Fractal geometry

Big ideas:

Fractals are often self-similar. This means they are made out of smaller copies of themselves

Fractals are usually infinitely detailed. Line and border fractals have lots of wiggles and can be infinitely long

There are lots of things in nature that are approximately self-similar or have more and more detail if you magnify them

You can find patterns in the way that fractals are made that help you figure out things about them.

On the quiz, you will be asked to draw and figure out the length or area of iterations of a fractal for a new fractal and for one we did in class:

Sample new question:

1. This fractal rule for the fractal Cantor dust is: take out the middle third or each segment in the previous iteration.

A. Draw the next iteration of the fractal

iteration 0
iteration 1
iteration 2

B. Complete the table below, and explain your rule for finding the numbers that go in the table

iteration number	number of pieces	length of each piece	total length
0	1	1	1
1	2	1/3	2/3
2
3

 

Old examples from class:

2. The fractal rule for the Koch curve is to replace each segment with a new shape (shown below) which is formed by erasing the middle third, and putting in two segments that would make an equilateral triangle in its place:

Segment was:	Changed to:

A. Draw the next iteration:

iteration 0
iteration 1
iteration 2

B. Complete the table below, and explain your rule for finding the numbers that go in the table

iteration number	number of pieces	length of each piece	total length
0	1	1	1
1	4	1/3	4/3
2
3

2. The fractal rule for the Sierpinski triangle is to replace each (point up) triangle with a new shape (shown below) which is formed by connecting the midpoints of its sides, and erasing the middle, leaving 3 smaller triangles:

triangle was	Changed to:

A. Draw the next iteration:

iteration 0
iteration 1
iteration 2

B. Complete the table below, and explain your rule for finding the numbers that go in the table

iteration number	number of sements	length of each segment	total length
0	3	1	3
1	9	1/2	9/2
2
3

C. Complete the table below, and explain your rule for finding the numbers that go in the table

iteration number	number of triangles	area of each triangle	total area
0	1	1	1
1	3	1/4	3/4
2
3

2. The fractal rule for the Sierpinski carpet is to replace each square with a new shape (shown below) which is formed by cutting the square into 9 smaller squares at the 1/3 points of the sides, and then removing the center square:

square was	Changed to:

A. Draw the next iteration:

iteration 0
iteration 1

B. Complete the table below, and explain your rule for finding the numbers that go in the table

iteration number	number of squares	area of each square	total area
0	1	1	1
1	8	1/9	8/9
2
3

 

Answers