Packaging of Spheres in 2-Dimension

Crystalline solids contain constituent particles in a regular and repeating pattern. The crystal lattice is a diagrammatic representation of the three-dimensional configurations of constituent particles in a crystal, in which each particle is represented as a point in space. The atoms in a crystal lattice are extremely densely packed, with very little space between them.

A sphere packing is a geometric arrangement of non-overlapping spheres contained within an enclosing space. Typically, the spheres evaluated are similar in size, and the space is three-dimensional Euclidean. However, sphere packing problems can be generalised to include unequal spheres, multidimensional spaces (where the problem becomes circle packing in two dimensions or hypersphere packing in higher dimensions), and non-Euclidean environments such as hyperbolic space.

A common sphere packing challenge is to design an arrangement in which the spheres take up the maximum amount of available space. The proportion of space occupied by the spheres is referred to as the arrangement’s packing density. Due to the fact that the local density of a packing in an infinite space varies depending on the volume measured, the objective is often to maximise the average or asymptotic density recorded over a sufficiently large volume.

The densest packing of equal spheres in three dimensions consumes around 74% of the volume. A random packing of equal spheres has a density of around 63.5 percent.

Close-packing in two dimensions

1. When a sequence of rows is piled on top of one another, a two-dimensional crystal plane is formed. There are two distinct methods for stacking the rows.

2. One possibility is for the rows to be stacked one on top of the other, with one sphere directly above the next.

The spheres are aligned horizontally as well as vertically within this arrangement.

3. In this configuration, each sphere is in contact with four more spheres, two on each side, one above and one below. As a result, the coordination number is increased to four. When the centres of these four spheres are connected, we have a square. As a result, this method of close packing is also known as square close packing.

4. Alternatively, the second row’s spheres can be staggered and seated in the first row’s depressions.

The densest packing of equal spheres in three dimensions consumes around 74% of the volume. A random packing of equal spheres has a density of around 63.5 percent.

Two-dimensional close packing involves stacking a row of closed packed spheres to create a two-dimensional pattern. This stacking is accomplished in one of two ways:

• Square close packing: The second row can be packed immediately beneath the first row. Thus, if we refer to the first row as an “A” type row, the second row, which is structured identically to the first, is also an “A” type row. Each sphere is in contact with four other spheres in this arrangement. As a result, it has a coordination number of four. We see that when the centres of four adjacent spheres are connected, a square is formed. In two dimensions, this sort of packing in crystalline materials is referred to as square tight packing.

• Hexagonal close packing: The second row can be staggered below the first row, with its spheres fitting into the depressions of the first row. Thus, if we refer to the first row as a “A” type row, the second row might be referred to as a “B” type row due to its arrangement. Again, the third row is of type “A.” This is known as the “ABAB” method of packing. Each sphere is in contact with six other spheres in this arrangement. As a result, it has a coordination number of six. We note that when the centres of six adjacent spheres are connected, a hexagon is generated. In two dimensions, this sort of packing in solids is referred to as hexagonal tight packing. In comparison to square tight packing, it has less empty space and thus a higher packing efficiency.

Conclusion:

The nineteenth century saw significant progress on the subject, when the famed German mathematician and physicist Carl Friedrich Gauss demonstrated that the orange-pile arrangement was the most efficient of all  those “lattice packing”. Phenomena that cause the component parts of atoms/molecules to be close to one another as possible in their crystal structure