Packing Circles in Containers

TL;DR: You cannot fill a box with circles without leaving gaps; the most efficient arrangement, known as hexagonal packing, still leaves about $9.3%$ of the container empty.

The Geometry of the Void

In our previous stop, we looked at how GPS satellites use sphere intersections to pinpoint locations in 3D space. Now, we are grounding that logic into the physical reality of a cardboard box. Have you ever noticed that when you buy a pack of tennis balls, they never quite fill the tube? There is always that annoying bit of air trapped between the curvature of the spheres and the flat walls of the container. That isn't a manufacturing defect; it is a fundamental rule of geometry.

When we pack circles into a rectangular frame, we are fighting against the fact that circles do not have edges. Squares fit together perfectly because their internal angles add up to ${360}^{\circ }$ at every vertex, leaving no room for error. Circles, however, are essentially infinite-sided polygons that refuse to sit flush against a straight line. If you line them up in a simple grid, you create massive pockets of wasted space. To optimize, we have to rethink the layout.

The Hexagonal Advantage

If you want to pack circles efficiently, stop thinking in rows and start thinking in triangles. This is called hexagonal packing. Instead of stacking a circle directly on top of another, you shift every second row so that the circles nestle into the gaps of the first row.

Mathematically, this creates a pattern where the center of every three touching circles forms an equilateral triangle. By pulling the circles closer together, you reduce the amount of empty space significantly. In a standard square grid, the packing density—the percentage of the box actually occupied by the circles—is about $78.5%$. By switching to a hexagonal arrangement, you can boost that efficiency. In an infinite plane, this density reaches roughly $90.7%$, meaning you are left with only $9.3%$ empty space.

Calculating the Waste

To calculate the efficiency of your own packing, you need to compare the area of the circles to the area of the container. If you have a box with width $W$ and height $H$, the area is simply $A_{box}=W\times H$. For $n$ circles with radius $r$, the total area taken up by the circles is $A_{circles}=n\times \pi r^2$. The percentage of empty space is calculated as:

$$\text{Empty Space}=(1-\frac{n\pi r^2}{W\times H})\times 100$$

This calculation is the secret language of logistics and manufacturing. Whether you are designing a battery pack for an electric vehicle or figuring out how many soda cans fit into a shipping crate, you are constantly balancing the weight of the objects against the volume of the wasted space. If you ignore these geometric constraints, you end up paying to ship air across the country.

When we move from the flat, two-dimensional world of circles into three-dimensional spheres, the math gets even more complex. We stop talking about area and start talking about packing fraction. The principles remain the same, but the stakes are higher because the gaps become harder to fill.

Because circles possess curved edges that cannot perfectly align with straight boundaries, a small percentage of empty space is mathematically inevitable in any physical container.

Now that you understand how to minimize the void in two dimensions, we will shift our focus to the natural world, where geometry is used not just for efficiency, but for beauty, as we explore the Golden Ratio in the next station.