| Now we introduce ways to quantify fractals. |
| We shall see that measuring a shape in the wrong dimension gives results
that at first seem peculiar. |
| Measuring in a dimension too low gives infinity: a filled-in
square has infinite length. |
| Measuring in a dimension too high gives zero: a filled-in
square has zero volume. |
| So what are we to make of the calculation, to be performed in the first
section below, that the Koch curve has infinite length (1-dimensional measure) and
zero area (2-dimensional measure)? |
| Does this mean the dimension of the Koch curve lies between 1 and 2?
How is this possible? What could it mean? |
| We motivate our study of dimensions, we look carefully at some peculiar aspects of
measurement. |
 |
| Box-counting dimension extends
the notion of dimension
to fractals. Arguing by analogy with Euclidean dimension, we develop an algorithm for
determining this dimension. |
 |
|
| Homework 3 |
| Practice homework |
| Homework 2 Solutions |