The Staircase Paradox

The Strange Case of the Shrinking Diagonal

Everyone knows it’s faster to cut across a field diagonally than to walk down one street and then the other. The good old Pythagorean theorem tells us that the distance from one corner of a square to the other diagonally is 1.4142… while getting there along the perimeter of the square takes 2, 1 up and 1 across.

But now imagine you are driving to your friend’s house diagonally across a grid of 10 by 10 city blocks. Even if you drive diagonally, it’s still the same distance as driving all the way up and all the way across. All of the streets add up to the same as 10 blocks up and 10 blocks across. Let’s say you could divide the same area into a 100 by 100 grid, you would still be driving the same distance as going around the perimeter. If you kept dividing the blocks smaller and smaller, even when you got to a microscopic level, the total distance would remain the same.

Now you can imagine that as you approach the limit of this, those little diagonal blocks (which look a lot like a staircase) get closer and closer to looking like a straight diagonal line. The paradox comes from the idea that the moment you go from the blocks being infinitesimally small to a straight line, the distance traveled suddenly drops from 20 (the size of the original 10 x 10 grid) to 14.4 (the length of the diagonal line).

I don’t remember why I randomly started thinking about this, but it seemed interesting, and of course I wasn’t the first person to find it interesting. As soon as we discovered the Pythagorean theorem and limits in calculus, the idea presented itself.

It reminds me of another paradox called the coastline paradox. Lewis Fry Richardson noticed that the Portuguese reported their border with Spain to be 613 miles, but the Spanish reported it as 754 miles. The issue, it turns out, is that the more accurately you measure a border, the longer it becomes. Imagine part of the border looks like a staircase as we mentioned above, but instead of drawing every single angle, you just draw a straight line through it. You would have a shorter distance than someone who took the time to trace out every angle.

Imagine mapping an island. There is a fractal nature to the perimeter. As you zoom in more and more, the little curves become intricate. While one person could just draw something close to a circle around it, someone else who traced every single nook and cranny would have a much larger circumference. If you keep zooming in, you would get to an atomic level where you would have to trace around each molecule. This effect is so extreme that an island with a circumference of 1 mile could turn into thousands of miles if you brought up the level of precision to ridiculous amounts.