Tutorial Chapter 4
Intro to Dimension
DIMENSION?...In 300 BC, Euclid began his Book I with several definitions, which included the
following:
1. A point is that which has no part.
2. A line is a breadthless length.
3. A surface is that which has length and breadth only.
In Book XI, he added:
4. A solid is that which has length, breadth, and depth.
The concept of dimension underlines these definitions. Everybody knows that a point has a dimension of 0, a line has a dimension of 1, a square is 2-dimensional, and a cube is 3-dimensional. Such dimension is called topological dimension. It was used for thousands of years, but proved to be inaccurate with the development of fractals.
WHAT’S WRONG WITH IT?
Consider a fractal called the Peano Curve. When generating it, one starts with a line segment and substitutes it with this shape:
Then, every line segment is substituted with the same shape again and the process is continued. By repeating it infinitely, we get a square at the end.
Now, we encounter a problem. The fractal is made up of lines, so the topological dimension is 1. However, it is inaccurate since the shape is a 2-dimensional square.
AND WHAT CAN WE DO ABOUT IT?
We cannot use topological dimension for fractals, but instead have to use what is called fractal dimension. In fact, the formal definition of a fractal is a figure in which the fractal dimension is greater than the topological dimension. Interestingly, the fractal dimension does not have to be an integer, but can be a fraction. As you can see from the table below, the complexity of a figure increases with dimension.You will learn how to calculate it exactly in the next three lessons.
| F | A finite number greater than 0 | ||||
| I | An infinite number | ||||
Dimension |
Num of Points |
Length |
Area |
Volume |
|
| D = 0 | F |
0 |
0 |
0 |
|
| 0 < D < 1 | I |
0 |
0 |
0 |
|
| D = 1 | I |
F |
0 |
0 |
|
| 1 < D < 2 | I |
I |
0 |
0 |
|
| D = 2 | I |
I |
F |
0 |
|
| 2 < D < 3 | I |
I |
I |
0 |
|
| D = 3 | I |
I |
I |
F |
|
|
|
