Complexity in Surfaces of Densest Packings for Families of Polyhedra

Packings of hard polyhedra have been studied for centuries due to their mathematical aesthetic and more recently for their applications in fields such as nanoscience, granular and colloidal matter, and biology. In all these fields, particle shape is important for structure and properties, especially upon crowding. Here, we explore packing as a function of shape. By combining simulations and analytic calculations, we study three two-parameter families of hard polyhedra and report an extensive and systematic analysis of the densest known packings of more than 55 000 convex shapes. The three families have the symmetries of triangle groups (icosahedral, octahedral, tetrahedral) and interpolate between various symmetric solids (Platonic, Archimedean, Catalan). We find optimal (maximum) packing-density surfaces that reveal unexpected richness and complexity, containing as many as 132 different structures within a single family. Our results demonstrate the importance of thinking about shape not as a static property of an object, in the context of packings, but rather as but one point in a higher-dimensional shape space whose neighbors in that space may have identical or markedly different packings. Finally, we present and interpret our packing results in a consistent and generally applicable way by proposing a method to distinguish regions of packings and classify types of transitions between them.

Popular Summary

Calculating the densest way to pack objects in space is known as the packing problem and has intrigued scientists and philosophers for millennia. The solution is notoriously difficult to prove mathematically; instead, researchers optimize numerical and analytical computations and can usually, at best, claim to have found the highest known packing density for a given shape. The difficulty and complexity of the calculations mean that little is known about how packing changes with shape. Such knowledge is important in industry, where tailoring the shape of particles for optimal packing efficiency can minimize costs. In addition, a better understanding of how shape affects the way particles pack could lead to new shape-shifting materials that are able to toggle between different structures and properties.

Here, we take a comprehensive approach to understanding packing density as a function of shape by using analytical and computational methods to calculate the densest known packing for over 55,000 polyhedra. Our approach is distinguished from earlier work in that we look at the effects of changing two parameters (as opposed to only one) in shape space, namely, the degree to which the edges and vertices of a polyhedron have been truncated. We are therefore able to explore the complexity and diversity of shape effects that may have been missed in earlier studies.

We represent our findings as surfaces of maximum densities for related shapes and find both smooth regions (indicating that the changes in packing density and packing structures evolve continuously with shape change) and highly corrugated neighborhoods (indicating a discontinuous evolution of the change in packing density and packing structures). Our key finding is that minor changes in particle shape can cause major changes in packing: both in how the shapes pack and how densely they pack. Finally, we present general principles for navigating this complex problem, where we advocate thinking of shape not as a static property of an object but as merely a point in a high-dimensional shape space.

Figure 1

The sum body (left) and difference body (middle) for a noncentrally symmetric polyhedron. A Kuperberg pair for the same polyhedron (right).

Figure 2

Example of Minkowski lattice types $G^{6-}$, $G^{7+}$, and $G^{6+}$ (left to right) for a noncentrally symmetric polyhedron. Lines connect the reference polyhedron at the origin to its neighbors’ centers. The lines indicate the color of the faces of the reference polyhedron that touches the neighboring polyhedra. The lines are darker shades of {magenta, cyan, and yellow} to aid the eye. The packings correspond to the $323·1$-family region ${\rho }_4$, the ridge ${\rho }_4\wedge {\rho }_0$, and the region ${\rho }_0$ (left to right).

Figure 3

Left column: Representative polyhedra for the families 523, 423, $323+$, and $323-$. The two pairs of diagonally opposite corners correspond to dual polyhedra. The 323 family also has reflection symmetry about the diagonal $a=c$. Middle column: The direction of face-normal vectors $\{\alpha ,\beta ,\gamma \}$, the parameter range $⟨a,c⟩$, and the names of the corner polyhedra. Right column: A color map of the surface-area fraction of face types. In all columns, the color {magenta, yellow, and cyan} indicates the type of face $\{\alpha ,\beta ,\gamma \}$.

Figure 4

Boundaries between adjacent regions of topologically distinct packings are classified into three boundary types: valley (left), ridge (middle), and tangent (right).

Figure 5

523-family optimal density surface ($3\times 3$ grid of cubes), parallel contacts (bottom left), face areas, and region size (bottom center).

Figure 6

423-family optimal density surface ($3\times 3$ grid of cubes), parallel contacts (bottom left), face areas, and region size (bottom center).

Figure 7

$323·1$-family optimal density surface ($3\times 3$ grid of cubes), parallel contacts (bottom left), face areas, and region size (bottom center).

Figure 8

$323·3$-family optimal density surface ($3\times 3$ grid of cubes) and face areas (bottom center).

Figure 9

$323·2$-family optimal density surface ($3\times 3$ grid of cubes), parallel contacts (bottom left), antiparallel contacts (bottom right), face areas, and region size (bottom center).

Figure 10

$323·4$-family optimal density surface ($3\times 3$ grid of cubes), parallel contacts (bottom left), antiparallel contacts (bottom right), face areas, and region size (bottom center).

Figure 11

Sum body $323+$ (top center) and difference body $323-$ (bottom center). 323-family face-face contacts: yellow to yellow between antiparallel neighbors (top left), magenta to magenta and cyan to cyan between antiparallel neighbors (top right), yellow to yellow between parallel neighbors (bottom left), and magenta to cyan and cyan to magenta between parallel neighbors (bottom right).

Figure 12

Contours of optimal packing density $\varphi$ for the 523 family, numeric (left) versus analytic (right). The color gradient {blue, magenta, red, yellow, green, cyan, and blue} represents the $\varphi$ gradient $\{\varphi -\frac{3}{60},\varphi -\frac{2}{60},\varphi -\frac{1}{60},\varphi ,\varphi +\frac{1}{60},\varphi +\frac{2}{60},\varphi +\frac{3}{60}\}$; yellow contours occur at $\varphi =\{\frac{8}{10},\frac{9}{10}\}$.

Figure 13

Contours of optimal packing density $\varphi$ for the 423 family, numeric (left) versus analytic (right). The color gradient {blue, magenta, red, yellow, green, cyan, and blue} represents the $\varphi$ gradient $\{\varphi -\frac{3}{60},\varphi -\frac{2}{60},\varphi -\frac{1}{60},\varphi ,\varphi +\frac{1}{60},\varphi +\frac{2}{60},\varphi +\frac{3}{60}\}$; yellow contours occur at $\varphi =\{\frac{8}{10},\frac{9}{10},1\}$.

Figure 14

Contours of optimal packing density $\varphi$ for the $323·1$ family, numeric (left) versus analytic (right). The color gradient {blue, magenta, red, yellow, green, cyan, and blue} represents the $\varphi$ gradient $\{\varphi -\frac{3}{60},\varphi -\frac{2}{60},\varphi -\frac{1}{60},\varphi ,\varphi +\frac{1}{60},\varphi +\frac{2}{60},\varphi +\frac{3}{60}\}$; yellow contours occur at $\varphi =\{\frac{4}{10},\frac{5}{10},\frac{6}{10},\frac{7}{10},\frac{8}{10},\frac{9}{10},1\}$.

Figure 15

Contours of optimal packing density $\varphi$ for the $323·3$ family, numeric (left) versus analytic along the symmetry diagonal (right). The color gradient {blue, magenta, red, yellow, green, cyan, and blue} represents the $\varphi$ gradient $\{\varphi -\frac{3}{60},\varphi -\frac{2}{60},\varphi -\frac{1}{60},\varphi ,\varphi +\frac{1}{60},\varphi +\frac{2}{60},\varphi +\frac{3}{60}\}$; yellow contours occur at $\varphi =\{\frac{7}{10},\frac{8}{10},\frac{9}{10},1\}$.

Figure 16

Contours of optimal packing density $\varphi$ for the $323·2$ family, numeric (left) versus analytic (right). The color gradient {blue, magenta, red, yellow, green, cyan, and blue} represents the $\varphi$ gradient $\{\varphi -\frac{3}{60},\varphi -\frac{2}{60},\varphi -\frac{1}{60},\varphi ,\varphi +\frac{1}{60},\varphi +\frac{2}{60},\varphi +\frac{3}{60}\}$; yellow contours occur at $\varphi =\{\frac{8}{10},\frac{9}{10},1\}$.

Figure 17

Contours of optimal packing density $\varphi$ for the $323·4$ family, numeric (left) versus analytic (right). The color gradient {blue, magenta, red, yellow, green, cyan, and blue} represents the $\varphi$ gradient $\{\varphi -\frac{3}{60},\varphi -\frac{2}{60},\varphi -\frac{1}{60},\varphi ,\varphi +\frac{1}{60},\varphi +\frac{2}{60},\varphi +\frac{3}{60}\}$; yellow contours occur at $\varphi =\{\frac{9}{10},1\}$.