might seem like a topic only a mathematician could love. Who else could get excited about finding the most efficient way to arrange circles in the plane, or spheres in space?
But right now, millions of people all over the world are thinking about this very problem.
Determining how to safely reopen buildings and public spaces under social distancing is in part an exercise in geometry: If each person must keep six feet away from everyone else, then figuring out how many people can sit in a classroom or a dining room is a question about packing non-overlapping circles into floor plans.
Graphics by Samuel Velasco/Quanta Magazine
Of course there’s a lot more to confronting COVID than just this geometry problem. But circle and sphere packing plays a part, just as it does in modeling crystal structures in chemistry and abstract message spaces in information theory. It’s a simple-sounding problem that’s occupied some of history’s greatest mathematicians, and exciting research is still happening today, particularly in higher dimensions. For example, mathematicians recently proved the best way to pack spheres into 8- and 24-dimensional space — a technique essential for optimizing the error-correcting codes used in cell phones or for communication with space probes. So let’s take a look at some of the surprising complications that arise when we try to pack space with our simplest shape.
If your job involves packing oranges in a box or safely seating students under social distancing, the size and shape of your container is a crucial component of the problem. But for most mathematicians, the theory of sphere packing is about filling all of space. In two dimensions, this means covering the plane with same-size circles that don’t overlap.
Here’s one example of packing circles in the plane. It might remind you of the side view of a case of soda cans:
You can imagine this pattern repeating in every direction, like a tiling of the plane. The little gaps between circles mean the plane isn’t entirely covered, but that’s to be expected with circle packings. Instead, we are interested in what percentage of the plane is covered. This is known as the “packing density” of the arrangement.
The above arrangement is called a square packing, and for good reason: We can imagine the centers of the circles as vertices of squares.
In fact, the squares themselves tile the plane.
The symmetry of this tiling makes our work easy. Since these squares cover the entire plane in a regular way, the percentage of the plane covered by circles is the same as the percentage of any one square covered by circles. So let’s take a closer look at one of those squares.
Suppose that each circle has radius r. That means the square has side length 2r. Each of the four vertices of the square is covered by a quarter-circle, so the percentage of each square covered is just the ratio of the area of one full circle to the area of one square:
Each square is about 78.54% covered by circles, so by our tiling argument, the entire plane is about 78.54% covered by circles. This is the density of the square packing. (Notice how the radius r drops out of our answer: This makes sense because no matter how big the circle is, the square will still contain four quarter-circles.)
Now, if you’ve ever tried to stack soda cans on their sides like this, only to watch them slip and slide into the gaps, you know there’s another way to pack circles in the plane.
Taking a similar approach to what we did above, we can imagine the centers of the circles in this arrangement as vertices of regular hexagons.
We call this a hexagonal packing. This arrangement seems to fill in the gaps more efficiently than the square packing. To verify, let’s compare their packing densities. Just like squares, hexagons tile the plane, so we can determine this arrangement’s packing density by analyzing a single hexagon.
How much of this hexagon is covered by circles? Since the interior angle of a regular hexagon is 120 degrees, there is a third of a circle at each of the six vertices of the hexagon. That adds up to two full circles, and the one in the middle makes three. So every hexagon is covered by three circles. If each circle has radius r, that’s a total area of 3πr².
How does that compare to the area of the hexagon? A hexagon of side length s is really six equilateral triangles of side length s, each with area
So now we can compute the percentage of the hexagon’s area that is covered by circles (by dividing the area of three circles by the area of the hexagon):
Each hexagon is about 90.69% covered by circles, making this a much more efficient packing than the square arrangement. (Notice how the radius of the circle again dropped out, as we should expect.) In fact, no arrangement is more efficient.
Proving this wasn’t easy: Famous mathematicians like Joseph Louis Lagrange and Carl Friedrich Gauss started the work in the late 18th and early 19th century, but the problem wasn’t completely solved until the 1940s, when all the possible arrangements — both regular and irregular — were rigorously dealt with. That it took so long to handle the problem in two dimensions, where things are relatively easy to visualize, is a warning of what’s to come in higher dimensions.
Packing spheres in three dimensions is a much more complicated problem, though it does share some features with its two-dimensional relative. For example, the two-dimensional packings we looked at are built from a single layer.
In the square packing, we put each new layer directly on top of the previous one.
In the hexagonal packing, we nestle each new layer in the gaps of the previous one.
We get different packings depending on how we put together copies of each layer.
In three dimensions, different fundamental packings arise from stacking layers like this.
