The meaning of dimension can be confusing. Fractals can for instance exist in a very real sense exist in a higher dimensional space without having the dimension of that space. If a fractal is made by repeated subdivision of a shape into
\[C\]
copies with scaling factor \[S\]
we nade define a fractal dimension by \[D=- \frac{logC}{logS}\]
, but there are examples of real situations where this does not work. No one would argue for example thgat a number line with one number missing does not have dimension 1. or that the universe with p[oints that are mathematically inaccessible to us - e.g. the singularities in black holes - does not thave dimension 4, but that insight leads to this fallacy. A torus has dimension 3, but the are poinys inside 3 dimensional space that cannot be reached via a torus and this goes for any 3 dimensional shape in a 3 dimensional space. More practically we can define dimension in terms of the properties of a set near to each of its points. This would then give each 3 dimensional shape in a 3 dimensional space dimension 3, and similar properties would extend to other shapes in other bspaces. A more rigorous definition results in Hausdorff dimension.