Give the students the student
packet so they can follow along.
Chaos Theory - “Chaos theory is the qualitative study of unstable aperiodic behavior in deterministic nonlinear dynamical systems” [1].
Fractal - A rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced/size copy of the whole.
Iteration - A computational procedure in which a cycle of operations is repeated, often to approximate the desired result more closely.
What’s the point?
Chaos Theory is a quite complex concept in Mathematics that can be simplified all the way down to: In the real world where things change all the time, sometimes some of these things are just random. The first step toward actually looking at Chaos Theory is an understanding of fractals. Fractals lend themselves very well to illustration on computers. We are going to use the process called iteration to let the computer draw fractals for us. The image of the fern that you see to the right is a classic example of an iterated fractal that is based in Chaos Theory.
Objectives
· The student will be able to use Geometer’s Sketchpad to generate well-known fractals.
· The student will be able to demonstrate basic construction in Geometer’s Sketchpad.
· The student will be able to use Geometer’s Sketchpad to create their own personalized fractal.
[1] http://www.duke.edu/~mjd/chaos/chaos.html
[2] Bourke, P. (1991). An Introduction to Fractals. http://astronomy.swin.edu.au/~pbourke/fractals/fracintro/
[3] Dictionary.com. http://dictionary.reference.com/search?q=iteration
If you have any problems with this section or any other section, feel free to ask the moderator of the group or the person sitting next to you. Be sure to have fun!
1. Open the “Start” menu in Windows.
2. Go to “Programs”
3. Select the program named “GSP” with the this icon:
Constructing Koch’s Curve uses a basic series of four steps. These four steps are to construct a line segment that is divided evenly into three pieces; construct an equilateral triangle resting on top of the center segment on your line; hide the unnecessary pieces; and use the iteration tool to produce the curve. We will step you through the entire process. Remember your geometry details.
1. Select the line segment drawing tool in the left bar.
2. Draw a line segment on your screen. Place it horizontally on your screen, only to make directions easier to follow.
3. Select the text tool in the left menu (it is the button with the capital A).
4. Double click each of the end points of your line segment and make sure they are named A and B.
5. Switch back to the arrow tool in the left menu.
6. Note: You can unselect everything by clicking in the white background while using the arrow tool.
7. Select the line segment AB.
8. From the “Construct” menu, choose “Point on Object”. Do this twice to create two points on your line segment. You can drag these points to any location that you like. This gives you three segments. If all three segments are about the same size, you will create Koch’s Curve.
9. Use the text tool to label these two points, C and D.
10. Select the arrow tool in the left menu again.
11. Select the line segment AB.
12. Go to the “Display” menu and choose “Hide Segment”. This will cause the segment to vanish from view, but not delete the points constructed on it.
13. Now, choose each pair of points in the sketchpad from left to right and go to the “Construct” menu and choose “Segment”.
14. Next, select all three segments and nothing else.
15. Go to the “Measure” menu and choose “Length”. Three lengths should appear on your screen. (Warning: They were already selected when you created them.)
16. You can now move the points around on your line segement and try to get all of your distances to be about the same.
We are going to use circles to create our triangle. Think about why this works and be ready to talk about it after we get through the construction process.
1. Select points C and D, in this order and the line segment CD.
2. Go to the “Construct” menu and choose “Circles by center + radius”.
3. With the two circles still highlighted, go to the “Construct” menu and choose “Intersections”.
4. Click somewhere in the white background to unselect everything.
5. Use the label tool, to label one of the new points created as E.
6. Switch back to the arrow tool and select the points C, D, and E.
7. Go to the “Construct” menu again and choose “Segments”.
Triangle CDE is an equilateral triangle. Why?
What characteristic(s) of circles ensured that we created an equilateral triangle?
For the actual creation of the fractal, we only need the four points that we currently have labeled and the line segments AC, CE, ED, and DB. We will hide everything else from view.
1. Select any piece or pieces that you want to hide.
2. Under the “Display” menu, choose the option that says “Hide”. It will have another word besides hide that tries to describe what you are hiding. For example, it could say “Hide Objects”, “Hide Circles”, “Hide Points”, or “High Segments”.
3. Now, make everything vanish except for those things that we said we needed.
Now we get to magically make our first fractal appear using the iteration tool.
1. Using the arrow tool, select points A and B.
2. Go to the “Transform” menu and select “Iterate…” from the menu.
3. In the box that comes up, we will now map the points A and B onto other points. This is telling Geometer’s Sketchpad how we want it to iterate our image. You will notice that the question mark next to the A and arrow is currently selected. This means that we are choosing what to map the point A to in our graph. Let’s go ahead and map it to A. Select the point A in your main window. The letter A will pop into the box and the next question mark will be highlighted. Let’s go ahead and map the B to our point C. Select the point C in your main window. This is telling Geometer’s Sketchpad that everything we did to the line segment AB, we want it to do to the line segment AC. Don’t close this window yet!
4. We are going to add more mappings. It would be cool to do the iteration and mapping to every piece of our picture. Select the drop down menu that says “Structure” and choose “Add New Map”. Now select points C and E so that we map what we did on AB to the segment CE.
5. Add two more maps and map in the other points. Map AB to ED and to DB.
6. Select the drop down menu that says “Display” and choose “Final Iteration Only”.
7. Now, press the button that says “Iterate”.
8. You can grab the table of values that is created and move it out of the way for now. Enjoy your first fractal.
What does this fractal look like? Describe it in words and try to draw it here.
Sierpiński’s Triangle begins with a simple triangle. It then constructs the midpoints of each segment of the triangle and creates another triangle inside of the first by connecting all the midpoints. It doesn’t look or sound all that interesting at first, but when you apply the iteration a number of times, what comes out is pretty neat.
1. The easiest way to do this is to use the point tool from the left menu and place three points onto your sketch.
2. Using the arrow tool, select the three points in order (clockwise or counterclockswise).
3. Go to the “Construct” menu and choose “Segments”.
4. Use the text tool to label your three points A, B, and C.
1. Select the three line segments that make up the triangle. Be sure not to have any points selected.
2. Go to the “Construct” menu and choose “Midpoints”.
3. Use the text tool to label your three midpoints D, E, and F. Label the points this way: D is the midpoint of _{ } . E is the midpoint of _{ } . F is the midpoint of _{ }.
1. Select the points A, B, and C.
2. Go to the “Transform” menu and choose “Iterate”.
3. This time we are going to use three mappings. Input the mappings like this:
A => |
A |
E |
D |
B => |
E |
B |
F |
C => |
D |
F |
C |
4. This time we are not going to use the final iteration feature. We are going to watch Sierpiński’s Triangle form right before our eyes. Make sure that you can see your triangle and press the “+” key on your keyboard. This will cause a single iteration to be shown to you. Pressing the “+” key again will add another iteration and pressing the “-“ key will take away an iteration. Watch what happens as you add iterations.
5. When you have a picture that you like, press the “Iterate” button.
What can you tell about the area of the new triangles being created? How does it relate to the area of the previous triangle?
The following table compares the number of iterations to the number of triangles pointing down:
Iteration |
Number of Triangles Pointing Down |
0 |
0 |
1 |
1 |
2 |
4 |
3 |
13 |
4 |
40 |
5 |
121 |
6 |
364 |
Do you notice a pattern? Is there any other mathematical triangle that looks similar? Try looking at just the odd numbers listed in Pascal’s Triangle.
Based on the above table, create a formula for determining the number of triangles pointing down per iteration.
This fractal begins with a square and a triangle “roof”. From there we will use iteration to create the fractal.
1. Use the Point tool to play two points on your sketchpad and then create a line segment. This line segment will be one edge of your square.
2. Select the line segment and a point.
3. In the “Construct” menu, choose “Perpendicular Line”.
4. Repeat this process with the other point.
5. Select the line segment and one endpoint again.
6. In the “Construct” menu, choose “Circle by center + radius”.
7. Select the circle and the line that intersects it.
8. In the “Construct” menu, choose “Intersections”.
9. Select your original line segment and one of your newly created intersection points.
10. In the “Construct” menu, choose “Parallel Line”.
11. The line that you just created should intersect the other line that you haven’t used yet. Select both lines and construct a point at their intersection.
12. You now have four points that represent the vertices of your square. We need to clear everything else away. Select everything except the four points and “Hide”them all.
13. After you have only the four points remaining, select them in order (clockwise or counter clockwise) around your square.
14. Construct line segments to connect the points.
15. Grab a vertex of your square and rotate it so it has one edge running horizontally on the page. This will make it easier to follow the steps below.
Can you think of another way to construct a square? Explain your idea here!
If you picture a very simple house with a square for the walls of the house and a little triangle right below the roof, that is what we are creating.
1. Select the “top” line segment and construct a midpoint.
2. Select the midpoint that you just constructed and one of the “top” corner points.
3. Use the “Construct” menu to build a circle.
4. Select the midpoint again and the “top” line segment.
5. Use the “Construct” menu to build a perpendicular line.
6. Construct a point at the intersection of the perpendicular line and your circle “above” your house.
7. Use the “Construct” menu to add the two more line segments needed to complete the triangle “roof”.
8. Select the circle, the line, and the two points that are not needed and hide them.
Through the magic of fractals, we will now transform your house into a blossoming tree.
1. Using the text tool to label the points on your house as follows:
A. Bottom left point
B. Bottom right point
C. Upper left point
D. Upper right point
E. Top-most point
2. Select points A and B and open up the iteration window.
3. Map A => C and B=> E.
4. Add another map and map A =>E and B =>D.
5. Use the “+” key on the keyboard to add more iterations and watch your tree grow.
What kind of tree does this look like?
Try and use geometric construction to give the make the base of the tree taller. Describe what you are doing here.
Now it’s time for you to experiment. Make sure you start with only points and use construction techniques to build up the images that you want to make fractals out of. Describe your ideas here.
Once you have your idea down and created, draw what you think your fractal will look like. Then, create your fractal and compare the two.