Here are the first few stages of the construction of the HILBERT CURVE. Notice that it begins as a one-dimensional segment. If we think of dimension as how much information you need to locate a point, we see we only need one piece here - how far along the curve we are. If iteration process is carried out infinitely many times the curve contains every point in the square; in other words it fills the square. But we know that a square is two-dimension, so what is this shape: one-dimensional, two-dimensional, both, neither?
Below is the CANTOR SET. It is created by beginning with a segment (one-dimensional), cutting it into three pieces and removing the middle third. This process is repeated at the next stage for all newly created segments. The final result is an object with no length, made up of intinitely many points. We know points are zero-dimensional. What is the dimensionality of the Cantor Set - one-dimensional (like the shape it began as), zero-dimensional (as a point is), both, neither?
Next we have the KOCH CURVE, in which we begin with a line segment, cut it into thirds and replace the middle third by two segments of the same length. At each new stage we do this for each new segment. In this way we get a curve in which the endpoints never get further apart, and which we could surround completely with a small oval, and yet that has infinite length. Also, any tiny piece of the final shape can be blown up to give us an EXACT copy of the original (see animation zoom).
