Purpose:  The purpose of this lab was to investigate whether the points created by choosing purely random numbers, generated by rolling a die, would lead to any specific pattern.


Procedure:  On the first day of the lab, each student pair received an overhead transparency with an equilateral triangle drawn on it, an overhead marker, a ruler, and a six-sided die.  The vertices (A, B, and C) of the triangle were numbered as follows:  A = 1 or 2, B = 3 or 4, C = 5 or 6.  We chose a random starting point within the triangle and made a dot with the marker on the transparency.  Then we took turns rolling the die.  I rolled first and rolled a three.  Then I took the ruler and found the halfway point between the original point and vertex B and marked a new point.  Next my partner rolled the die and rolled a five.  He took the ruler and found the halfway point between my last point and vertex C and made a new point.  Next it was my turn again.  I rolled a two and used the ruler to mark the halfway point between my partner's last point and vertex A.  We took turns repeating this activity until we had each had 15 turns.  The transparency just looked like a mess of randomly placed dots.  When we looked at the work of the other partners in our class, their work was also just a random mess.  Our teacher put a blank transparency with just the original triangle on the overhead and had each pair place their transparency on top, being careful to match up vertices A, B, and C.  We ran out of time the first day, so when we came to class the next day each group was asked to realign their transparency.  When all eleven groups had placed their transparencies on the overhead, the teacher turned it on and asked us if we saw any pattern.   After each person recorded their impressions of the pattern, we linked graphing calculators to download the "Chaos" program from the computer.  When everyone had a copy of the program on their calculator, we ran the programs, chosing a random starting number.   Within a few minutes, everyone started to see the same pattern on their calculator, except that the patterns on the calculators were much more clear than the transparencies had been.  The teacher had each pair of partners chose and post five differnent starting numbers on the board and then run the program with their starting numbers.  Although the class used over 100 different starting numbers, the ending pattern was always the same.  Weird, huh?

Data:  Data generated on day one was turned in to the teacher on our transparency.  On day two, my partner and I used the starting numbers of 5, 27, 43, 119, and 523.  All of these numbers generated the Sierpinski triangle on the graphing calculator.

Results:  It was kind of hard to tell on from the hand drawn data on day one, but when all of the transparencies were placed on top of one another and lined up, definite clear (empty) and dark spaces appeared with in the triangle.  Most of the empty spaces were sort of triangle shaped and they varied in size.  It was fuzzy but it sort of looked like the Sierpinski triangle fractal we learned to draw at the beginning of this unit.  The "Chaos" program always generated the same result no matter what starting number you used or how many iterations you told it to repeat.  Every time it made the Sierpinski triangle.  

Discussion:  If the layered transparencies with the whole classes hand-drawn data had not shown results similar to the Chaos program on the graphing calculator I probably would have thought the program was rigged.  After all, I don't know anything about programming a calculator or what the program makes the calculator do.  But since the hand drawn data of everyone put together and the program (no matter what starting number anyone chose to use) made Serpinski triangles, I have to conclude that completely random data can produce an ordered pattern and strange attractors really do happen.

Conclusion:  Regardless of the starting position chosen, with enough repetitions finding and marking the midpoints of the random data results in a drawing of the Sierpinski triangle, so random data can form a non-random pattern.