Close-packing of equal spheres

In geometry, close-packing of equal spheres is a way to arrange the same size balls very closely together in a repeating pattern. This pattern fills up as much space as possible with the balls. A famous mathematician named Carl Friedrich Gauss showed that the best way to fill space in a regular pattern gives a packing density of about 0.74, meaning about 74% of the space is filled by the balls.

This same high density can also be reached by stacking layers of balls in different ways, even if the layers are not always stacked the same. A big question in math, called the Kepler conjecture, asked if this was the best possible way to pack balls, no matter how you arrange them. This was finally proven true by a mathematician named Thomas Hales.

Close-packing is very important in nature. Many crystal structures, like the way atoms are arranged in metals and rocks, use this packing method. Sometimes bigger atoms or ions pack together, with smaller ones fitting in the spaces between them. The two most common ways to pack—cubic and hexagonal—are very similar in energy, making it tricky to know which one will form naturally.

FCC and HCP lattices

There are two simple ways to arrange spheres closely together to fill space efficiently. These are called face-centered cubic (FCC) and hexagonal close-packed (HCP), named after their patterns. Both use layers of spheres placed at the corners of triangles. They differ in how these layers stack on top of each other.

The story of packing spheres tightly began with stacking cannonballs on ships. People wanted to know the best way to pile these balls into pyramids. This led to important math discoveries about how to arrange spheres in space.

Lattice generation

Thomas Harriot thought about the math behind stacking spheres like cannonballs a long time ago. This way of stacking is called a face-centered cubic (FCC) lattice.

When we stack spheres close together, we notice something interesting. If two spheres touch, we can draw a straight line from the center of one sphere to the center of the other. This line will pass through the point where the spheres touch. The distance between the centers of two touching spheres is always twice the radius of one sphere.

Simple HCP lattice

To stack spheres in a special way called hexagonal close packing (HCP), we can imagine placing the centers of the spheres at certain points in space. We start by placing a row of spheres along a straight line. The centers of these spheres are spaced apart by twice the radius of a sphere.

Next, we add another row of spheres next to the first row. This new row is shifted so that each sphere touches two spheres from the first row. By doing this, the centers of the spheres in the new row form triangles with the centers of the spheres they touch.

We can keep adding rows in this way, shifting them each time, to build up a pattern that repeats. This creates a neat, close-packed structure of spheres.

Miller indices

Main article: Miller index

In crystals, special patterns can be shown using a system called Miller index. This system uses four numbers ( hkil ) to describe directions and planes in the crystal. One of these numbers, i, is linked to the first two and always equals minus the sum of h and k. The first three numbers are spread out at 120° from each other, while the last number stands straight up compared to the others.

Filling the remaining space

The FCC and HCP packings are the densest ways to arrange identical spheres while keeping the pattern very regular. To completely fill space without gaps, we would need shapes that are not spheres, like honeycombs.

When we connect the centers of touching spheres with lines, we can form shapes like tetrahedrons and octahedrons. The FCC pattern creates one special shape called the tetrahedral-octahedral honeycomb, while the HCP pattern creates another called the gyrated tetrahedral-octahedral honeycomb.

Tiny round bubbles in soapy water often arrange themselves in FCC or HCP patterns, but these patterns are not very stable. More stable bubble patterns, like the Kelvin foam and the Weaire–Phelan foam, use different shapes that use less energy.

There are small empty spaces left between spheres in these patterns, called interstitial holes. These holes can be either tetrahedral, where four spheres surround the space, or octahedral, where six spheres surround the space. Many simple chemical compounds have smaller atoms that fit into these holes in larger structures made of bigger atoms.