geometer%27s sketchpad fractals presentation

Using The Geometer’s Sketchpad to Explore Fractals Images

and Iterated Function Systems

ME2 by the Sea June 9, 2006

Richard Rupp

Del Mar College

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INTRODUCTION

According to the Key Curriculum Press website, the publisher of The Geometer’s Sketchpad, “Sketchpad is a dynamic construction and exploration tool that enables students to explore and understand mathematics in ways that are simply not possible with traditional tools—or with other mathematics software programs. With a scope that spans the mathematics curriculum from middle school to college, The Geometer’s Sketchpad® brings a powerful dimension to the study of mathematics.” Understanding this, we will see that Sketchpad can be an incredibly useful tool for the study of fractal images. In simplest terms, fractal images are geometric figures that exhibit self-similar patterns in continually decreasing sizes. One of the most common ways of creating fractal images is to use a recursive definition, i.e. an iterated process. When using recursion, we must define a specific starting image and a process by which we will transform the first image into a second image. The starting image is called the initiator or seed and the second image is called the generator or first iteration. Infinite repetition of the recursive process ultimately creates an infinitely complex image…our fractal. Fractal images defined in this way are also referred to as Iterated Function systems. While finite iterations of the process will not create an exact image of the fractal, they can give us a reasonable approximation and The Geometer’s Sketchpad excels in this application. To draw fractals with Sketchpad, one merely needs to be able to construct a generator from an initiator and successfully apply the iterate function. We will use this process to create an approximation of a very simple fractal, Lévy’s curve, and a more complex fractal, the Koch curve, and its beautiful, pseudo-fractal counterpart, the Koch Snowflake.

Time permitting; we will discuss fractal or similarity dimension and applications of fractals in real world settings.

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CREATING LÉVY’S CURVE WITH SKETCHPAD

Initiator Generator Approximation of fractal

1.) Start with a blank sketch and construct a line segment using the straightedge tool. This is the

initiator of Lévy’s curve. 2.) Using the text tool, click on endpoints of our segment to label the points as “A” and “B”

A B

3.) Using the selection arrow tool, select the line segment. When selected, it will highlight. Now from the construct menu, select midpoint to construct a midpoint on the segment

A B

4.) First select both the midpoint and AB , then, from the construct menu, select perpendicular to create a line perpendicular to the segment passing through the line.

A B

5.) Using the selection arrow tool, deselect all objects. Now double click on point A to set it as a point of rotation. You should see a bulls-eye animate briefly around point A. Now select AB and from the transform menu choose rotate. Make sure the fixed angle radio button is selected (it should be by default) and enter 45 into the degrees entry space. You should see a lighter image of the rotated segment in the sketch. If everything looks correct, press the rotate button and obtain the following image.

A B

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6.) Now, select the rotated segment and perpendicular line. Using the construct menu, choose intersection to create the point of intersection.

A B

7.) We now want to hide the portions of the construction that aren’t needed. So, using the selection arrow tool, deselect all objects by clicking in blank space and then select the rotated segment, the original segment, the perpendicular line, and the midpoint. Now, in the display menu, choose hide objects. Your image should look like this.

A B

8.) Using the text tool, click on top point to label it as “C”. Then, using the arrow selection tool, select points A and C and from the construct menu choose segment.

C

A B

9.) Now select points C and B and from the construct menu choose segment. C

A B

We have now constructed the generator of Lévy’s curve. To approximate the fractal we now need to iterate the generator construction process.

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10.) Using the arrow selection tool, deselect all objects and then select points A and B. In the

transform menu, select iterate. A new menu appears. We have to tell Sketchpad where we want the iterated process mapped or placed. We want point A to be mapped to point A and point B to be mapped to point C. This will iterate the process on AC . So with the A ⇒ box highlighted, click on point A in the sketch. Now the B ⇒ and point B should highlight. We want to map point B to point C, so now click on point C in the sketch. Note: you may have to move the iterate menu box to have access to point C in your sketch. When you have done this you should see a lighter image that looks like the following

C

A B

11.) We still need to map the iterated process onto CB. In the iterate menu box, click on structure and then select add new map. This time map A to C and B to B. So first click on point C and then on point B in the sketch. Now press the iterate button to obtain the following picture.

C

A B

12.) Pressing the “+” or “-” keys will increase or decrease the number of iterations displayed. Also by right clicking on the fractal and selecting properties and then the iteration tab, one can select whether all iterations of the fractal are displayed “full orbit” or just the “final iteration only”.

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CREATING THE KOCH CURVE AND SNOWFLAKE WITH SKETCHPAD


1.) Start with a blank sketch and construct a line segment using the straightedge tool. This is the

initiator of Koch’s curve. 2.) Using the text tool, click on endpoints of our segment to label the points as “A” and “B”

A B

We need to trisect segment AB. To do this, we will utilize the proposition that parallel lines intersecting a triangle will partition the sides into proportional lengths. Thusly, we will construct a triangle such that one side is our original segment and a new side consists of three collinear congruent segments. The third side of the triangle will be the segment connecting the remaining endpoints.

3.) Using the straightedge tool, construct a new segment starting at A and creating an acute angle with vertex A. The acute angle is not necessary, but does help to keep the construction more compact. Use the text tool to label the new endpoint of the segment as “C”.

A B

C

4.) We now will mark a vector for a segment translation. Using the selection arrow tool,

deselect all objects, and then in order select points A and then C. Now from the transformation menu, choose mark vector. You should see dots race briefly from point A to point C.

5.) Now select the AC segment and point C. From the transform menu, choose translate. When the translate menu box appears, make sure the marked radio button is selected (it should be by default) and then press the translate button.

A B

C

6.) Without changing the current selections, from the transform menu, choose translate and press

the translate button. Your image should now look as follows.

A B

C

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7.) Use the text tool to label the new points, in order, as C′ andC′′ . Then, using the selection arrow tool, deselect all objects, and then select C′′ and B. From the construct menu, choose segment.

C''

C'

A B

C

8.) Now select point C′ , note: C B′′ is already selected, and, from the construct menu, choose

parallel line. C''

C'

A B

C

9.) Now select point C, note: the previously constructed parallel line is still selected, and, from

the construct menu, choose parallel line. C''

C'

A B

C

10.) We now want to select AB , note: the previously constructed parallel line is still selected, and from the construct menu choose intersection. Using the text tool label this new point D.

D

C''

C'

A B

C

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11.) Using the selection arrow tool, deselect all objects, and then select AB and the parallel line passing through C′ . From the construct menu choose intersection and then, using the text tool, label this new point E.

ED

C''

C'

A B

C

12.) Using the selection arrow tool, deselect all objects, and then select all objects other than the

points A, D, E, B, and from the display menu choose hide objects. EDA B

13.) Congratulations! You have successfully trisected AB . If you want to verify this, select

points A and D and from the measure menu choose distance. Repeat this for points D, E and E, B. Note: all three distances are equal, but may not necessarily be same as those shown here.

EB = 2.13 cm

DE = 2.13 cm

AD = 2.13 cm

EDA B

14.) If you choose to measure the above distances, select them and press the delete key as we don’t really need them and they will just clutter up the image. If you didn’t measure the distances, just go on to the next step.

15.) Using the selection arrow tool, deselect all objects, and then select points A and D. From the construct menu, choose segment.

EDA B

16.) Using the selection arrow tool, deselect all objects, and then select points E and B. From the

construct menu, choose segment.

EDA B

17.) Now double click on point D to make it as a center of rotation for a transformation. You should see a bulls-eye animate briefly around point D. Using the selection arrow tool, deselect all objects, and then select point A and AD . From the transform menu choose rotate. Make sure the fixed angle radio button is selected (it should be by default) and enter -120 into the degrees entry space. You should see a lighter image of the rotated segment in the sketch. If everything looks correct, press the rotate button and obtain the following image.

EDA B

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18.) Using the text tool, label the new point as A′ . Now, using the selection arrow tool, deselect all objects, and then select points A′ and E. From the construct menu, choose segment.

A'

EDA B

We have now constructed the generator of Koch Curve (Snowflake). To approximate the fractal we now need to iterate the generator construction process.

13.) Using the arrow selection tool, deselect all objects and then select points A and B. In the transform menu, select iterate. A new menu appears. We have to tell Sketchpad where we want the iterated process mapped or placed. We want point A to be mapped to point A and point B to be mapped to point D. This will iterate the process on AD . So with the A ⇒ box highlighted, click on point A in the sketch. Now the B ⇒ and point B should highlight. We want to map point B to point D, so now click on point D in the sketch. Note: you may have to move the iterate menu box to have access to point D in your sketch. When you have done this you should see a lighter image that looks like the following

A'

EDA B

14.) We still need to map the iterated process onto DA′ , A E′ , and EB . In the iterate menu box, click on structure and then select add new map. This time map A to D and B to A′ . So first click on point D and then on point A′ in the sketch.

15.) Now click on structure and then select add new map a second time. This time map A to A′ and B to E. So first click on point A′ and then on point E in the sketch.

16.) Finally, click on structure and then select add new map a third time. This time map A to E and B to B. So first click on point E and then on point B in the sketch.

17.) Press the display button in the iterate menu box and choose final orbit only. Now press the iterate button to obtain the following picture.

A'

EDA B

19.) Using the selection arrow tool, deselect all objects, and then select AD , DA′ , A E′ , and EB . From the display menu, choose hide segments. Your picture will now appear as follows.

A'

EDA B

This is an approximation of the Koch curve.

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20.) Using the selection arrow tool, select the Koch curve and press the “+” keys to increase the number of iterations displayed. It should work twice. You now have this image.

A'

EDA B

There are many ways to finish the snowflake. This is an easy one.

21.) To finish the snowflake, from the edit menu choose select all, then copy, and then paste twice. The image will be a mess and look like this. Note: if you repeat and undo the copy and paste process, Sketchpad will rename the points differently. Use the point labels given here as references only and do not worry if the points in your copied images are labeled differently.

S'2

U2 T2

N' 2

P2 O2

A'

EDA BM2N2

R2S2

22.) Now move one copy near the end of the original so that points B and N2 are very close to

each other as in the following diagram. Move the second copy near the other end of the original so that points R2 and A are very close to each other as in the following diagram. When moving the copies, select the Koch curve, rather than the labeled points, and the curve will move as a unit and not deform.

23.) Now, using the arrow selection tool, deselect all objects and then select points B and N2.

From the edit menu select merge. One of the points, in my case B, will move and become one with the other point. Repeat this process to merge points R2 and A.

S'2

U2 T2

N' 2

P2 O2

A'

EDA B M2N2R2S2

S'2

U2 T2

N' 2

P2 O2

A'

EDA M2N2S2

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24.) Using the arrow selection tool, deselect all objects and then select point S2. Drag this point

to a location directly underneath A′ so that the distance from S2 to A is preserved. Don’t worry if the distance isn’t exact, we can fix it later. Now, deselect all objects and then select point M2 and move it next to S2. It should look as follows.

S'2 U2

T2

N' 2

P2

O2

A'

EDA

M2

N2

S2

25.) Using the arrow selection tool, deselect all objects and then select points S2 and M2. From the edit menu select merge.

M'2 U2

T2

N' 2

P2

O2

A'

EDA

M2

N2

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26.) Using the point selection tool, deselect all objects, then select points A and N2. From the measure menu select distance. Repeat this process to also measure the distances from A to M2 and from N2 to M2. Now slightly adjust the locations of point M2 to try to equalize all three distances. Don’t worry, you probably won’t be able to make them perfectly equal.

N2M2 = 6.68 cm

AM2 = 6.68 cm

AN2 = 6.67 cm

M'2 U2

T2

N' 2

P2

O2

A'

EDA

M2

N2

27.) Now to clean up the picture, select only the three measurements and from the display menu

choose hide distance measurements. 28.) Here is a neat trick to get rid of the points. Click on the point tool. Now from the edit menu

choose select all points. Only the points will be selected. (Using the tools and select all also works with straight objects and circles!) Finally, from the display menu choose hide points. Viola! The Koch Snowflake! Note: The Koch Snowflake lacks true self similarity and, thus, is not a true fractal.

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PICTORIAL CATALOG OF FRACTAL INITIATORS AND GENERATORS Lévy’s curve


Koch curve


Quadric Koch curve


Minkowski’s fractal


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Box curve


Cantor set


Sierpinki gasket


Sierpinski carpet


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Calculation of Fractal or Similarity Dimension Fractal or similarity dimension is a calculation of how densely a fractal image fills a 2-dimensional plane. The calculation is quite simple.

log1log

ND

r

=⎛ ⎞⎜ ⎟⎝ ⎠

N is defined to be the replacement ratio or the number of copies created in each iteration. r is defined to be the scaling factor of the transformation involved in each iteration. For example, consider the Koch curve

Initiator Generator

Here, N = 4 since 4 copies of the initiator are created. Also, 13

r = since each copy is 13

of the length of

the initiator. Thus, log 4 log 4 1.262log3

1log 13

D = = ≈⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

In Lévy’s curve,

Initiator Generator

N = 2 since 2 copies of the initiator are created. Also, 12

r = since each copy is 12

of the length of

the initiator. Thus, 12

log 2 log 2 log 2 log 2 21log 2 log 2log 21 2log 12

D = = = = =⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

.

Thus, Lévy’s curve more densely fills a 2-dimensional plane than the Koch curve.

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Applications of Fractal Geometry Fractals have many applications ranging from computer science, to cartography, to advanced antenna design. In computer science, fractals as iterative processes have been used to develop image compression routines based upon self-transformations that have been demonstrated to be superior to traditional formats like .gif and .jpg. Fractals have been used to study the length of coastlines, especially the coastline of Great Britain. Most interestingly, it has been shown by Nathan Cohen and Robert Hohlfield that “for an antenna to work equally well at all frequencies, it must satisfy two criteria. It must be symmetrical about a point. And it must be self-similar, having the same basic appearance at every scale – that is, it has to be a fractal.”1 Another antenna fact is that, generally, the larger or longer the antenna, the more powerful it is. With this in mind, the Koch curve has been utilized for antenna design because, interestingly, the more iterations used in the curve, the longer the curve becomes…and this process is unlimited. Thusly, the Koch snowflake is a bizarre geometric object as it has infinite perimeter, but only contains a finite area. Finally, the increased reception capabilities of fractal antennae, in some cases up to 25% greater than traditional cell phone antennae2 have been a driving force in including fractal antennae inside the bodies of cellular phones and allowing these phones to incorporate GPS capability.

Pictures of fractal cellular phone antennae featured in Mathematical Excursions by Auffmann, Lockwood, Nation, and Clegg, page 535.

1 Musser, George. Technology and Business: Wireless Communications. Scientific American, July, 1999. 2 Ibid.

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POSSIBLE TEKS CORRELATIONS FOR PRESENTATION

§111.22. Mathematics, Grade 6. (3) Patterns, relationships, and algebraic thinking. The student solves problems involving

proportional relationships. The student is expected to: (A) use ratios to describe proportional situations; (C) use ratios to make predictions in proportional situations. (13) Underlying processes and mathematical tools. The student uses logical reasoning to make

conjectures and verify conclusions. The student is expected to: (A) make conjectures from patterns or sets of examples and nonexamples; and (B) validate his/her conclusions using mathematical properties and relationships. §111.23. Mathematics, Grade 7. (3) Patterns, relationships, and algebraic thinking. The student solves problems involving

proportional relationships. The student is expected to: (B) estimate and find solutions to application problems involving proportional

relationships such as similarity, scaling, unit costs, and related measurement units.

(4) Patterns, relationships, and algebraic thinking. The student represents a relationship in numerical, geometric, verbal, and symbolic form. The student is expected to:

(A) generate formulas involving conversions, perimeter, area, circumference, volume, and scaling;

(B) graph data to demonstrate relationships in familiar concepts such as conversions, perimeter, area, circumference, volume, and scaling; and

(C) describe the relationship between the terms in a sequence and their positions in the sequence.

(6) Geometry and spatial reasoning. The student compares and classifies shapes and solids using geometric vocabulary and properties. The student is expected to:

(D) use critical attributes to define similarity. (15) Underlying processes and mathematical tools. The student uses logical reasoning to make

conjectures and verify conclusions. The student is expected to: (A) make conjectures from patterns or sets of examples and nonexamples; §111.24. Mathematics, Grade 8. (6) Geometry and spatial reasoning. The student uses transformational geometry to develop

spatial sense. The student is expected to: (A) generate similar shapes using dilations including enlargements and

reductions; (10) Measurement. The student describes how changes in dimensions affect linear, area, and

volume measures. The student is expected to: (A) describe the resulting effects on perimeter and area when dimensions of a

shape are changed proportionally (16) Underlying processes and mathematical tools. The student uses logical reasoning to make

conjectures and verify conclusions. The student is expected to: (A) make conjectures from patterns or sets of examples and nonexamples;

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§111.34. Geometry (One Credit). (b) Geometric structure: knowledge and skills and performance descriptions. (2) The student analyzes geometric relationships in order to make and verify conjectures.

Following are performance descriptions. (A) The student uses constructions to explore attributes of geometric figures and

to make conjectures about geometric relationships. (c) Geometric patterns: knowledge and skills and performance descriptions.

The student identifies, analyzes, and describes patterns that emerge from two- and three-dimensional geometric figures. Following are performance descriptions.

(1) The student uses numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, and angle relationships in polygons and circles.

(2) The student uses properties of transformations and their compositions to make connections between mathematics and the real world in applications such as tessellations or fractals.

(f) Similarity and the geometry of shape: knowledge and skills and performance descriptions. The student applies the concepts of similarity to justify properties of figures and solve problems. Following are performance descriptions.

(1) The student uses similarity properties and transformations to explore and justify conjectures about geometric figures.

(2) The student uses ratios to solve problems involving similar figures.

MATERIALS AND PRICING INFORMATION

The only materials required to implement this activity into a K-12 classroom are a computer (or computer lab) and the The Geometer’s Sketchpad software. Please visit the website of Key Curriculum Press (http://www.keypress.com/catalog/products/software/Prod_GSP.html#Anchor-The-44867) for information on pricing of the software.

BIBLIOGRAPHY OF SUGGESTED READING The Geometer’s Sketchpad Resource Center (http://www.keypress.com/sketchpad/)

- This is a part of the Key Curriculum Press website and features classroom activities, project ideas, bibliographies, information on pricing of the software, and more.

Mathematical Excursions by Auffmann, Lockwood, Nation, and Clegg, Houghton Mifflin Company,

2004. The Fractal Geometry of Nature by Benoit B. Mandelbrot, W. H. Freeman and Company, 1982. Fractals Everywhere by Michael F. Barnsley, Academic Press, 1988.

geometer%27s sketchpad fractals presentation

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