In geometry, a point has no dimension, since it has no length, no width and no depth.

A line is one-dimensional because it has length.

Figure 7: A line.

A plane is two-dimensional, since it has length and width.

Figure 8: A plane.

A box is three-dimensional: it has length, width and depth.

Figure 9: A cube.

Up to now, we are referring to the concept we ordinarily associate with dimension (also called Euclidean or topological dimension). Fractals, on the other hand, have fractional dimensions, usually with non-integer values as 1.7, 0.5326478 or 3.28. How can that be?

If we divide a one-dimensional object in two smaller equal parts, we get two small versions of the same object.

Figure 10: Division of a line.

If we divide a 2 dimensional object in half its length and width, we get four copies of the same object.

Figure 11: Division of a plane.

If we divide a 3 dimensional object in half its length, width and depth, we get eight copies of the same object.

Single small cube

Figure 12: Division of a cube.

Looking closely will show us what can be termed, to our present purpose, as geometrical duplication (a.k.a. exponential growth) in which duplication occurs at an exponential rate 2, 4, 8 and so on. Arithmetically, these numbers can also be expressed as:

2 = 2^1

4= 2^2

8 = 2^3

Examining the exponent in each case, we find that it is equal to the dimension of each object: 1, 2 and 3.

Now, let us do likewise with a fractal object as the Sierpinski triangle. If we divide it in half its height and base, we only get three copies (remember that the central portion do not belong to the triangle). Then, we need an exponent z such that 2^z = 3.

Figure 13: Dividing a Sierpinski triangle.

The Sierpinski triangle is not one-dimensional because 3 is greater than 2, but it isn't two-dimensional because 3 is less than 4. So, its dimension must lie between those two dimensions (1 and 2). Actually, it is close to 1.58496250072115618145373894395.