Figure 1
Figure 2
Figure 3

Consider a square in the plane. Is it possible to
draw a curve in the square that touches every point
inside the square?
It seems that this should not be possible...
after all, lines and planes are different dimensions.
Surprisingly, such a curve is possible!
This counterintuitive
object is called a spacefilling curve. You
cannot draw it yourself, because it would take forever.
However, it exists as the limit of a bunch of curves that
you can draw.
In the first stage draw a straight line from the
lower left to upper right corner. In the second stage,
replace that straight line by the zigzaggy curve in Figure 1.
(By the way, the zigzaggy curve does meet itself at 2
places, but we've drawn them separate so you can resolve
the curve.) Then, for every straight line segment in
the second stage, replace it by a exact copy of the
second stage, but scaled down by a 1/3 in each direction!
See Figure 2 and Figure 3. Now repeat this process over and over,
and the limiting object you obtain will be a continuous
curve that fills the whole square!
Presentation Suggestions:
Of course, draw lots of pictures. Colored chalk may be
helpful.
The Math Behind the Fact:
Which sets of curves "converge" to a continuous limiting
curve is an issue tackled in a fun course
called real analysis.
How to Cite this Page:
Su, Francis E., et al. "Spacefilling Curves."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
