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## Bruce Wayne Fractals

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### Bruce Wayne Fractals Logo Programming It is not just for the kids. Big kids can have fun with it as well !!! Fractals with Sketchpad Now that you have experimented ... – PowerPoint PPT presentation

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Title: Bruce Wayne Fractals

1
Bruce Wayne Fractals
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What is a Fractal?
3
According to Benoit Mandelbrot A fractal is
by definition is a set for which the
Hausdorff-Besicovitch dimension strictly exceeds
the topological dimension. So the concept
of dimension is very important as we are
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Fractals in Nature Clouds are not spheres,
mountains are not cones, coastlines are not
circles, and bark is not smooth, nor does
lightning travel in a straight line." -- Benoit
Mandelbrot
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large glacial lake in Finland.
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Fractal Clouds
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Tajikistan
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Fractal History
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Helge Von Koch
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Waclaw Sierpinski
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Georg Cantor
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Gaston Julia
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Benoit Mendelbrot
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Richard Swearingen
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Sunny
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Dianne Clark
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Fractal Terminology
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• Important Characteristics of Fractals
• They are recursive that is, the process of
their creation gets repeated indefinitely
• They are self-similar that is, copies of the
entire fractal may be found, in reduced form,
within the fractal.

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• Ways to Create Geometric Fractals
• use a base shape and replace it with a recurring
motif shape (we did this when we created the Koch
Triangle for homework, the initial triangle was
the base and the shape that we replaced each side
with was the motif)
• play the chaos game
• method of successive removals

23
Introducing XAOS Software Lets look at this
really neat fractal software and keep in mind
those ideas about recursion and self-similarity.
24
Lets Play The Chaos Game
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Dimension 1 dimensional 2 dimensional 3
dimensional
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Dimension Definition (1) A measure of spatial
extent, especially width, height, or length.
(2) The least number of independent coordinates
required to specify uniquely the points in a
space.
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Dimension Definition The first formal definition
was stated by Dutch mathematician L E J Brouwer
(1881-1966) in 1913. A (solid) cube has the
topological dimension of three because in any
decomposition of the cube into smaller bricks
there always are points that belong to at least
four (31) bricks.
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Definition Self-similarity Dimension D
log ( number of pieces ) log (
magnification factor )
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Easy example What is the self-similarity
dimension of a cube that has a length 3, a
width 3, and a height 3 ?   We can break the
cube up into 27 smaller cubes, or "pieces".
Also, if we take one of the smaller cubes and
"magnify" the sides by 3, we end up with a cube
that is the same size as the original.  Hence,
the "magnification factor" is 3. Self-similarity
dimension log( number of pieces )
log( magnification factor
) Self-similarity dimension log (27) log(3)3
3 log(3) 3 log (3)
log(3) log(3)
30
What is the fractal dimension of the Koch
Snowflake ? Self-similarity dimension log(
number of pieces ) log(
magnification factor )
31
What is the fractal dimension of the Koch
Snowflake ? Self-similarity dimension
log(4) 1.26 log(3)
32
What would the "self-similarity dimension" be for
the Koch Island Fractal ?
33
Self-similarity dimension log ( number of
pieces ) log (
magnification factor ) Self-similarity dimension
log (8) log(2)3 3 log(2) 1.5
log(4) log(2)2 2 log(2) What is the area of
the Koch Island fractal ? What is the perimeter
of the Koch Island fractal ?
34
Logo Programming It is not just for the kids.
Big kids can have fun with it as well !!!
35
36
Now that you have experimented with creating the
Hat Curve Fractal, its time to make your own.
• Go to FILE, then DOCUMENT OPTIONS.
• Choose the ADD PAGE tab, then BLANK PAGE.
• Click on OK.
•
• Use the same procedure as you did for the Hat
Curve Fractal to create your own.
•
• Decide upon a rule to use. Creativity counts
here! For example, two rules you have seen are
to replace the middle third of the segment with a
triangle or with a square. Type your rule on
your page with a text box.
•
• After you have created your fractal, copy and
fill in the table below on sketchpad. Pick some
convenient starting length for your segment
(other than 1).
•

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Sierpinski Pyramid
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Fractal Cards
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Fractals in the K-16 Curriculum
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Fractal References
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A Fractals Unit for Elementary and Middle School
Students, by Cynthia Lanius, Rice
University, http//math.rice.edu/lanius/frac/inde
x.html 1996-2007 Build a Sierpinski Pyramid, by
Paul Kelly, Mathematics Teacher, 92, 384. 1999.
Chaos Game Applet by Trevor Stone
http//trevorstone.org/applets/ChaosGame.html
Exploring Geometry, by Dan Bennett Emeryville,
CA Key Curriculum Press. 2002. Fractal Cards
A Space for Exploration in Geometry and Discrete
Mathematics. Simmt, Elaine Davis, Brent
Mathematics Teacher, 91, 102. 1998.
57
Fractals  A Toolkit of Dynamics Activities, by
Jonathon Choate, Robert Devaney, and Alice
Foster, Key Curriculum Press, 1999 Fractint, a
free fractal generator http//spanky.triumf.ca/ww
w/fractint/fractint.html GNU XaoS, a free
interactive fractal zoomer at http//wmi.math.u-s
zeged.hu/xaos/doku.php Interactivate Website by
Shodor  http//www.shodor.org/interactivate/activ