Naturally Algebra

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Naturally Algebra

• G. Whisler

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NATURALLY ALGEBRA
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What is a Fractal?

• A Self Similar Pattern
• 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)

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A section of one of the most famous fractals
created..
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A fractal in nature

• Exchange profiles
• An example of an exchange profile is a radiator
• Root systems are good natural examples
• Picture by Greg Vogel

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and so are branches
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Fractals in Nature

• Many times exchange profiles are solutions to
  problems faced by nature.
• These exchange profiles are created by iteration.
• ITERATION Repeating a process

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Fractal Tree Activity

• Logon to the computers and
• Launch GSP 4.07
• Start a new sketch
• Follow me
• The Geometer's Sketchpad

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RESULTS
STEP NEW Branches TOTAL BRANCHES
0 1 1
1 2 3
2 4 7
3 8 15
4 16 31
Nth 2N 2(New)-1
16 65536 131071
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Now it is your turn!

• 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 dilation cards as
  branches to set the ratio for each branch,
• Last, pick as many rotation cards as you have
  branches for the angle of rotation for each
  branch.
• DATA CHART

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Group DataOur Forest!
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and since we have time

• We are going to look at other patterns.
• You might even find this one familiar!

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More Challenging Patterns

• Not all patterns are obvious or have easy to
  write rules for describing them,
• Now a more CHALLENGING puzzle

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The Great Domino Wall

• How many Ways..
• Can we build the Wall

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Instructions for the GREAT DOMINO WALL

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

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RESULTS
Length N Number of Dominos Ways to Build The Wall
0 0 1
1 1 1
2 2 2
3 3 3
4 4 5
5 5 8
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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!

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Iterating expressions with GSP

• Start a new sketch in Geometers Sketchpad, and
• Follow Me
• 1, 1, 2, 3, 5, 8, 13, 21,

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Technique of the future?

• Two advances in MODERN MATHEMATICS and SCIENCE
  are FRACTAL GEOMETRY and CHAOS THEORY. They were
  developed from iterating functions
• Fractals f(z) z2 c
• Chaos f(x) ax(1- x)

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Fractal research started with an iterated
function made by Mandelbrot
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This is an image created using fractal technology
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Chaos Theory started with an investigation into
weather patterns
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CHAOS THEORY
THE BUTTERFLY ATTRACTOR
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FRACTALS and CHAOS

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

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THANK YOU

• Ive enjoyed spending the time with you!

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Motivation for Heros Method

• Iteration can be used to solve other problems,
  such as
• How does a calculator evaluate v12 ?
• One way is to use Heros Method

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Heros Method

• For finding square roots
• A special case of Newtons 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 Heros Method!

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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).
• The expression iterated is
• xnew xold (sqr root/xold )/2
• Example For v12
• xnew 3 (12/3)/2
• Xnew 7/2 0r 3.5 this goes back as xold

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FRACTALS
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