1. Naturally Algebra G. Whisler

2. (c) MathScience Innovation Center, 2007 NATURALLY ALGEBRA

3. (c) MathScience Innovation Center, 2007 What is a Fractal? A Self Similar Pattern A Self Similar Pattern Formed by recursion (iteration or repeated application of a process on its output) Formed by recursion (iteration or repeated application of a process on its output) Has fractal dimension (dimension that is not always in whole number scale) Has fractal dimension (dimension that is not always in whole number scale)

4. (c) MathScience Innovation Center, 2007 A section of one of the most famous fractals created…..

5. (c) MathScience Innovation Center, 2007 A fractal in nature… Exchange profiles Exchange profiles An example of an exchange profile is a radiator An example of an exchange profile is a radiator Root systems are good ‘natural’ examples Root systems are good ‘natural’ examples Picture by Greg Vogel Picture by Greg Vogel

6. (c) MathScience Innovation Center, 2007 …and so are branches

7. (c) MathScience Innovation Center, 2007 Fractals in Nature Many times exchange profiles are solutions to ‘problems’ faced by nature. Many times exchange profiles are solutions to ‘problems’ faced by nature. These exchange profiles are created by iteration. These exchange profiles are created by iteration. ITERATION : Repeating a process ITERATION : Repeating a process

8. (c) MathScience Innovation Center, 2007 Fractal Tree Activity Logon to the computers and… Logon to the computers and… Launch GSP 4.07 Launch GSP 4.07 Start a new sketch Start a new sketch Follow me… Follow me… The Geometer's Sketchpad The Geometer's Sketchpad

9. (c) MathScience Innovation Center, 2007 RESULTS RESULTS STEP NEW Branches TOTAL BRANCHES 011 123 247 3815 41631 “Nth” 2N2N2N2N2(New)-1 1665536131071

10. (c) MathScience Innovation Center, 2007 Now it is your turn! First pick a GREEN (this is the number of branches your tree will have: 2, 3 or 4), First pick a GREEN “branches card” (this is the number of branches your tree will have: 2, 3 or 4), Then pick as many BLUE as branches to set the ratio for each branch, Then pick as many BLUE “dilation cards” as branches to set the ratio for each branch, Last, pick as many as you have branches for the angle of rotation for each branch. Last, pick as many “rotation cards” as you have branches for the angle of rotation for each branch. DATA CHART DATA CHART DATA CHART DATA CHART

11. (c) MathScience Innovation Center, 2007 Group Data Our Forest! Group Data Our Forest!

12. (c) MathScience Innovation Center, 2007 …and since we have time… We are going to look at other patterns. You might even find this one familiar!

13. (c) MathScience Innovation Center, 2007 More Challenging Patterns Not all patterns are obvious or have easy to write rules for describing them, Not all patterns are obvious or have easy to write rules for describing them, Now a more CHALLENGING puzzle… Now a more CHALLENGING puzzle…

14. (c) MathScience Innovation Center, 2007 The Great Domino Wall How many Ways….. How many Ways….. Can we build the Wall… Can we build the Wall…

15. (c) MathScience Innovation Center, 2007 Instructions for the GREAT DOMINO WALL Each group has been tasked to … Each group has been tasked to … Build a wall ‘n’ units long and two units high. You will model this with dominos. A domino has dimensions of 2 units by 1 unit (2x1). Build a wall ‘n’ units long and two units high. You will model this with dominos. A domino has dimensions of 2 units by 1 unit (2x1). Find the number of ways you can build the “Great Wall of Dominos” using 1, 2, 3, 4 and 5 dominos. Find the number of ways you can build the “Great Wall of Dominos” using 1, 2, 3, 4 and 5 dominos.

16. (c) MathScience Innovation Center, 2007 RESULTS Length N Number of Dominos Ways to Build The Wall 001 111 222 333 445 558

17. (c) MathScience Innovation Center, 2007 Summary of results Does anyone recognize this pattern? Fibonacci !! Problem - No easy formula for the Nth term BUT… We can use the power of iteration to find bigger N!

18. (c) MathScience Innovation Center, 2007 Iterating expressions with GSP Start a new sketch in Geometer’s Sketchpad, and Start a new sketch in Geometer’s Sketchpad, and Follow Me… Follow Me… 1, 1, 2, 3, 5, 8, 13, 21,…

19. (c) MathScience Innovation Center, 2007 Technique of the future? Two advances in MODERN MATHEMATICS and SCIENCE are FRACTAL GEOMETRY a aa and CHAOS THEORY. They were developed from iterating functions: Fractals: f(z) = z2 + c Chaos: f(x) = ax(1- x)

20. (c) MathScience Innovation Center, 2007 Fractal research started with an iterated function made by Mandelbrot

21. (c) MathScience Innovation Center, 2007 This is an image created using fractal technology

22. (c) MathScience Innovation Center, 2007 Chaos Theory started with an investigation into weather patterns

24. (c) MathScience Innovation Center, 2007 THE “BUTTERFLY” ATTRACTOR CHAOS THEORY

25. (c) MathScience Innovation Center, 2007 FRACTALS and CHAOS… Are helping investigate and explain complex systems in the world around us… Are helping investigate and explain complex systems in the world around us…

26. THANK YOU I’ve enjoyed spending the time with you!

27. (c) MathScience Innovation Center, 2007

28. Motivation for Hero’s Method Iteration can be used to solve other problems, such as… Iteration can be used to solve other problems, such as… How does a calculator evaluate √12 ? How does a calculator evaluate √12 ? One way is to use Hero’s Method One way is to use Hero’s Method

29. (c) MathScience Innovation Center, 2007 Hero’s Method For finding square roots For finding square roots A special case of Newton’s Method used by calculus students to find roots of many equations A special case of Newton’s Method used by calculus students to find roots of many equations No longer the most efficient method (by hand) it was replaced by tables, then the tables were replaced by calculators, but the calculator can quickly perform Hero’s Method! No longer the most efficient method (by hand) it was replaced by tables, then the tables were replaced by calculators, but the calculator can quickly perform Hero’s Method!

30. (c) MathScience Innovation Center, 2007 How it works: Goal: Get as close as you desire to the answer by the iteration of an expression, starting with an approximation (the seed). Goal: Get as close as you desire to the answer by the iteration of an expression, starting with an approximation (the seed). The expression iterated is: The expression iterated is: x new =[ x old + (“sqr root”/x old )]/2 x new =[ x old + (“sqr root”/x old )]/2 Example: For √12 x new = [3 + (12/3)]/2 x new = [3 + (12/3)]/2 X new = 7/2 0r 3.5 this goes back as x old

31. (c) MathScience Innovation Center, 2007 FRACTALS