As we see in the Koch snowflake example, curves which exhibits "self-similarity"
seem to have different lengths at different scales of magnification.
In mathematical terminology, these curves have non-integer "dimension".
Roughly speaking, their geometry lies somewhere between between a line and a plane.
The Kock snowflake has dimension:
\[
\displaystyle \log_3 4 \approx 1.2619.
\]
The Sierpinski triangle, which may be constructed by "hollowing out"
a filled-in triangle (which has dimension 2),
has dimension:
\[
\displaystyle \log_2 3 \approx 1.5850.
\]
One way to compute the dimension of a set $S$ is the so-called "Box Counting" method:
Position the set $S$ in question in a rectangular box.
Subdivide the rectangular box into squares of the same width $s$.
Count the number $N$ of boxes which has non-empty intersection with $S$
under examination.
Compute $\displaystyle \log_{\frac{1}{s}} N$.
Repeat the procedure with smaller and smaller boxes.
In many cases, as the size of the boxes tend to zero,
the quantity computed in Step 4 approaches a fixed number.
This number is the (fractal) dimension of the set $S$.