The Peano curve — a line that fills the square

What just happened?

As you increase the number of iterations, you watch successive terms of a sequence of curves that wind through the square ever more densely. For small $k$ you see a simple, angular path; the larger $k$, the finer the curve folds, until it looks as if it covered the whole area. The key point: at every step it remains one continuous line — running without a break from corner to corner (green start, red end).

A continuous map of the interval onto the square:

In the limit (infinitely many iterations) the Peano curve is a continuous surjection of the interval onto the square: every point of the square has its own moment $t\in[0,1]$, at which the curve passes through it. A one-dimensional “line” really does fill a two-dimensional area.

The length grows without bound, the spacing tends to zero:

With a $2^k\times 2^k$ grid the curve has $4^k-1$ segments of length $1/2^k$, so its length doubles every iteration and is divergent. At the same time the spacing between neighbouring “strands” shrinks to 0 — which is why in the limit no point is left unvisited.

Where does this curve come from?

The Italian mathematician Giuseppe Peano constructed it in 1890 — it is the first curve that fills space — a space-filling curve. The discovery shook the contemporary understanding of the notions of dimension and continuity: earlier Cantor had shown that an interval has “just as many” points as a square, and Peano gave this a continuous realization. Here we draw the Hilbert variant — a curve that is easier to render recursively but has the same square-filling property.

Why this matters:

Space-filling curves have real applications today: in computer science (indexing of multidimensional data, image compression, spatial algorithms), computer graphics and measure theory. It is a beautiful example of how precise mathematics can overturn the “obvious” intuition that a line (dimension 1) cannot fill an area (dimension 2). (A topic beyond the standard curriculum — a fascinating curiosity about dimension and continuity.)

Remember: there is a continuous curve that fills the entire square — its length tends to infinity, while the spacing between strands tends to zero. A one-dimensional line passes through every point of a two-dimensional region.