It seems paradoxical, but is nevertheless true that there are continuous curves which completely fill up higher dimensional spaces such as squares or cubes. The first examples were constructed by Giuseppe Peano (1858-1932) and thus curves of this kind are known as Peano curves. The construction shown above is due to David Hilbert (1862-1943). This space-filling curve which completely fills up the unit square is the limit of a sequence of curves defined iteratively. The first six iterations in this process are shown above.

See the following sites for some variants of Hilbert's construction:

It was later shown that a wide variety of spaces can be completely filled up by continuous curves. A characterization of all such spaces was given by Hans Hahn (1879-1934) and by Stefan Mazurkiewicz (1888-1945):

Hahn-Mazurkiewicz Theorem A nonempty Hausdorff topological space can be completely filled up by a continuous curve if and only if the space is compact, connected, locally connected and metrizable.

Although at each finite stage of Hilbert's construction the curve obtained is an arc (ie. the curve doesn't cross itself), the limiting curve is not an arc: it has double points, triple points and quadruple points. It is easy to show that no arc can be a space-filling curve: there have to be uncountably many multiple points. It can also be shown that any square-filling curve has to have points of multiplicity at least three.

The space-filling curve shown above also is nowhere differentiable - if we write Hilbert's curve in parametric form, x=f(t), y=g(t), then the graph of y=f(x)looks like this:



and the graph of y=g(x) looks similar. It is fairly easy to see, using methods of calculus, that any space-filling curve has to be non-differentiable on an uncountable set. You might also take a look at the following brief discussion of continuous nowhere-differentiable functions.