naturally algebra g. whisler. (c) mathscience innovation center, 2007 naturally algebra

Naturally AlgebraNaturally Algebra

G. WhislerG. Whisler

(c) MathScience Innovation Center, 2007(c) MathScience Innovation Center, 2007

NATURALLY ALGEBRA


What is a Fractal?What is a Fractal?

A Self Similar PatternA Self Similar Pattern Formed by recursion (iteration or Formed by recursion (iteration or

repeated application of a process on repeated application of a process on its output)its output)

Has fractal dimension (dimension Has fractal dimension (dimension that is not always in whole number that is not always in whole number scale)scale)


A section of one of the most A section of one of the most famous fractals created…..famous fractals created…..


A fractal in nature…A fractal in nature… Exchange profilesExchange profiles An example of an An example of an

exchange profile is a exchange profile is a radiatorradiator

Root systems are good Root systems are good ‘natural’ examples‘natural’ examples

Picture by Greg VogelPicture by Greg Vogel


……and so are branchesand so are branches


Fractals in NatureFractals in Nature

Many times exchange profiles are Many times exchange profiles are solutions to ‘problems’ faced by solutions to ‘problems’ faced by nature.nature.

These exchange profiles are created These exchange profiles are created by iteration.by iteration.

ITERATION : Repeating a processITERATION : Repeating a process


Fractal Tree ActivityFractal Tree Activity

Logon to the Logon to the computers and…computers and…

Launch GSP 4.07Launch GSP 4.07 Start a new sketchStart a new sketch Follow me…Follow me…

The Geometer's SketThe Geometer's Sketchpadchpad

A''

A''

A'

A

B


RESULTSRESULTS

STEPSTEP NEW NEW BranchesBranches

TOTAL TOTAL BRANCHESBRANCHES

00 11 11

11 22 33

22 44 77

33 88 1515

44 1616 3131

““Nth”Nth” 22NN 2(New)-12(New)-1

1616 6553665536 131071131071


Now it is your turn!Now it is your turn!

First pick a GREEN First pick a GREEN “branches card” (this (this is the number of branches your tree will is the number of branches your tree will have: 2, 3 or 4),have: 2, 3 or 4),

Then pick as many BLUE Then pick as many BLUE “dilation cards” as branches to set the ratio for each as branches to set the ratio for each branch,branch,

Last, pick as many Last, pick as many “rotation cards” as as you have branches for the angle of you have branches for the angle of rotation for each branch.rotation for each branch.

DATA CHARTDATA CHART


Group DataGroup DataOur Forest!Our Forest!


……and since we have time…and since we have time…

We are going to look at other patterns.We are going to look at other patterns.

You might even find this one familiar!You might even find this one familiar!


More Challenging PatternsMore Challenging Patterns

Not all patterns are obvious or have Not all patterns are obvious or have easy to write rules for describing easy to write rules for describing them,them,

Now a more Now a more CHALLENGINGCHALLENGING puzzle… puzzle…


The Great Domino WallThe Great Domino Wall

How many Ways…..How many Ways…..

Can we build the Wall…Can we build the Wall…


Instructions for theInstructions for the GREAT DOMINO WALL GREAT DOMINO WALL

Each group has been tasked to …Each group has been tasked to … Build a wall ‘n’ units long and two Build a wall ‘n’ units long and two

units high. You will model this with units high. You will model this with dominos. A domino has dimensions dominos. A domino has dimensions of 2 units by 1 unit (2x1). of 2 units by 1 unit (2x1).

Find the number of ways you can Find the number of ways you can build the “Great Wall of Dominos” build the “Great Wall of Dominos” using 1, 2, 3, 4 and 5 dominos.using 1, 2, 3, 4 and 5 dominos.


RESULTSRESULTSLength NLength N Number of Number of

DominosDominosWays to Build Ways to Build

The WallThe Wall

00 00 11

11 11 11

22 22 22

33 33 33

44 44 55

55 55 88


Summary of resultsSummary of results

Does anyone recognize this pattern?Does anyone recognize this pattern? FibonacciFibonacci !! !! Problem - No easy formula for the Problem - No easy formula for the

Nth termNth term BUT…BUT… We can use the power of iteration to We can use the power of iteration to

find bigger N!find bigger N!


Iterating expressions with GSPIterating expressions with GSP

Start a new sketch in Geometer’s Start a new sketch in Geometer’s Sketchpad, andSketchpad, and

Follow Me…Follow Me…

1, 1, 2, 3, 5, 8, 13, 21,…1, 1, 2, 3, 5, 8, 13, 21,…


Technique of the future?Technique of the future?

Two advances in Two advances in MODERN MODERN MATHEMATICS and SCIENCE are MATHEMATICS and SCIENCE are FRACTAL GEOMETRY andand CHAOS THEORY. They were developed from They were developed from iterating functions:iterating functions:

Fractals: f(z) = zFractals: f(z) = z22 + c + c Chaos: f(x) = ax(1- x)Chaos: f(x) = ax(1- x)


Fractal research started with an Fractal research started with an iterated function made by iterated function made by

MandelbrotMandelbrot


This is an image created using This is an image created using fractal technologyfractal technology


Chaos Theory started with an Chaos Theory started with an investigation into weather patternsinvestigation into weather patterns


THE “BUTTERFLY” ATTRACTOR

CHAOS THEORY


FRACTALS and CHAOS…FRACTALS and CHAOS…

Are helping investigate and explain Are helping investigate and explain complex systems in the world around complex systems in the world around us…us…

THANK YOUTHANK YOU

I’ve enjoyed spending the I’ve enjoyed spending the time with you!time with you!


Motivation for Hero’s MethodMotivation for Hero’s Method

Iteration can be Iteration can be used to solve other used to solve other problems, such problems, such as…as…

How does a How does a calculator evaluate calculator evaluate √12 ?√12 ?

One way is to use One way is to use Hero’s Method Hero’s Method


Hero’s MethodHero’s Method

For finding square rootsFor finding square roots A special case of Newton’s Method A special case of Newton’s Method

used by calculus students to find used by calculus students to find roots of many equationsroots of many equations

No longer the most efficient method No longer the most efficient method (by hand) it was replaced by tables, (by hand) it was replaced by tables, then the tables were replaced by then the tables were replaced by calculators, but the calculator can calculators, but the calculator can quickly perform Hero’s Method!quickly perform Hero’s Method!


How it works:How it works:

Goal: Get as close as you desire to the Goal: Get as close as you desire to the answer by the answer by the iterationiteration of an of an expression, starting with an expression, starting with an approximation approximation (the seed).(the seed).

The expression iterated is: The expression iterated is:

xxnew new =[ x=[ xoldold + (“sqr root”/x + (“sqr root”/xoldold )]/2 )]/2Example: For √12Example: For √12

xxnew new = [3 + (12/3)]/2 = [3 + (12/3)]/2

XXnewnew = 7/2 0r 3.5 this goes back as x = 7/2 0r 3.5 this goes back as xoldold


FRACTALSFRACTALS

naturally algebra g. whisler. (c) mathscience innovation center, 2007 naturally algebra

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