(Spoiler alert: I have not broken geometry)
Suppose you have a square with side-length, L, completely surrounding a circle of radius, R, such that the edge of the circle just touches the inside of the square, like so:
Those of you that remember your geometry will recall that the circumference of the circle – that is the length of the red curve that defines the circle – is given as follows:
C = 2⋅𝜋⋅R
and the perimeter (i.e. length of the curve) of the square can be written as:
P = 4⋅L = 8⋅R
since the radius of the circle is half the length of the side of the square. Note: here I am using (and will continue to use) the mathematical meaning of the word “curve”. That means that while the black lines that make up the square are straight segments, they are still defined by a “curve” in the mathematical sense.
Now, suppose you look at the corners of the black curve that makes the square and cut out two little line segments of length, s, in order to rearrange them so that the tips of the corners are now just touching the edge of the circle, like so:
Now we have two curves, a circle and something that is not quite a square. The keen-minded among you might have already noted that even though the black curve is no longer a true square, the perimeter is still exactly the same. All we have done is swap around some horizontal and vertical line segments to make the corners touch the circle without changing their length.
Let’s say that doing this process of rearranging parts of the black curve so that the corners pointing outwards now touch the edge of the circle is an algorithm with one step. Now, let’s count the iterations of this algorithm. By doing this once, as above, we have N=1.
Suppose we iterate the algorithm one more time, so that we have N=2 and make all eight of the outward pointing corners point inward, like so:
Just as before, the length of the black curve is still unchanged; all we’ve done is rearrange it. I’ve zoomed in on one quadrant so the details are more visible, but imagine this is being done on all the other corners.
It may be clear at this point that the black curve is beginning to converge to the red curve. In fact, as the number of iterations, N, goes to infinity, we can say that the black curve would lie almost exactly on the red curve of the circle. However, we know that the circumference (perimeter) of the red curve is 2⋅𝜋⋅R, whereas the perimeter of the black curve is unchanged at 8⋅R.
If the two curves are laying right on top of each other, how can this be?
This may see a bit counterintuitive, but a large part of the weirdness comes from a (somewhat intentional) mixing of mathematical concepts. Here, the simplest explanation why the perimeter of the black curve is not the same as a circle (the red curve) is that the black curve is simply NOT a circle – it’s a completely separate shape.
When we talk about mathematical curves that enclose space like this, it is important to distinguish the lengths of the curves from the areas that they enclose. It is correct to say that as we iterate our algorithm towards N→∞, the area confined within the black curve will converge to the area contained within the circle. Intuitively this follows, as the square has visibly more surface area than the circle at the start, and each iteration chops away a small amount.
However the shape of the black curve will be nothing like a circle. In fact, the black curve will be taking on a fractal nature. As we iterate this process infinitely many times and because mathematical curves have infinite resolution, we could hypothetically zoom in further and further and see the same repeating pattern of corners in the black curve – it would never truly converge to a circle. This is a similar phenomenon to one of my favorite mathematical concepts, the coastline paradox.
This phenomenon is perhaps more easily illustrated if we consider a single line that doesn’t enclose any area. Suppose we are trying to make the blue curve converge to the red line in the (hastily illustrated) image below. We can do a similar procedure of flipping the corners of the blue curve N times so that they touch the red line. As N→∞, the blue curve will lay directly over the red line but its overall length will be unchanged.
I hope that this illustration makes it clear that the initial blue curve could be arbitrarily long and stay that way, yet it can be made to converge to the straight red line. This difference in the lengths of the two lines/curves is described by the concept of taxicab geometry, a line of geometric thinking that considers the absolute differences of Cartesian coordinates between points.
Another way of thinking about the fractal nature of this converging curve is to think about the number of vertices or corners that emerge. In the above example, the number of corners that don’t touch the red line doubles with each iteration. As N→∞, the number of vertices will also approach infinity. A similar example from mathematics is the Cantor set, one of the earliest descriptions of fractal geometry. To visualize the Cantor set, imagine a line segment, divided into thirds, then remove the middle segment. At each iteration, this process is repeated on the remaining line segments, like so:
At each step, the number of remaining line segments doubles. At the same time, the total length of the remaining segments is decreasing. Eventually as N→∞, the total size – that is, the sum of the remaining lengths – goes to zero, while the number of segments approaches infinity. What’s more, as you zoom in on any of the smaller segments, each “level” of resolution will exactly resemble the level above and below it. This is an example of the fractal idea of self-similarity.
So, to sum up: the lengths of the curves in both examples are different because the curves are different and it is important not to confuse curve length with confined area. Furthermore, the fractal nature that the curve can take on has some non-intuitive properties. Finally, no I did not break geometry – I just explored it a little.
Until next time.