Dimensions
What is meant by the word dimension? You can probably name one-, two- and three-dimensional objects easily. For instance, a cube has three dimensions, length, width and height. But are there objects which do not have a whole number of dimensions? Some of the greatest mathematical discoveries have come about by looking at things from a new perspective. This article gives a different way of thinking about dimensions which enables us to find such strange objects, and we are left with many exciting questions.
Think about a rectangle. If we double the lengths of the sides of the rectangle, how many copies of the smaller rectangle are required to make the larger one?
We need four of them. So doubling the lengths requires four copies. How many copies does trebling the lengths require?
If you multiplied the lengths by x, how many copies of the small rectangle would be required?
We see that for the rectangle, we need x² copies.
What about a brick? If you wanted to make a brick each of whose sides was twice as long as an ordinary brick, how many ordinary bricks would you have to join together?
What about three times the lengths? And xtimes?
For a brick, we require x³ copies of the original brick.
By the usual definition of dimensions, we would say that a rectangle is two-dimensional and that a brick is three-dimensional.
Suppose that in order to double (or multiply by x) the lengths of an object, we required 2n (or xn) copies. Our new definition of dimension says that the object is n -dimensional. Does this agree with the usual definition? Well, the 2-dimensional rectangle needed x² copies, and the 3-dimensional brick needed x³ copies. Check out other shapes. Doubling the length of a line, for instance, requires two copies of the original line. 2¹ = 2 and a line is one dimensional, so the definitions agree there.
Now, equipped with our new idea of dimension, can we find shapes whose dimension is not a whole number? The answer is yes. The snowflake curve is generated by repeating a design, the generator of the fractal, on an ever decreasing scale. The snowflake curve is generated by replacing each line segment by the shape below.
Here, to remind you, are the first three applications of the process – the snowflake curve will by produced if this process is continued indefinitely.
What is the dimension of the snowflake curve?
Notice that at each stage of the process, each line segment is divided into three and the middle piece is replaced by two line segments of the same length. So the length of each line segment is multiplied by 4/3. Knowing this, suppose we wanted to make each line segment of the curve 3 times longer. As the diagram shows, we need four of our original curves to make it.
