The Many Dimensions of Dimensions II

Now that we know from the previous posting why the usual coordinate based definition of dimension is not always valid, we need to explore the alternative.

One possibility is to consider the scaling of length/area/volume. For example, consider first a line that is one unit in size. How do I double its size - by gluing a second unit length line to one end. So a factor of two increase in size requires two units. To increase by a factor of three takes three unit lines. We define all objects with this property to be one-dimensional.

What about a square? Start with a 1x1 square, and then to create a 2x2 square requires four of the 1x1 squares, or 2^2 squares(note that here and in all posts, 2^n represents 2 multiplied by itself n-times). A 3x3 square requires 9 or 3^2 squares. So we define this to be 2-dimensional.

You can do this for any simple objects (with the obvious requirement that you can tile the small pieces to make larger copies) and define dimension to be the log n/log m where an m-fold increase in the size of the object requires m^n tiles (or in non-mathematical terms, if an m-fold increase in size requires m^n tiles, the space is n-dimensional)

And now the fun begins!

Consider the Koch snowflake. It doesn't tile in the right way to define a dimension, but try taking 1/3 of the snowflake - or one side of it. If you make four copies, and line them up with the center two meeting at an angle as shown, you would triple the size of the object. So the dimension must satisfy 4 = 3^n, or n = 1.262 ! So here we have an object with a well-defined dimension which is not a whole number!

Consider the Siepinski triangle. and the Sierpinski carpet. In the first case three copies are needed to double the size, so n = 1.585. In the second case eight copies triple the size, so n = 1.893.

Which leads one to ask - with all of the recent discussion of extra dimensions in physics, could some of them have non-integer dimensions? (Before anyone gets too excited about this possibility, remember that such spaces either have (1) no easy way of defining derivatives and therefore no particle momenta or (2) require an infinite number of singularities in spacetime itself )