Prairie Dog Town Relocation

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Fractals and Fractal Geometry
Teacher Guide
This activity introduces students to modern
mathematics. Fractal geometry is just over
thirty years old, and studying it shows students
that math is a dynamic subject. This activity
explores the quasi-dimensional aspects of
fractals and has students use GIS to make a
real-world application.
Overview
Oklahoma PASS Objectives
Process Standard 1(1), 4(1, 2, 3), 5(1)
National Council of Geographic Education Standards
World in Spatial Terms (1, 3) Uses of Geography
(17, 18)
National Council of Teachers of Mathematics
Geometry Standard 1(1), 2(2), 4(1, 5)
High School
Grade Level
Beginning requires changing the level of scale
and using the measurement tool
GIS Skill Level
This activity will require students to draw
fractals, to cut and fold paper to create
fractals, and use a ruler to estimate distance
along a river.
Other Skills
Should be completed in three 50-minute classes.
Part I introduces fractals and 1.X dimensional
fractals. Part II requires students to create
2.X dimensional fractals. Part III uses GIS for
measuring the bank of a river at different scales.
Time
Fractals and Fractal Geometry
Teacher Guide p. 1
2
Fractals and Fractal Geometry
Teacher Guide
Materials
Triangle grid paper for Sierpinskis Triangle and
Kochs Snowflake (see website below) Diego
Uribes book, Fractal Cut-Outs. Colored
paper. Aerial photo of Cimarron River within
Payne County.
Sources
http//math.rice.edu/lanius/fractals http//www.i
ts.caltech.edu/atomic/snowcrystals/photos/photos.
htm http//enwikipedia.org/wiki/Fractal http//cla
sses.yale.edu/Fractals http//www.crystalinks.com/
fractals.html http//www.fractalwisdom.com/fractal
.html http//www.answers.com http//www.imho.com/f
rae.chaos/chaos.html http//www.archive.ncsa.uiuc.
edu/Edu/Fractal/Fgeom.html http//library.thinkque
st.org/3493/frames/fractal.html Uribe, Diego.
Fractal Cut-Outs.
Lesson Preparation
This lesson is largely taken from lessons and
sources on the internet. Activity 1 is from a
lesson available online and Activity 2 is from
the Fractal Cut-Outs book. Activity 3 is an
original real-life application. The aerial
photos should be downloaded and placed on the
C\\ drive prior to the activity. A starting and
finishing point should be drawn along the river.
Adapted and developed by Pamela D. Jurney.
Fractals and Fractal Geometry
Teacher Guide p. 2
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Fractals and Fractal Geometry

• What is a Fractal?
• Self-similar structure magnified version
  virtually indistinguishable from unmagnified
  version
• Generated by repeating patterns at different
  scales (iterations)
• Mandelbrot fractal at different scales
• http//en.wikipedia.org/wiki/ImageMandelbrot-simi
  lar-x1.jpg

• Famous Fractals
• Mandelbrot Set
• - http//classes.yale.edu/Fractals/MandelSet/Mand
  el1.gif
• - http//en.wikipedia.org/wiki/ImageMandelpart2.
  jpg
• Sierpinskis Triangle
• - http//en.wikipedia.org/wiki/ImageSierpinsky_t
  riangle_28evolution29.gif
• Kochs Snowflake
• - http//math.rice.edu/lanius/fractals

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Fractals and Fractal Geometry
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Fractals and Fractal Geometry

• Fractals in Nature
• Snowflakes
• - http//www.its.caltech.edu/atomic/snowcrystals
  /photos/photos.htm
• Ferns
• - http//classes.yale.edu/Fractals/Panorama/Natur
  e/NatFracGallery/Gallery/
• RealFern.gif
• Broccoli
• Rivers

• The History of Fractals
• Chaos Theory 1960s
• Meteorologist, Edward Lorenz
• Finding the underlying order in apparently random
  data
• Butterfly effect
• Benoit Mandelbrot
• How Long is the Coast of Britain? Statistical
  Self-Similarity and Fractal Dimension Science,
  1967
• Coined term fractal in 1975 from Latin fractus
  meaning broken or irregular

Fractals and Fractal Geometry
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Fractals and Fractal Geometry

• Measuring the Length of a Coastline
• Mandelbrot thought to answer the question,
  how long is the coastline of Britain?
• New degrees of detail at each scale.
• - Pictures of the coastline of Britain

• Classical Geometry
• Euclidean geometry Integer dimensions
• - Zero dimension point
• no length, no width, no height
• - One dimension line
• no width and no height, but infinite length
• - Two dimensions polygons circles
• length and width, no depth
• - Three dimensions cubes, spheres, polyhedra
• length, width, and depth

• Fractal Geometry
• Modern mathematicsjust over 30 years old
• Fractal geometry Non-integer dimensions
• - One . X dimension fractal curve
• How much space does it take up as the line twists
  and curves?
• More space is closer to two-dimensional
• - Two . X dimension fractal landscape

Fractals and Fractal Geometry
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Fractals and Fractal Geometry
Integer Dimensions
Doubling of Copies
Dimension
Figure

Self-Similar Figure
n 2d
d
Taken from http//math.rice.edu/lanius/fractals
Fractals and Fractal Geometry
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Fractals and Fractal Geometry
Fractal Dimension
Sierpinskis Triangle
Doubling of Copies
Dimension
Figure
d
Sierpinski Triangle
3 2d
Lets try 21.5
2.83
What about 21.6?
3.03
d something between 1.5 and 1.6
21.58 2.99
Fractals and Fractal Geometry
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Fractals and Fractal Geometry
Activity 1 Drawing 1.X Dimensional Fractals
Kochs Snowflake Step One Start with a large
equilateral triangle. Step Two Divide one
side of the triangle into three equal parts.
On the middle third, draw an equilateral
triangle. Erase the base of the new
triangle. Do this to all three sides of the
triangle. Step Three For each straight
edge, divide into three equal parts. Draw
an equilateral triangle on the middle third.
Erase the base of the each new triangle. Step
Four Repeat the process to as many
iterations as possible.

Sierpinskis Triangle Step One Draw a large
equilateral triangle on the triangle grid paper.
Make sure that it takes up most of the page.
Find the mid-points of each side.
Connect the mid-points to make an inverted
triangle. Then, shade the inverted
triangle. Step Two Find the mid-points of
each side of the three non-shaded triangles.
Connect the mid-points to make three inverted
triangles, and then shade them. Step Three
Find the mid-points of the nine non-shaded
triangles. Connect the mid-points and then
shade in the inverted triangles. Step Four
Repeat the process to as many iterations as
possible.
Triangle Grid Paper available on line at
http//math.rice.edu/lanius/fractals
Activity 2 Building 2.X Dimensional
Fractals Fractal Cut-Outs by Diego Uribe
Fractals and Fractal Geometry
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Fractals and Fractal Geometry
Activity 3 Measuring the Cimarron River
In this activity you are going to measure a
segment of the Cimarron River at different
scales. To maintain consistency, follow the path
of the north bank of the river. The first scale
is 140,000. This means that one meter on the
map equals 40,000 meters in real life. You will
be given aerial photos at three different scales.
You will measure the length of the segment in
centimeters then solve a proportion to find the
actual distance. Since you are measuring in
centimeters, you need to convert the lengths to
meters. Since 1 meter 100 centimeters, move
the decimal over two places to the
________________.
Exploring Different Scales
Scale 140,000
_______cm 1
m 40,000 x x
_____________ meters
Scale 124,000
_______cm 1
m 24,000 x x
_____________ meters
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Fractals and Fractal Geometry
Scale 110,000

_______cm 1
m 10,000 x x
_____________ meters
Note these photos have been copied and pasted,
losing their exact scale dimensions.
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Fractals and Fractal Geometry
Using the Measure Tool in ArcMap Go to My
Documents and navigate to Local Disk C// drive.
Go to Documents and Settings and then to All
Users. Open the river map under the Fractals
folder. You will be using the measure tool in
ArcMap to determine the same segment of the river
that you measured previously Follow the north
bank of the river for consistency. Make sure
that the measurement is as detailed as possible,
click on every twist and turn.

Scale box
Measure tool
The measurement tool is very sensitive. If you
make a mistake, you will need to click on the
arrow and start over. If the map does not open
to the scale 140,000, highlight it and change
it. Give the map time to redraw. You will
measure from the right edge of the start box to
the left edge of the finish box. The total
measurement will appear in the bottom left corner
of the screen.
When you have completed the 140,000 measurement,
record your result below. Change the scale to
124,000 and re-measure. Do the same for
110,000 and record both results. Then, answer
the questions regarding the fractal applications
of these activities.
Fractals and Fractal Geometry
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Results River segment at the scale of 140,000
_____________________ meters River segment at
the scale of 124,000 _____________________
meters River segment at the scale of 110,000
_____________________ meters Questions Imagine
you had to measure the length of the Cimarron
River that runs through Payne County. 1. You
could measure the length at different scales.
Why might you choose the 140,000 over the
110,000? Remember, the 140,000 scale has less
detail. 2. Why might you choose the
110,000 over the 140,000? Why would more
detail be important? 3. Now that you have
worked with measuring a coastline (or a bank-line
in our case), explain how a fractal coastline
could be considered to have an infinite
length. Infinity goes on forever and ever.
Every time you zoom in, you find more and more
detail to measure. An accurate measurement can
never really be obtained because you the detail
of the coastline could be considered infinite.

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