A complicated technical definition of the
Hausdorff dimension is possible, which applies to a large class of
metric spaces and objects embedded as sets of points within them. But, with the aid of some of the worst ascii-art you have ever seen, the following, simpler, illustration hopefully gives some idea of what is meant by the term.
Consider the Koch curve - we show here two successive stages in generating it:
/\
/ \
______/ \______
_/\_
| |
__/\__/ \__/\__
Well, it may not be obvious from the bad ascii representation, but we generate it by taking each line segment:
__________________
and adding two additional lines, in the form of a triangular bump:
/\
/ \
______/ \______
This gives us four lines instead of one, and please note the length of each of these lines is (or should be) exactly
one third the length of the original line.
Each of the resulting four lines is then modified the same way in the succeeding generation. This continues forever to produce the full, infinitely wiggly (i.e. fractal) Koch curve.
The idea of a fractal dimension is to provide a measure of this wiggliness - intuitively this will be between 1 and 2, since an infinitely wiggly line can be thought of as "more than a (1D) line, but less than a (2D) plane".
Now, imagine that we have a coin whose diameter is equal to the length of one of these four segments. In order to completely cover the line, we will need to use four such coins:
,.
/\ `
.-. /| \ |
|___|/ \_\_____
`._.'
(well, because of the difficulties of
ascii, I only show two.
Pretend that they are four nice neat equal-sized circles, and the lines they cover are their
diameters, please!)
Now, since the lines produced by the next modification are each one third the length of the lines shown above, we can see that if we reduce the size of the coin so that its diameter is one third the size, we will need four times as many coins to cover the resulting object.
This still applies on the completely constructed Koch curve: since the bumpy triangles aren't big enough to stick out over the top of coins whose diameters are the 'parent' lines (the lines that they are 'bumps' in), if we count how many coins of a certain diameter we need to cover the curve, and then how many coins we'd need when we reduce the diameter of the coin to a third of the previous size, we find we always need four times as many coins for each reduction.
And so the 'coin covering dimension' of the Koch curve can be expressed as 4/3 (or, 1.333...) - it's the ratio of the extra coins needed to the reduction of diameter of the coins: four times as many coins per threefold reduction in diameter.
The Hausdorff dimension is simply a generalised version of this idea, which works in a metric space of any integer dimension, with a few logarithms thrown in. Which is why the Hausdorff dimension of the Koch curve is log 4 / log 3, as stated in the above writeup.