I was wondering what dimension can mean in physics?
I know it can mean the dimension of the space and time.
But there is dimensional analysis. How is this dimension related to and different from the previous one? How is it related to and different from units (e.g. kilometer)?
Answer
A dimension (in dimensional analysis) is defined by the transformation law of an object under changes in scale. If I have an object which is twice as big, it has 4 times the surface area and 8 times the volume, so the surface area has dimension of length-squared, and the volume has dimension of length-cubed. Dimensional analysis is applied whenever you have a quantity where there is a scale that you can change.
There are two different notions of dimension of space which took a while to be disentangled. The topological dimension is defined inductively by the cutting properties of the space. If a space can be cut in two by a point, it's 1 dimensional. If it can be cut in two by a 1 dimensional shape, it's two dimensional. This type of definition requires care for wild shapes, but it produces an integer dimension of the space.
The scaling dimension, or fractal dimension, is defined differently, in terms of distances on the space. The scaling dimension counts the number of boxes of size A required to cover the space, and sees how this goes up as A gets small. The exponent is the scaling dimension.
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