Solutions:
1. I have a mystery shape that is self similar. I have a specific smaller similar copy of my mystery shape where the enlargement scale factor (the scale factor to make the larger copy from the smaller one) is 5.
A. If the mystery shape was 1-D, how many copies of the smaller shape would you need to build the larger shape?
51=5
B. If the mystery shape was 2-D, how many copies of the smaller shape would you need to build the larger shape?
52=25
C. If the mystery shape was 3-D, how many copies of the smaller shape would you need to build the larger shape?
53=125
2. I have a mystery shape that is self similar. I have a specific smaller similar copy of my mystery shape where the enlargement scale factor (the scale factor to make the larger copy from the smaller one) is 5. If I need 25 copies of the smaller shape to build the larger shape, what dimension is my shape?
2-dimensional because 52=25
3. I have a mystery shape that is self similar. I have a specific smaller similar copy of my mystery shape where the enlargement scale factor (the scale factor to make the larger copy from the smaller one) is 5. If I need 15 copies of the smaller shape to build the larger shape, what dimension is my shape?
A dimension in between 1 and 2 because 15 is in between 5 and 25.
You can get 1 point extra credit if you can use the formula on page 509 to write down exactly what the dimension is.
5. In this Koch curve, circle two pieces that are the same shape as the whole curve, but at a different magnification. For each of your pieces, tell how much (length scale factor) you would need to magnify them to get the whole thing, and how many of them it takes to make the whole thing:
Many choices for solutions. Here are two:
 |
If I magnify this x3, I should get the whole thing (the enlargement scale factor to change small -> large is 3). It takes four pieces this size to make the whole |
 |
If I rotate this, and magnify it x9 it will be the same as the whole thing (magnifying x3 would give me a piece the same size as in A); the enlargement scale factor is 9. It takes 16 pieces this size to get the whole thing |
5A. For the circled similar part of this Sierpinski triangle, tell the scale factor needed to enlarge it to the whole triangle, and the number of copies needed to create the whole triangle.
The scale factor is 8, and the number of copies is 27:
The total length is as long as 8 widths of this sized triangles, so the enlargement scale factor is 8. |
There are 3 this size in the red circle (1, 2, 3). There are 3 red sized in a purple/pink, so that makes 9 so far (3+3+3). Then there are 3 of those to make a whole: 3x9=27 (9+9+9). |
B. Compare the number of copies for this scale factor to the copies needed for a 1D and 2D shape. What does this tell about the fractal dimension of the Sierpinski triangle?
A 1D shape with scale factor 8 would only need 8 copies (less than 27), and a 2D shape with scale factor 8 would need 8x8=64 copies (more than 27) so the Sierpinski triangle has a dimension between 1 and 2.
6A. For the smaller, similar part circled in this fractal, tell the scale factor to enlarge it to the larger whole curve, and the number of copies needed to create the larger whole curve.
The enlargement scale factor is 4
B. Compare the number of copies for this scale factor to the copies needed for a 1D and 2D shape. What does this tell about the fractal dimension of this fractal?
A 1D shape with scale factor 4 would only need 4 copies (less than 9), and a 2D shape with scale factor 4 would need 4x4=16 copies (more than 9) so the fractal has a dimension between 1 and 2.
7. A. The biggest, and easiest similar shape to circle would be this one:
The scale factor for this one would be 2:
And the number of copies needed would be 3:
B. with a scale factor of 2, if it was 1D, you would only need 2 copies, and if it was 2D, you would need 4 copies, so the dimension of this fractal is between dimensions 1 and 2.
(more about fractal dimension here: not required for the Math 126 quiz, geeks only)
