For the past week or so I’ve been obsessed with the following party trick that I used to use to keep my kids entertained in Italian restaurants. Flip over your paper placemat and grab a pencil. Draw a straight line from the top left-hand corner to a point at distance d down from the top of the right-hand edge, where d is a small distance like a finger-width. This is shown in Step 1 below. Then, draw a straight line from where it hits the right-hand edge to a point at distance d left from the bottom of the right-hand edge. This is shown in Step 2 below Next, draw a straight line from where it hits the bottom edge to a point at distance d up from the bottom of the left-hand edge. This is shown in Step 3 below. Finally, draw a straight line from where it hits the left edge to a point at distance d to the right of the left edge, but this time measured along the line you drew in Step 1. This is shown in Step 4 below.
Now repeat this process, each time measuring the distance d along your old line. Keep going until you run out of space.
See my Straight Line Curves Webpage for an account of my weeklong obsession with these curves, which starts by having fun with a paper placemat and a pencil, ends with the Four Ant Problem, and has some math and programming in between. Oh, and there’s homework if you’re up to it.