Project: fractals
Koch Snowflake / sierpinski’s triangle / cantor dust
Fractals are often formed by what is called an iterative process. Here's what I mean.
To make a fractal: Take a familiar geometric figure (a triangle or line segment, for example) and
operate on it so that the new figure is more "complicated" in a special way. Then in the same way,
operate on that resulting figure, and get an even more complicated figure.
Now operate on that resulting figure in the same way and get an even more complicated figure. Do it
again and again...and again. In fact, you have to think of doing it infinitely many times.
Koch curve
or
Koch Snowflake
Assignment #1: Draw a Koch Snowflake
Step One.
Use the triangular grid paper provided. Start with a large equilateral triangle with sides 9 units
in length. See the attached example at the end.
Step Two.
Make a Star.
1. Divide one side of the triangle into three equal parts and remove the middle section.
2. Replace it with two lines the same length as the section you removed.
3. Do this to all three sides of the triangle.
Do it again and again.
To view an animation of Koch Curve iteration or inflating Koch Snowflake, go to the following
internet sites:
o http://www.miqel.com/fractals_math_patterns/visual-math-iterative-fractals.html
o http://demonstrations.wolfram.com/InflatingAKochSnowflake/
You need to click on the little + in the upper right hand corner in order to turn on the Autorun.
o http://fractalanimation.com/
Iteration 0
Iteration 1
Iteration 2
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Iteration 3
Sierpinski’s triangle
The Opposite of the KOCH CURVE is the SIERPINSKI TRIANGLE (or gasket) produced by
subtracting triangles from the interior, instead of adding them to the surface as in the Koch curve.
Iteration 0
Iteration 3
Iteration 1
Iteration 4
Iteration 2
Iteration 5
Fig. 1: Steps in creating Sierpinski's Triangle
Assignment #2: Draw Sierpinski’s Triangle
Probably, the simplest fractal is the Sierpinski's Triangle, as seen above.
The way Sierpinski created this triangle is quite simple.
Step One.
Begin with the (sort of) equilateral triangle sheet provided. Use the image above and below to help (fig. 1,
Iteration 1; fig. 2, first image). It doesn't necessarily have to be equilateral but for our purposes we will use
one.
Step Two.
Connect the midpoints of each side to form four separate triangles (fig. 1, Iteration 1; fig. 2, second
image).
Step Three.
For each of the three exterior triangles, perform this same act. You only need to take your diagram to the
5th iteration (see above).
Fig. 2: Steps in creating Sierpinski's Triangle
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cantor dust
Also known as the Cantor set, possibly the first pure fractal ever found – by Georg Cantor around
1872. To produce Cantor Dust, start with a line segment, divide it in to three equal smaller
segments, take out the middle one, and repeat this process indefinitely.
Although Cantor Dust is riddled with infinitely many gaps, it still contains unaccountably many
points. It has a fractal dimension of log 2/log 3, or approximately 0.631.
http://www.daviddarling.info/encyclopedia/C/Cantor_dust.html
The Cantor Set
Consider a line segment of unit length. Remove its middle third. Now remove the middle thirds from the
remaining two segments. Now remove the middle thirds from the remaining four segments. Now remove the
middle thirds from the remaining eight segments. Now remove ... well, you get the idea. If you could continue
this construction through infinitely many steps, what would you have left?
What remains after infinitely many steps is a remarkable subset of the real numbers called the Cantor set, or
“Cantor’s Dust.”
At first glance one may reasonably wonder if there is anything left. After all, the lengths of the intervals we
removed all add up to 1, exactly the length of the segment we started with:
Yet, remarkably, we can show that there are just as many “points” remaining as there were before we began!
This startling fact is only one of the many surprising properties exhibited by the Cantor set.
http://www.mathacademy.com/pr/prime/articles/cantset/
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Assignment #3: Project: Self-Similarity via the Cantor Set
See the attached worksheet.
Follow the instructions numbered 1 through 5 on page 119 of the worksheet. Record your information on a
sheet of notebook paper turned sideways (landscape view). Your answers should look something like the
diagram and table shown below.
Project: Self-Similarity
Worksheet Chapter 8, pg.119
I (0)
I (1)
I (2)
I (3)
I (4)
I (5)
Iteration
0
1
2
3
4
5
n
Length of each
segment
1
1/3
1/9
No. of segments
in row
1
x
=
2
Total length
shown in row
1
2/3
Assignment #4: Project: Self-Similarity worksheet
Creating a Fractal Card
Follow the instructions on the bottom half of the same worksheet, page 119, steps 1-6. You will need
a sheet of notebook paper to begin with. Once you are satisfied with your creation, transfer your design
onto an index card to create a fractal pop-up card.
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Sierpinski meets Pascal
Whose triangle is this, anyway?
Have you ever seen the triangular pattern of numbers named after the famous French
mathematician Blaise Pascal? Do you see the pattern?
Assignment #5: Fill in Pascal’s Triangles
Step One.
Begin with the (sort of) equilateral triangle grid sheet provided.
Step Two.
Start with an equilateral triangle with sides 8 units in length.
Label each ROW of your triangle beginning at the top with ROW 0.
Fill in each triangle THAT POINTS UP with the appropriate numbers beginning with the
top triangle as 1. (HINT: Two consecutive triangles add together to make the triangle beneath.)
Step Three.
Color all BLANK triangles and all EVEN NUMBERED triangles.
DO NOT COVER NUMBERS!!
You are shading all the little triangles in Pascal's
Triangle except the odd numbered ones. What do you see?
Step Four.
Write out an explanation of how Pascal’s Triangle works. If you need help, you may go to the website:
All you ever wanted to know about Pascal’s Triangle. (http://ptri1.tripod.com/)
Check your answers at http://math.rice.edu/~lanius/fractals/pasc.html
PASCAL EXPLAINED:
Prime Numbers
If the 1 element in a row is a prime number, all the numbers in that
row are divisible by it
st
If the 1 element in a row is a prime number (remember, the 0th element of every row is 1), all the
numbers in that row (excluding the 1's) are divisible by it. For example, in row 7 (1 7 21 35 35 21
7 1) 7, 21, and 35 are all divisible by 7.
st
The Sums of the Rows
The sum of the numbers in any row = 2n
The sum of the numbers in any row is equal to 2 to the nth power or 2n, when n is the number of
the row. For example:
20 = 1
21 = 1+1 = 2
22 = 1+2+1 = 4
23 = 1+3+3+1 = 8
24 = 1+4+6+4+1 = 16
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Now, what does Pascal’s Triangle have to do with Sierpinski's Triangle?
Try shading all the little triangles in Pascal's Triangle except the odd numbered ones, and see what
happens. (That includes shading the triangles with no numbers and the even numbered ones.)
Now compare it to the triangle below. (Hidden way below so you won't cheat!)You can get
Sierpinski's Triangle from Pascal's!
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Sierpinski Meets Pascal
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