**Chaotic Crystals**

**Problem**:

How can a large number of independent unpredictable events, combine together to yield a predictable group behavior?

How can apparently identical initial conditions for a system yield greatly different results?

**Materials:**

50 ml of CuSO4 solution, petri dish, plastic petri dish

thin copper wire(18 gauge or smaller) (2 pieces: one long piece and one short piece)

5-10 V power supply

Each lab group will be given a specific concentration of copper sulfate solution and a gauge of bare copper wire from which you are to make the copper loop anode.

**Procedure:**

1. Bend the long copper wire into a loop and place it inside the large petri dish. Twist the ends of the loop together so that they stick out above the top of the petri dish.

2. Pour approximately 5 ml of CuSO4 into the glass petri dish.

3. Set the plastic petri dish down into the glass one, open side up, so that it pinches the solution on the wire between them. Make sure that there are no air bubbles trapped between the dishes, if necessary add more solution.

4. Fasten the dishes together with scotch tape. Do not tape under the glass petri dish.

5. Attach the positive lead from the voltage source to the exposed twist in the looped copper wire. Attach the negative lead from the voltage source to the copper wire in the center. See the diagram below:

6. Set the power supply to a voltage as directed by your instructor and then turn it on.

7. Grow crystals until 50% of the petri dish is covered or the growth stops. Observe the characteristics of the crystal growth i.e.: general shape, rate of growth, time for crystal growth.

8. Turn off the power supply and disconnect the electrodes. Remove the cathode.

9. Gently carry your crystal and photocopy the petri dish crystal sandwich.

10. Place the number of your crystal on the photocopy with the data describing concentration of CuSO4, wire gauge, and growth time.

**Summing Up:**

Use class results to answer the following questions:

1. How did the radius change as the crystal grew?

2. In what ways are the crystals alike?

3. In what ways are the crystals different?

4. Examine crystals that were grown under identical conditions. Why do you think that the crystals are not identical?

5. How did modifications in the procedures cause differences?

6. What characteristics do all of the crystals share?

**Teacher's Notes-**

Preparation time for this lab is fairly lengthy. Once the dishes and electrodes are prepared they can be used repeatedly.

Set up specific combinations for concentration, anode gauge, and voltage. This will allow enough class data so that individual variables can be examined, i.e.:

Same concentration of CuSO4 (1 molar); Same voltage; 18, 20, 22 gauge wire.

Same concentration of CuSO4 (1 molar); Same gauge wire; Voltage 2,4,6 volts

0.5 molar, 1 molar, 2 molar CuSO4; Same gauge wire; Same voltage

As an exploratory or prelab activity, you may show crystal growth in a petri dish on an overhead projector.

For alternate concept development and application activities for fractals use any activities from The Fractals in the classroom using Sierpinski Triangles M 114 through M 131.

M114

Fractals in the Classroom

**Sierpinski Triangle and Variations**

This construction process, when repeated over and over, generates a well known fractal image called the Sierpinski triangle (or gasket).

*Construction : *Connect midpoints on the sides as shown, keeping
only the three corner subtriangles formed.

Apply the construction process on the newly formed corner subtriangles through a second, third, and fourth stage. Remember, at every stage, each remaining triangle is transformed into three new subtriangles with sides half as long. Three times as many triangles appear at each successive stage.

Count dots carefully. Every vertex of every subtriangle at each of the
four stages is on a dot on the grid paper. The resulting figure should contain
81 small triangles, representing the fourth stage in the construction of
the *Sierpinski triangle. *Shade in these triangles.

1. Imagine repeating the process. Visualize and describe how the figure changes. If the process continues on without end, a Sierpinski triangle emerges.

2. What would remain of the original large triangle after four iterations if the algorithm were changed to keeping only the inner triangle?

M115

**Triangle Variation**

When repeated over and over, this construct variation generates another fractal.

*Construction *: Connect trisection points on the sides as shown,
keeping only the six border subtriangles.

In this variation, the sides of the triangle are divided into thirds. Repeat the process through a second iteration using exactly the same procedure in each of the six border subtriangles shown in this first stage. Count dots carefully. Each vertex of each of the 36 congruent subtriangles that emerge at the second stage are on dots of the grid paper. Shade in these triangles.

1. Imagine repeating the process over and over. At each stage, each triangle is transformed into six new subtriangles with sides one-third as long. Describe what you would see of the original triangle if the process were continued on without end.

2. Change the algorithm from keeping the six border subtriangles to keeping the three inner ones. What kind of figure would emerge after two iterations?

M116

**NUMBER PATTERNS WITH VARIATIONS**

This activity explores some of the number patterns found in the Sierpinski triangle.

DIRECTIONS: The first four stages of the construction of the Sierpinski triangle are shown below. In subsequent stages, the subdivision continues into smaller and smaller triangles. Use these figures to explore number patterns that emerge as the Sierpinski triangle is developed through successive iterations.

NUMBER OF TRIANGLES

1. Count the number of shaded triangles at each stage 0 through 4.

STAGE .....0...1...2...3...4

NUMBER .1

2. Extend the pattern to predict the number of triangles at stage 4. What constant multiplier can be used to go from one stage to the next?

3. Generalize to find the number of triangles for level *n*.. As
*n *becomes large without bound, what happens to the number of triangles?

AREA OF TRIANGLES

4. Let the area at stage 0 be 1. Find the total shaded areas at stages 1 through 4.

STAGE ...0... 1... 2... 3... 4... 5 ... . . . *n*

AREA .....1

5. Extend the pattern to predict the total area at stage 5. What constant multiplier can be used to go from one stage to the next?

6. Generalize to find the total area at stage *n*.. As *n .*becomes
large without bound, what happens to the shaded area?

**SQUARE CARPET**

Repeating this construction over and over generates a fractal based on a square.

*Construction : *Connect trisection points on the sides as shown,
keeping only the eight boundary subsquares.

Repeat the process through three successive iterations, the first being shown below. Remember, use only the border subsquares at each stage. The result will be a square carpet made from 512 small subsquares. Shade in these stage-3 squares. There should be 73 square holes of three different sizes in the carpet.

1. Imagine repeating the process over and over. At every stage, each square is transformed into eight new subsquares with sides one-third as long. Describe what you would see of the carpet if the process were to continue without end. How many holes will there be? How much of the original square will remain?

2. Suppose the algorithm were changed from keeping the eight border subsquares to keeping only the four corner subsquares. What figure emerges after two iterations?

**NUMBER PATTERNS IN THE SQUARE CARPET 1.3B**

This activity focuses on number patterns found in generating successive stages leading to the square

carpet.

DIRECTIONS The first three stages of the construction of the square carpet are shown below. In

subsequent stages, the subdivision continues into smaller and smaller subsquares. Use these figures

to explore number patterns that emerge as the square carpet is developed.

NUMBER OF SQUARES

1. Count the number of shaded subsquares at each stage 0 through 3.

STAGE .....0... 1... 2... 3... 4... . . . *n*

NUMBER .1

2. Extend the pattern to predict the number of shaded subsquares at level 4. What constant multiplier can be used to go from one stage to the next?

3. Generalize to find the number of shaded subsquares for level *n*.
As *n *becomes large without bound, what happens to the number of shaded
subsquares?

AREA OF SQUARES

4. Let the area at stage 0 be 1. Find the total shaded area at stages 1 through 3.

STAGE ..0... 1... 2... 3... 4... . . .

nAREA ....1

5. Extend the pattern to predict the shaded area at stage 4. What constant multiplier can be used to go from one stage to the next?

6. Generalize to find the shaded area for stage *n*. As *n *becomes
large without bound, what happens to the shaded area?

7. Imagine successive figures generated by using only the four corner squares at each level. What would be the new answers for questions 1 through 6?

**TREES**

As trees grow, they branch out. From big branches grow smaller ones. From these grow smaller ones still, and so on. Use this dot paper to draw a mathematical tree with some of the same properties as the live ones.

*Construction : .*From the endpoint of each branch, draw two new
branches half as long growing off at 60° in opposite direction.

1. Stage 1 of the tree has already been drawn. Draw the four new branches for stage 2 by connecting endpoints to the appropriate dots on the grid. Draw the eight new branches for stage 3. Repeat again for stage 4. Endpoints should still be on the dots of the grid. Continue the growing process until the branches become too small to draw.

Suppose the tree starts with an initial vertical segment of 1 unit as the trunk. Imagine further that the tree continues growing branches, over and over by the process given, until fully grown. Visualize this completed tree.

2. How many branches have lengths of 1/4? of 1/16? What is the sum of the lengths of all branches 1/4 long? 1/16 long?

3. What is the total length of all branches of the completed tree?

4. Are there parts of the completed tree that look like the entire tree? Using the tree just drawn as a model of a fully grown tree, draw a hexagon around a part that would be an exact image of the tree itself. Draw another using a hexagon of a different size.

One interesting shape found on the completed tree is a *spiral. *Start
at the base of the tree and turn right at each and every junction point.
Note how these particular branches trace out a spiral.

5. Find another spiral that is a reflection of the one just described. What is the length of this spiral?

6. Find four spirals with half the length of the one just described. How many spirals in the tree have one-quarter the length of the original?

7. Consider all such spirals of all sizes that can be found on the tree. They all do not have the same length. Are they all exact replicas of each other except for size?