### 11/14/2021

Cylinder Packings** **

Before jumping into describing polystix arrangements, let’s first take a quick look at some simple cylinder arrangements, to get familiar with some of the geometric themes that these blog posts will explore. Packing arrangements, where all the cylinders are parallel to just one direction, can be understood as comparable to 2D circle packings. The densest possible circle and cylinder packing is the hexagonal packing arrangement, shown here in Figure 1, left. The other regular packing is the simple square/ tetragonal packing, Figure 1 right.

Figure 1: Hexagonal/ honeycomb and tetragonal cylinder packings.

The two packings above can also be made with hexagonal or tetragonal cross section rods to fill 100 percent of space, just like their respective hexagonal and square 2D tilings. However, if using circles or regular** **right circle cylinders, the density is not as obvious. The fraction of space the structure would occupy, if it were to extend infinitely, can be calculated by selecting a small repeating section from the overall pattern (a unit cell), and then for the hexagonal packing, dividing the area of the circle by the area of the hexagon or π by 2**√**3 = .90689. For the square packing, density can be calculated by** **dividing the square by the inscribed circle or using π*.5^2 = .78539.

Figure 2: Hexagonal and tetragonal unit cells.

Moving forward from only 1-direction, packings can be made from layers of cylinders with axes that are not all parallel, but do all lay** **in parallel planes (coplanar). Below in Figure 3, the simplest 2-layer, tetragonal cylinder packing, it has the exact same density (.78539) as the 1-directional tetragonal packings.

Figure 3: 2-layer cylinder packing.

In the next blog post, we will add a dimension and look at some non-coplanar cylinder arrangements that have cubic symmetry and 3 directions!

Copyright 2021 Anduriel Widmark