Project AMP Dr. Antonio Quesada – Director, Project AMP

FRACTALS ROCK: INVESTIGATING THE DRAGON CURVE FRACTAL

Keyword: Fractal (Dragon Curve) Rotation

Fractals: A figure generated by repeating a special sequence of steps infinitely often. Fractals often exhibit self-similarity (the fractals clone themselves).

Introduction: Exploring nature can be an intriguing and challenging topic for all students. Nature is all around us and provides students with many interesting designs. The designs that make up objects in nature are polygons. However, the polygonal designs may not be recognized as the traditional polygons many students have studied and are familiar with from their mathematics classes. The designs in nature are given the general name of fractals. A branch of geometry called fractal geometry explores the designs within nature and the patterns that exist in the designs. A few examples of such fractals are the Koch Snowflake, the Fern Fractal, the Tree Fractal, the Sierpinski Triangle and our topic of investigation, the Dragon Curve. The Dragon Curve will challenge students as they try to create the various levels of the fractal.

Existing knowledge: Students should be familiar with the following concepts: length of a segment, rotating objects about a point, polygons and their characteristics, and fractals (Koch Snowflake, Tree Fractal and the Sierpinski Triangle). They should also have experience using Dynamic Geometry Software.

NCTM Standards: Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships

Specify locations and describe spatial relationships using coordinate geometry and other representational systems

Apply transformations and use symmetry to analyze mathematical situations

Use visualization, spatial reasoning, and geometric modeling to solve problems

Learning Objectives: 1. Students will be able to define a polygon on a plane. 2. Students will be able to rotate a defined polygon about a point on the polygon.

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Materials: 1. Piece of paper (for the pre-lab paper folding activity) 2. Worksheet from optional paper folding activity 3. Laboratory worksheet from Lab 1 4. Laboratory worksheet from Lab 2 5. Computer and/or graphing calculator equipped with Cabri Dynamic Geometry Software.

Procedure: 1. Students should be grouped in pairs. Method of grouping is left to the individual instructor. 2. Pose the following challenge to students: “Suppose your school district is sponsoring a contest to design a new school logo. The logo will be placed on all school communication and apparel. The challenge is to use fractals, specifically the Dragon Curve. How could you take the following figure to create the Dragon Curve and a potentially creative design that will rock the fashion industry?”

3. Optional: Have students complete the pre-lab paper folding activity (see lab worksheet). 4. Have students complete Lab 1 (rotating objects about vertices) 5. Have students complete Lab 2 (Dragon Curve Fractal: Creating the Dragon Curve) 6. Choices for the types of assessment are left to the discretion of the instructor. Some examples might include: the completed lab questions, class participation (student explanations of the lab, conjectures, answer and/or processes in finding solutions to the extension problems or student created extensions), peer or self-evaluation, evaluations (quizzes and/or tests) to replicate the lab results. Several forms of assessment should be utilized when lessons are inquiry/discovery based.

Bonus: For interested students, the winning design would be placed on t-shirts, brought from home, using an iron-on print.

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PAPER FOLDING INVESTIGATION FOLDING A FRACTAL – CREATING THE DRAGON CURVE

Team Members: ______File Name: ______

Introduction: The first dragon curve fractals were constructed by folding paper in a prescribed manner. Each added crease doubles the number of folds and complexity. When the paper is opened, the outline of the dragon curve fractal appears. As a result of getting the outline of the dragon curve, students will have a visual and a preview of the dragon curve they will create while completing the lab, using Cabri.

Directions: Follow the procedure below to create a fractal pattern with at least three repeats, or iterations, using a sheet of paper (3 inch by 5 inch recommended). The sheet of paper represents level 0, or the generator level.

Procedure: 1. Create level 1 by folding the paper in half. The folded paper reveals the outline for the basic seed: two edges.

2. Create level 2 by folding the paper in half again. This fold creates level 2 because each line in level 1 is divided in half. How many edges are in the outline of the dragon curve now, if you were to look at it opened up from the side?

3. Create level 3 by folding the paper in half again. This fold creates level 3 because each line in level 2 is divided in half. How many edges are in the outline of the dragon curve now if you were to look at it, opened up from the side?

4. Unfold the paper. Look at the distinctive dragon curve emerging. Carefully trace the outline of the dragon curve below.

5. Extension: If you were to fold the paper one more time, how many edges are in the outline of the dragon curve now, if you were to look at it, opened up from the side?

By folding the paper a fourth time, you would have created level 4. Can you draw the outline of this fourth level of the dragon curve? Give it your best attempt.

In theory, you could continue until it is impossible to continue folding the paper.

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Fractals Rock: Dynamic Geometry Lab – Creating the Dragon Curve

Team Members: ______File Name: ______

Lab I: Rotating Objects about Vertices.

Setup: Open Cabri Geometry II. Set it so that it fills the screen. Choose “Show Attributes” under the Options menu. Click on the color option (top button on the side). The color palette should appear on the screen. While holding the mouse button down, move the cursor to the middle of the screen and release the button. The color palette should now appear on the screen in its own window. Move the palette to the right side of the screen. [You can add the other option tools to the palette window by clicking on them – however, the other tools are not needed for this task]

1. Rotating a line segment around an end point. a. Show the axis. Using the Numeric Edit Tool create a measure of 4.00 centimeters (assign the units using CTRL-U). Move this number (using the Pointer Tool) below the palette. Using the Numeric Edit Tool create a measure of 25 degrees. Move this number below the 4.00 centimeter number. b. Using the Measurement Transfer Tool create a point located 4 cm from the origin along the x- axis. Click on the origin (when it indicates “this point”), then click on the number 4.00, then move the dashed line that appears until it lies on the x-axis and click to put the point there. Using the Segment Tool create a segment between the point at the origin and the point at the 4 cm mark on the x-axis. Change the color of this segment to Green. c. Now using the Rotation Tool rotate the segment 25 degrees about the point at the origin. Click on the segment (“this segment”), click on the point at the origin (“around this point”), and click on the 25 degree number (“using this number”). A new line appears. Change the color of that line to purple. Describe this line compared to the original line.

d. Using the Numeric Edit Tool, click on the 25 degree number. Using the up and down arrows at the side, change the number. What happens to the line as the numbers change?

Page 4 Project AMP Dr. Antonio Quesada – Director, Project AMP

e. Where is the line when the number reads 165 degrees? f. Set the number back to 25 degrees. Using the Angle Tool measure the angle between the two lines. Using the Numeric Edit Tool, change the number to 125 degrees. What happens to the measure of the angle?

g. Change the number back to 25 degrees. Grab the point at the 4 cm mark on the axis (the end point of the original segment) and move the first segment to the 4 cm mark on the y-axis. What happened to the measure of the angle?

h. Now change the number to 80 degrees. What happened to the measure of the angle and where is the rotated line relative to the original line?

i. How would you define the rotation of a line that is rotated 120 degrees about a point?

2. Rotating a polygon about a vertex.

a. Clear the screen (Select All from the Edit Menu and press the delete key). Put a point in the middle of the screen using the Point Tool (2nd button). Using the Triangle Tool, create a triangle with one vertex at the point you created and the other vertices wherever you like. Keep the triangle fairly small. Label each vertex (A, B, and C) using the Label Tool (10th button). Color the perimeter of this triangle green.

b. Using the Numeric Edit Tool create a 30 degree number and put it below the palette. Then, using the Rotate Tool rotate the triangle 30 degrees about vertex A. Color the perimeter of this triangle purple and label the two new vertices D and E where D is the vertex above the AB side of the green triangle and E is the vertex above the AC side of the green triangle.

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Using the Angle Tool measure and label ÐDABand ÐEAC . What are the measures of these angles?

c. Using the Numeric Edit Tool change the degree number under the palette to 60 degrees. What happened to the two triangles?

d. Now change the degree number to 0 degrees. What happened?

e. Using the scroll bar on the degree number, change the number up and down from zero. Describe what happens as the number becomes more positive.

f. Describe what happens when the number becomes less positive or becomes negative.

g. How would you characterize the rotation of a triangle 80 degrees about a vertex?

Extension: Repeat steps a though h for a polygon of your choosing (use the Polygon Tool). How would you characterize the rotation of any polygon 130 degrees about a vertex?

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Lab II: Creating Dragon Fractals

Setup: Select the Pointer Tool and using the option buttons color to dark green and the point size to the largest solid value. Select the Segment Tool and using the option buttons set the color to purple and the line size to the smallest value. Select the Polygon Tool and using the option buttons set the color to light green and the line size to the middle value. Show the Axes Define the Grid for the Axes

1. Create a Dragon Fractal

a. The First Seed: creating the starting position.

- Construct three points in an “L-shape” on a small grid square in the bottom left quadrant of the grid. Label the points 1, S, and 2 as in the picture at the right.

- Construct segments connecting 1 and S, and 2 and S.

This is the first seed.

b. The Second Seed: the first iteration.

- Construct a polygon connecting points 1, S, and 2 using the Polygon Tool. - Rotate the polygon 270 degrees about point 2. - Delete the original polygon (The line segments in purple will become visible.) - Draw line segments connecting the points on the new polygon except for the diagonal line. - Delete the new polygon and label the point at the end of the new figure, 3.

This is the second seed and it should look like the picture at the right.

c. The Third Seed: the second iteration.

- Using the Polygon Tool, start a polygon someplace not on the figure (for example, somewhere above points 1 and 3, Point P in the figure at the right) and then construct a polygon joining all of the points of the second seed and the first point. - Rotate the polygon 270 degrees about point 3. - Delete the original polygon and the segment lines of the second seed should be visible.

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- Construct segments joining the points of the new polygon except for the lines to the new point (i.e., the diagonal lines.) - Delete the new polygon. - Label the last point of the figure, 4.

- This is the Third Seed and it should look like the figure at the right.

d. The Fourth Seed: the fifth iteration.

- Using the Polygon Tool, start at a point not on the figure and construct a polygon joining all of the points of the third seed and the new point. - Rotate the polygon 270 degrees about point 4. - Delete the original polygon and the segment lines of the third seed should be visible. - Construct segments joining the points of the new polygon except for the lines to the new point (i.e., the diagonal lines). - Delete the new polygon. - Label the last point of the figure, 5.

- This is the Fourth Seed and it should look like the figure at the right.

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Extensions:

1. Create a Fifth Seed and a Sixth Seed. Do your Results look like these?

Fifth Seed

Sixth Seed:

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2. This is a really fun extension and helps demonstrate your ability to create and modify fractal designs. This extension will help answer some of the following questions.

- What would happen if the lengths of the segments of the starting seed were not the same? - What would happen if the angle between the segments of the starting seed was not a 90-degree angle? - What would happen if the angle of rotation was not 270 degrees? - What would happen in the starting seed had more than two lines? - What would happen if the starting seed had a curved line? - Setup - Close your current work (save if you like) and open a completely new work sheet. - Select the Pointer Tool and using the option buttons color to dark green and the point size to the largest solid value. - Select the Segment Tool and using the option buttons set the color to purple and the line size to the smallest value - Select the Polygon Tool and using the option buttons set the color to light green and the line size to the middle value. - Create a degree measure of 270 degrees using the Numeric Edit Tool. - Do not show the axes or the grid this time.

a. As before, create a starting seed by constructing and labeling 3 points. (Label them as before). Then construct the two line segments. Using the Pointer Tool try to arrange the line segments so that they look like the starting seed from the previous lab. b. Construct Seeds up through Seed Six.

1. Using the Numeric Edit Tool, edit the 270-degree number by using the scroll buttons.

What happens when the angle of rotation decreases?

What happens when the angle of rotation increases?

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2. Using the Pointer Tool, change the length of the line segment joining points 1 and S. What happens?

3. Using the Pointer Tool, change the length of the line segment joining points 2 and S. What happens?

4. Using the Pointer Tool, change the angle joining the two initial line segments by moving point S. What happens?

5. How would you define a dragon fractal based on what happened when you adjusted all these starting conditions?

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