Fractal Fundamentals
Fractal-A complex geometric pattern exhibiting self-similarity in that small details of its
structure viewed at any scale repeat elements of the overall pattern.
Properties of Fractals
1. Self Similarity
Shapes that are similar have the same shape and the corresponding sides are in
proportion.
For example:
Similar Squares
SImilar Rectangles
Shapes that are self similar are like copies of similar shapes that form a bigger shape
that is similar to the smaller copies.
Examples
Questions on Self-Similarity
Question 1: If the red image is the original figure, how many
similar copies of it are contained in the blue figure?
Question 2: Are squares self-similar? (Can you form bigger squares out of smaller
ones?) Are hexagons? (Can you form larger hexagons out of smaller ones?) Draw
examples to justify your answer.
Question 3:Are circles similar? Are they self-similar?(Can you form larger circles out of
smaller ones? Draw examples to justify your answer.
2.
Question 4: Experiment with designing another self-similar figure. Fractal
DImension
A point has no dimensions - no length, no width, no height.
.a
A line has one dimension - length. It has no width and no height, but infinite length.
l
A plane has two dimensions - length and width, no depth.
Sp
ac
e,
thin
k of
it
as
a
hu
ge
It's an absolutely flat tabletop extending out both ways to infinity.
em
pty
box
,
has
1 unit
2 units dimensions, length, width, and
three
depth,
extending to infinity in all three
P
directions.
1unit x 1 unit
3. Fractal Dimension Investigation
In order to determine fractal dimension lets consider dimension in general are how it
relates to self similarity.
Take a self-similar figure like a line segment, and double its length. Doubling the length
gives two copies of the original segment.
1unit x 1 unit x 1unit
Take another self-similar figure, this time a square 1 unit by 1 unit. Now multiply
the length and width by 2. How many copies of the original size square do you
get? Draw the self similar square.
Take a 1 by 1 by 1 cube and double its length, width, and height. How many
copies of the original size cube do you get? Draw the self similar cube
Now consider the Sirpinski Triangle. If we double the length of the sides how many
copies of the original triangle do we have.
Finish filling in the table below
Figure
Dimension
Number of Copies
Line Segment
1
2=21
Square
2
Cube
3
Sierpinski Triangle
d=
What do you notice about the dimension of the fractal?
Is there anything special about us doubling the lengths here? Could you have tripled
them and derived the formula as well?
Challenge Question: Consider the Koch Fractal whose length reduces by 1/3 can you
figure out the dimension of this fractal? (Hint: start with investigating the number of selfsimilar copies)
The Sierpinski Triangle
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 operate on that figure in the same way and get an even more complicated
figure. Then do it again and again...and again.
Let's make a famous fractal called the Sierpinski Triangle. You will need the triangular
graph paper at the end of this packet.
Step One
Draw an equilateral triangle with sides of 2 triangle units each. Connect the midpoints of
each side.
How many equilateral triangles do you now have? Shade out the triangle in the center.
Step Two
Draw an equilateral triangle with sides of 4 triangle units each. Connect the midpoints of
the sides and shade the triangle in the center as before. Notice the three small triangles
that also need to be shaded out in each of the three triangles on each corner.
Step Three
Draw an equilateral triangle with sides of 8 triangle units each. Follow the same
procedure as before, making sure to follow the shading pattern. You will have 1 large, 3
medium, and 9 small triangles shaded.
Step Four
How about doing this one on a poster board? Follow the above pattern and complete
the Sierpinski Triangle. Use your artistic creativity and shade the triangles in interesting
color patterns. Does your figure look like this one? Then you are correct!
Invent a Fractal
Use the square or pentagonal graph paper in this packet. Invent a fractal based on
either squares or
pentagons.