It’s easiest to answer this question with an example. Look at the following sequence of images:
The Koch Snowflake after 0, 1, 2, 3 and 6 iterations
The Koch Snowflake
The sequence shows the evolution of a fractal called the Koch Snowflake after a number of iterations. The fractal is formed thus:
Despite its simplicity, the Koch Snowflake shows the essential characteristics that define a fractal:
The term “fractal” was coined by Benoit Mandelbrot in a 1975 book Fractals: Form, Chance and Dimension
In a trivial sense, a straight line has infinite detail: you can remove the tiniest segment and you get a straight line. But note the word “trivial”. No matter how tiny a section of the Koch Snowflake you remove, you never end up with a straight line; instead, you end up with something that still has infinite detail.
The tiniest section of a Koch Snowflake scaled up is indistinguishable from a larger section. We can say that the curve displays exact self-similarity. Exact self-similarity is not a requirement for a fractal. The Mandelbrot Set, for example, shows quasi self-similarity where repeats of the basic form are found at all scales. Naturally occurring fractals show statistical self-similarity where the essential characteristics of the form are preserved at all scales. An obvious example is clouds: they look “cloudlike” no matter where you view them from: the ground, the top of a mountain or from above in an airplane.
This requires a bit more explanation. For the moment, it is sufficient to know that the differential of a function gives you the instantaneous rate of change of the function. It works on the principle that no matter how sharply curved a function is, if you get in close enough, the graph of the function will appear to be a straight line whose slope you can measure. But remember that with the Koch snowflake, you never find a straight line, no matter how close you get in.
A curious property of the Koch Snowflake is that at every iteration, we remove one third of its perimeter but add two thirds back in. This means that the length of the perimeter increases by one third every iteration. This increase in length is unbounded so that the length of the perimeter (and, indeed, the tiniest section of the perimeter) is infinite. Yet the area is clearly finite since one can draw a box of finite area that entirely encloses it. Now imagine that you remove a 3 cm long section from the side of the Snowflake. If you only had a ruler that was 3 cm long, you would measure the length of this section to be 3 cm. But if you had a 1 cm ruler, you could make four measurements and find the length to be 4 cm. There is a formula that relates dimension to the number of units measured at different scales:
N ∝ ϵ-D
N is the number measured, ϵ is the scaling factor and D is the dimension. For a 3 cm long straight line, we measure 3 cm when we use a 1 cm ruler, so the above equation gives us:
3 = ⅓-D
From which we can readily see that D = 1, since ⅓-1 = 3. So a straight line has 1 dimension. No surprise there. However, a section of the Koch Snowflake gives us a measurement of 4 when we use a ruler one third the scale. Plugging these values into the formula gives:
4 = ⅓-D
Rearranging the values gives:
D = -log⅓4
⅓ isn’t a particularly useful base for logarithms, so we can rearrange some more to get the values in terms of natural logarithms that we can readily compute:
D = -log 4 / log ⅓ = -1.3862943611198906 / -1.0986122886681098 ≈ 1.26186
So it seems that a section of the Koch Snowflake has a dimension > 1. The fact that we had an infinitely long perimeter enclosing a finite area suggested that the familiar rules of geometry no longer applied. The non-integer dimension is another way of saying that the dimension of a fractal is higher than the space it is embedded in.