Mathematics often guides us through amusing and surprising paths. In the realm of geometry, much of the allure comes from the diverse visuals we can create while tackling problems. One captivating example is the study of circle packing. The fundamental question is straightforward: in a two-dimensional area, how many circles can be accommodated? This simple query has sparked extensive exploration, unveiling numerous applications in real-world packaging and logistics.

The most basic approach to solve this is known as the square-lattice method. Imagine connecting the centers of circles with lines, forming a square lattice. This method's simplicity makes it appealing. During the COVID pandemic, schools often arranged desks in a square lattice to maintain distance, reflecting the radius of each circle. However, despite its simplicity, this is not the most efficient technique.

Research has demonstrated that a hexagonal lattice is the most effective way to pack circles into a space. This can be assessed using a concept known as packing density, which requires a lattice that extends uniformly throughout the plane.

In the hexagonal packing method, we can visualize a hexagon since this pattern repeats throughout the packing. The density is calculated by comparing the hexagon's area to the space occupied by the circles within it, yielding a ratio of approximately 0.9062, or ? / (2?3). So far, we've only examined infinite planes, which can help build intuition, but other approaches exist. What happens when we pack circles into a confined area? To explore this, we turn to one of the 20th century's most brilliant minds: Ronald Graham.

Ronald Graham was a prolific mathematician, authoring six books and over 400 papers. He is best known for Graham's Number, the largest number derived from a mathematical proof, surpassing a Googleplex. His work spanned various fields, including Ramsey Theory and computer science, and he was also an enthusiastic juggler, contributing new techniques to the community.

For this discussion, we will focus on Graham’s contributions to sphere packing. While the hexagonal tiling method for infinite spaces was established in the 18th century, Graham's work has significant implications for packing circles into finite spaces. Let's first consider packing circles within a larger circle, an area where Graham made substantial advancements.

This problem involves packing n circles of equal size as closely as possible within a larger circle. The optimal configurations for one to five circles are straightforward, but the fifth arrangement required more intricate work. In 1968, Graham proved that this configuration achieves a density of about 0.68, establishing a foundation for proofs of similar types with larger numbers.

As more circles are added, the arrangements become increasingly unusual. For six spheres, there are two distinct configurations that maintain the same density of approximately 0.6 repeating.

Among the optimal packings for a prime number of circles, the arrangements tend to be the most peculiar. Since prime numbers cannot be evenly divided, they often yield configurations, such as the optimal arrangement for 17 circles, that lack radial symmetry.

Despite their complexity, these patterns exhibit some degree of order. What occurs when we try to pack circles into a square?

Arrangements for two to six circles appear quite organized, but the configuration for seven circles is rather bizarre. In this image, each orientation's smaller square denotes the smallest square containing the center of each circle. Similar to circles, prime numbers often produce unconventional shapes, as they cannot be divided into symmetric groups. Even composite numbers can yield strange arrangements, as seen with the optimal setup for 15 circles.

This arrangement may seem random, yet it has been definitively shown to be the most efficient. Mathematicians utilize various methods to uncover these optimal configurations.

Earlier, I mentioned that the maximum achievable density for circle packing is around 0.9062. While this is relatively low for a two-dimensional shape, it is not the lowest. Research indicates that the smooth octagon has the lowest maximum packing density, approximately 0.902.

As previously noted, packing methods have real-world applications for enhancing packaging efficiency. While the two-dimensional examples provide insight, real-life scenarios often involve three dimensions.

We also have documented efficient methods for packing spheres. Beginning with n equally-sized spheres, the objective is to arrange them within the smallest possible larger sphere. The arrangement for five spheres shown above has been proven to achieve a maximum density of 0.74, or ? / (3?2). The symmetry between two-dimensional and three-dimensional bounds is remarkable.

Visualizing these optimal arrangements can be more challenging than their 2D counterparts, yet they often relate to one another. For instance, consider the optimal arrangement for seven spheres. To me, it resembles the setup for seven circles. The three central spheres (3, 4, and 7 in pink) mirror the three circles forming the diagonal from bottom left to top right, surrounded by two different pairs. Similar connections can be observed for other configurations as well.

Mathematicians can also simulate random sphere packing. Depending on the method employed, a random packing of spheres typically achieves a density of about 64%.