Figure 1

The first four densest sphere packings in a cylinder.Reuse & Permissions

Figure 2

(Top) The first five solutions of the circle packing problem. (Bottom) Five columnar structures corresponding to the first five solutions of the circle packing problem. The unit cells are shown in yellow (light gray) and blue (dark gray).Reuse & Permissions

Figure 3

Volume fraction of the densest packing as a function of $D/d$. Discontinuities in the derivative are indicated by vertical lines. Red dotted lines indicate solutions to the circle packing problem. The main graph shows results in the range $1.8<D/d<2.71486$; the inset are continuations of the same graph showing the regions $1<D/d<1.8$ (left) and $2.71486<D/d<2.873$ (right). Structures above $D=2.71486$ include internal spheres while those below do not; the division between these two regions is denoted by a heavy blue dashed line.Reuse & Permissions

Figure 4

The numerically computed derivative (finite difference) of the volume fraction curve, as shown in Fig. 3.Reuse & Permissions

Figure 5

A schematic representation illustrating variations in the volume fraction of the densest packing arrangements as a function of $D/d$ (see text).Reuse & Permissions

Figure 6

A plot of the chiral index $\chi (D/d)$, which measures the degree of mismatch between a packing and its mirror image (see text). Vertical lines indicate structural transitions as in Fig. 3. The main graph on the left shows results in the range $1.8<D/d<2.71486$ while the graph on the right is a continuation but focuses on the region $2.71486<D/d<2.873$.Reuse & Permissions

Figure 7

Maximal contact structures, with the corresponding “rolled-out” pattern of contacts. The vector V (corresponding to the perimeter of a cylinder cross section) is indicated by the red arrows.Reuse & Permissions

Figure 8

Line slip structures at arbitrarily chosen values of $D/d$ within their respective ranges, with corresponding rolled out patterns of contacts. The vector V is indicated by an arrow.Reuse & Permissions

Figure 9

(a) symmetric packing [5, 4, 1], the black arrow represents the periodicity vector; (b) the square packing [5, 4]; (c) the symmetric packing [9, 5, 4] which is connected by an affine shear to [5, 4, 1]; (d & e) show a line slip connecting [5, 4, 1] to the symmetric packing [6, 4, 2], which is shown in (f).Reuse & Permissions

Figure 10

Surface density $\Pi$ of disk packings on the plane which are consistent with wrapping onto a cylinder of diameter $\left|\mathbf{V}\right|/\pi$. Asymmetric packings are shown by (red) dashed curves; line-slip solutions are (black) continuous curves. Symmetric structures labeled $[l,m,n]$ correspond to points of maximum density while intermediate asymmetric packings are labeled using the rhombic notation $[p,q]$.Reuse & Permissions

Figure 11

A plot of Eq. (4) for various values of $D/d$, showing the approach toward elliptical, and eventually circular, packing as $D/d$ increases beyond 2.Reuse & Permissions

Figure 12

Comparison of simulated and analytic volume fractions for the densest packings—upper and lower curves, respectively. Dotted lines are a guide for the eye to the detailed correspondence. In the analytic results, the line-slip structures are identified by black continuous lines, and the red dashed-dotted lines denote asymmetric packing structures. The numbers 1, 2, and 3 (see text) denote the type of line slip observed (upper curve) or predicted (lower curve) with $1\setminus 2$ denoting a degenerate case. The shaded region on the right is the continuation of the single layered structures for $D/d>2.71486$.Reuse & Permissions

Figure 13

Structures with internal spheres. In the case of structures which are not maximal contact the images are produced from numerical results for arbitrarily chosen values of $D/d$ within their ranges. The central image shows each structure viewed side-on. The image on the right shows the inner layer by making the outer layer transparent. The image on the left, for each packing, is the rolled out structure generated by the outer layer of spheres which contact the cylindrical boundary - the periodicity vector $\mathbf{V}$ is indicated by the red arrow.Reuse & Permissions

Figure 14

In the dry foam equivalent of the zigzag structure (often called the staircase structure) equal-volume gas bubbles are separated by thin liquid films. (a) Computer simulation using the surface evolver software of Ken Brakke [23]. (b) Photograph of bubbles in a cylinder of about 1 cm in diameter.Reuse & Permissions