# Packing circles into a square

In the last two weeks – completely by coincidence – I’ve done two fun project with each of the boys related to circle packing.

The first was with my younger son and inspired by a Robert Kaplinsky tweet:

Sharing Robert Kaplinsky’s pipe stacking problem with my younger son

The second was with my older son and inspired by one of the problems on the Iowa State “problem of the week” collection.

The problem was about the limit of the area of circles packed into an equilateral triangle: A terrific problem to share with calculus and geometry students from the Iowa State problem collection

Today I wanted to combine the two ideas and look at two ways of packing circles into a square. First I introduced the problem and we looked at the problem of packing circles in a square stacked directly on top of each other. As in the Iowa State problem, we found a surprise in the area covered by the circles as the number of circles approaches infinity:

Now we moved on to the problem of “staggered stacking.” In this video I introduce the problem and let the boys try to figure out why this problem is a little bit harder than the stacking problem from the last video:

Now we began to try to solve the “staggered stacking” problem. Turns out this problem is really tough! There are a lot of things about – the number of circles is much harder to calculate – but we were able to make some progress on some of the easy cases:

Now we tried to calculate how tall the stack of circles is. I think the algebra here is close to the edge of my younger son’s math knowledge. But he does a great job of explaining how to calculate the height. The nice thing is that he remembered the main idea from the project inspired by Robert Kaplinsky:

Finally – we put all of the ideas together. There are a lot of them, but with a little math magic, they all fit together really well!

So, a fun project following the neat coincidence of seeing to other problems related to circle packing recently. I think all of these problems are great ones to share with kids learning geometry and also learning calculus.