F = 3 and F = What if there is a remainder? So in full generality PROOF: There are a myriad of Fibonacci identities that perhaps can be proved via path walking. (Care to try?) CHALLENGE FOR TODAY: Weeks of fun to be had all with the POWER OF A PICTURE! THANK YOU! SOME FIBONACCI SURPRISES The Power of Visualization James Tanton MAA Mathematician-at-Large Curriculum Inspirations:Mathematical Stuff:Mathematical Courses:
