We are all familiar with the concept of dimension: a point is zero-dimensional, a line is one-dimensional, a plane is two-dimensional and the space around us is three-dimensional. A position on a line can be specified by a single number, such as the distance from a fixed origin. In the plane, a point can be located by giving its Cartesian coordinates , or its polar coordinates . In space, we may specify the location by giving three numbers .

In 1872, Georg Cantor made a remarkable discovery. He found a one-to-one mapping between the one-dimensional unit interval and the two-dimensional unit square . His idea is easily explained: any point on the unit interval may be expressed in decimal form, . We use the odd and even digits of to construct two numbers

giving us a point in the unit square . Any point in can be obtained from some .

It is clear that this argument can be reversed: given the two coordinates of any point in ,

we can form the number in the interval , thereby mapping into (there are some “sticky points” that have been ignored in this article. For more detail, see Gouvêa, 2011). In a letter to Dedekind in 1877, Cantor wrote of the result “I can only say: *je le vois, mais je ne le crois pas”.* The letter was in German, but the famous phrase “I see it, but I don’t believe it” was in French.

Cantor’s result was remarkable: it meant that a point in the two-dimensional unit square could be indicated by a single number in . Indeed, this was also true for a point in for any . So, there were far-reaching consequences for the concept of dimensionality.

In 1878, Cantor proved that any two smooth manifolds of *any finite dimensions* have the same cardinality. But could there be a *continuous* mapping between them? In particular, could there be a *one-to-one continuous* function from to ? The answer is no, as was proved by Eugen Netto in 1879.

Dropping the requirement of the mapping being one-to-one, mathematicians sought a continuous surjective mapping . The first to find one was Giuseppe Peano. The following year, 1891, the great David Hilbert found a simpler map and provided a clear description of its construction and its properties.

**The Hilbert Curve**

The Hilbert curve can be constructed as the limit of a sequence of functions of increasing detail. The zero-order map is constant: . We move from to by a recursive process: four copies of are scaled by and mapped to the four quadrants , , and of the square (see Figure above). They are arranged in such a way that the end of each one lines up with the beginning of the following one. This requires us to reflect the curves in the south-west and south-east quadrants about the diagonals through the centre of . Finally, connecting line-segments are added to link the four quadrants in a single curve. The first six approximations are shown in the Figure below.

The approximating curves give an impression of the Hilbert curve, which we may denote by or just . However, it is far from clear how to calculate any particular value . It is also far from obvious that the mapping is surjective. Does this curve really pass through every point in the unit square. We will return to these questions in Part II.

** Sources **

Gouvêa, Fernando Q., 2011: Was Cantor Surprised? *Amer. Math. Monthly*, **118**.

Sagan, Hans, 1994: *Space-filling curves.* Springer-Verlag, New York. 193pp. ISBN: 0-387-94265-3.

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