PHYSICAL REVIEW X 4, 011024 (2014)

Complexity in Surfaces of Densest Packings for Families of Polyhedra

† ‡ Elizabeth R. Chen,1,2,* Daphne Klotsa,2, Michael Engel,2 Pablo F. Damasceno,3 and Sharon C. Glotzer2,3,4, 1School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA 2Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA 3Applied Physics Program, University of Michigan, Ann Arbor, Michigan 48109, USA 4Department of Materials Science and Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA (Received 10 September 2013; revised manuscript received 20 December 2013; published 25 February 2014) 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.

DOI: 10.1103/PhysRevX.4.011024 Subject Areas: Materials Science, Soft Matter

I. INTRODUCTION over 400 years ago [26], was proven by Hales in 2005 [1,27,28]. Besides the sphere and objects that tile space The optimization problem of how to pack objects in (i.e., fill space completely without gaps or overlaps), no space as densely as possible has a long and colorful history mathematical proofs have been found and results are [1–3]. Packing problems are both easy to grasp and obtained numerically, which means that other, denser notoriously hard to solve mathematically, qualities that packings may exist (e.g., aperiodic, large primitive unit have made them interesting recreational math puzzles [4]. cells). Throughout this paper, all of the densest packings Recent work on nanoparticle and colloidal self-assembly reported or cited refer to the densest known packings. [5–7], micrometer molecule analogs [8], reconfigurability Motivated by the great diversity of nanoparticle shapes [9–17], and jammed granular matter [18,19], as well as that can now be synthesized [7,29,30], many groups have biological cell aggregation [20,21] and crowding [22,23], studied dense packings of highly symmetric polyhedra has motivated further the study of packing. Packing in (Platonic solids, Archimedean solids, and others) [31–38]. containers has a broad range of applications in operations Yet, finding the densest (optimal) packing is challenging research, such as optimal storage, packaging, and trans- even for seemingly simple shapes such as the tetrahedron portation [24,25]. [32–35,39,40]. Recent experiments have gone even further Despite significant progress in the study of packing, by synthesizing nanoparticles—specifically, nanocubes knowledge remains patchy and focuses on a few selected (superballs) [13], whose shape can be tuned from a cube shapes with high symmetry. The densest packing of the to an octahedron via a sphere—generating homologous sphere, known as the Kepler conjecture and formulated families of shapes whose packings may vary with shape. To date, most theoretical and/or computational studies have *[email protected] reported the densest known packings for shape deforma- † [email protected] tions of one-parameter families (i.e., one “axis” in “shape ‡ [email protected] space”). Examples include ellipsoids [41], superballs [13,42], puffy tetrahedra [43], concave n-pods [36] and Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distri- bowls [44], convex shapes characterized by aspect bution of this work must maintain attribution to the author(s) and ratios [45], as well as truncated polyhedra, such as the the published article’s title, journal citation, and DOI. tetrahedron-octahedron family [37], the octahedron-cube

2160-3308=14=4(1)=011024(23) 011024-1 Published by the American Physical Society CHEN et al. PHYS. REV. X 4, 011024 (2014) family [46], and tetrahedral dimers [38]. Many of these objects Ξ in a container or in infinite Euclidean space. It is, recent studies report a diversity of densest packings as a in general, an intractable problem that does not allow a functionofshape,resultinginatopographicallycomplexline rigorous analytic treatment or numerical search. When through what is actually a high-dimensional shape space. packing in containers, the problem depends on a finite The behavior of higher-dimensional maximum-density sur- number of object positions and orientations and can often faces in shape space obtained by varying two or more shape be solved via a brute-force search [50]. Packing in infinite parameters simultaneously, as we do in this paper, affords a space requires the optimization of an infinite number of more in-depth look at the role of shape in packing. In variables. Here, we focus on the packing of identical convex particular, such a study allows for the identification and objects (convex particles) in infinite Euclidean space. definition of topographical features (regions, types of boun- Because all known densest packings of convex objects are daries). These definitions characterize the density surfaces periodic, we restrict our search to those. The packing density and facilitate the comparison among different packing of a periodic packing is ϕ ¼ nU=V, where U is thevolume of studies. the object, V is the volume of the unit cell of the lattice, and n In this paper, we investigate the packing problem for is the number of objects in the unit cell. three two-parameter families of symmetric convex poly- In this section, we give a theoretical background on hedra. Our families interpolate between edge-transitive mathematical concepts, tools, and theories that have been polyhedra via continuous vertex and/or edge truncations. developed to solve packing problems and that we use in our The interpolation goes through various solids (Platonic, approach in this paper. Specifically, we introduce (i) def- Archimedean, Catalan), thereby including some of the initions for the sum and difference bodies (from set theory), above-referenced one-parameter studies as linear paths where polyhedra are defined as sets of points comprising (subfamilies) on our two-parameter surfaces. The maxi- the shape, (ii) Minkowski’s theory on packings [47,48] that mum packing density forms surfaces in shape space that have one particle in the primitive unit cell, and (iii) the reveal great diversity in richness and complexity. Our Kuperbergs’ extension of Minkowski’s theory to include results demonstrate that some not previously studied paths packings that have two antiparallel particles in the primitive through shape space give a plethora of consecutive distinct unit cell [49]. packing structures through a series of transitions (as the shape deforms), whereas other paths give the same or similar packings with no or few transitions. Given the A. Sum and difference bodies richness of the surfaces of the densest packings, we aim to Given two particles Ξ and Z, the sum body and the standardize the way packing results are presented and difference body are the sets of sums and differences of all interpreted in the community by doing the following. points in the particles: Based on the theories of Minkowski [47,48] and the Kuperbergs [49], we define regions of topologically equiv- Ξ þ Z ¼fξ þ ζ∶ ξ ∈ Ξ; ζ ∈ Zg; alent packings from their intersection equations (contacts Ξ − Z ¼fξ − ζ∶ ξ ∈ Ξ; ζ ∈ Zg. with nearest neighbors) as opposed to their Bravais lattice type or symmetry group. We thus define and classify three See Fig. 1 for an example of the sum and difference bodies types of boundaries between adjacent regions (valley, ridge, tangent). The classification is general for continuous for a noncentrally symmetric polyhedron. For any particle Ξ, we define positive orientation as families of shapes. We analyze previous works and argue þΞ ¼ Ξ and negative orientation −Ξ by inversion at the that packing problems can be treated consistently using our Ξ Z framework. origin. Two particles and are parallel, if there exists a vector ς such that Z ¼ ς þ Ξ. They are antiparallel, if there The paper is organized as follows: Section II introduces ς Z ς − Ξ Ξ some of the theoretical concepts and mathematical tools that exists a vector such that ¼ . Two particles and Z are in contact, if their intersection is equal to the have been formulated over the years for packing problems. Ξ∩Z ∂Ξ∩∂Z We construct three two-parameter families of symmetric intersection of their boundaries ¼ . From polyhedra in Sec. III. Section IV describes our analytical, basic set theory, we know the following. Ξ numerical, and computational methods. We show results for (i) If a convex particle (centered at 0) and a parallel neighbor ς þ Ξ (centered at ς) touch, then ς lives on the surfaces of maximum packing density in Sec. Vand close Ξ − Ξ with a comparison with other studies in Sec. VI, and a the surface of the difference body (see the Appendix, Fig. 11, top left and right). summary of our main points and conclusion appear in Ξ Sec. VII. (ii) If a convex particle (centered at 0) and an anti- parallel neighbor ς − Ξ (centered at ς) touch, then ς lives on the surface of the sum body Ξ þ Ξ (see the II. THEORETICAL BACKGROUND Appendix, Fig. 11, bottom left and right). The packing problem is an optimization problem that (iii) The sum body of particle Ξ is always convex, even searches for the densest possible packing arrangement of if Ξ is not.

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FIG. 1 The sum body (left) and difference body (middle) for a noncentrally symmetric polyhedron. A Kuperberg pair for the same polyhedron (right).

(iv) For a convex particle Ξ, the sum body has the same of three possible types with either six or seven pairs shape but twice the size: Ξ þ Ξ ¼ 2Ξ (see Fig. 1, left). of neighbor contacts: (v) The difference body of particle Ξ is always centrally symmetric, even if Ξ is not (see Fig. 1, middle). G6− ¼fχ; ψ; ω;ψ − ω; ω − χ; χ − ψg; (vi) For a centrally symmetric particle Ξ, the difference G6þ χ; ψ; ω;ψ ω; ω χ;χ ψ ; body equals the sum body. ¼f þ þ þ g Examples of parallel and antiparallel neighbors of a G7þ ¼fχ; ψ; ω;ψ þ ω; ω þ χ;χ þ ψ; χ þ ψ þ ωg. noncentrally symmetric polyhedron, whose centers are on the difference and sum bodies, are shown in the Appendix, Fig. 11. Examples of the Minkowski lattice types G6−, G7þ, and G6þ for a noncentrally symmetric polyhedron are shown B. Minkowski lattices in Fig. 2. A lattice packing is a packing with one particle in the We make two comments regarding the relation of the primitive unit cell (n ¼ 1). It has been observed that the Minkowski lattices (and all packings reported in this paper) densest known packing for many convex particles with to Bravais lattices employed in crystallography. First, in central symmetry is a lattice packing [33,36–38], but this is connection to the unit cell, we choose vectors that form a not generally true [41]. The densest packings of non- basis for the lattice with a minimum positive determinant centrally symmetric shapes frequently require two or more (lattice volume), and thus we always use the primitive particles in the primitive unit cell. unit cell (smallest repeated unit). In crystallography, it is common to define basis vectors so that they are close to a The theory of lattice packings was originally developed π π by Minkowski [47,48], who described packings consider- cubic lattice (angles between 3 and 2) but do not necessarily ing the contacts of a particle with its neighbors. Chen [51] correspond to the primitive unit cell; see Fig. S1 used this method to study the densest packings of various (Supplemental Material [52]). Second, in connection to 6− 6 7 shapes. In a lattice packing, all particles are parallel and the lattice, the Minkowski lattice type G , G þ,orG þ have identical neighborhoods related by translations, which refers to the number and positions of the reference-particle are linear combinations with integer coefficients of the contacts with its neighbors. The Bravais lattice type is not lattice vectors fχ; ψ; ωg. Minkowski proved the following. equivalent because it refers to particle centers. We can draw (1) The densest lattice packing (Minkowski lattice) of a a connection if we consider the space-filling packing of convex particle Ξ is always identical to the densest Voronoi polyhedra for the face-centered-cubic (fcc) and the lattice packing of its difference body Ξ − Ξ. body-centered-cubic (bcc) packings, which are the rhombic This theorem is important because it is easier to dodecahedron and the truncated octahedron, respectively. (i) mathematically manipulate centrally symmetric We can then think of the fcc packing as an example of G6− shapes and (ii) write down intersection equations in (with 12 contacts) and the bcc packing as an example of terms of the difference body [see Sec. IV B and the G7þ (with 14 contacts). Supplemental Material [52] (p. 6–42)]. We note that Betke and Henk [31] developed another (2) For all centrally symmetric convex shapes (in three method to calculate the densest lattice packings that makes dimensions), a Minkowski lattice can always be use of Minkowski’s result (see point 1 in Sec. II B). Their chosen such that each particle Ξ is in contact with algorithm systematically checks feasible combinations of either 12 or 14 neighbors and the lattice satisfies one vectors on the faces of the difference body of the reference

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FIG. 2 Example of Minkowski lattice types G6−, G7þ, and G6þ (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 ρ4, the ridge ρ4∧ρ0, and the region ρ0 (left to right). shape, where the neighbors’ centers live, and is thus a construction are that (i) it gives us a continuous function deterministic, not probabilistic. of shape, so we can then use as fine a grid as we wish to study packing properties, and (ii) the maximum-density C. Kuperberg pairs surface that we calculate is also a continuous function of shape and thus easier to manipulate mathematically. A double-lattice packing of a convex particle Ξ has two antiparallel particles in the primitive unit cell (n 2) Ξ ¼ þ A. Spheric triangle groups and δ − Ξ that form a Kuperberg pair [49]; see Fig. 1 (right). The Kuperbergs extended Minkowski’s theory to Weintroducefamiliesofpolyhedrathatinterpolatebetween double-lattice packings. They showed that the pair often various symmetric solids (Platonic, Archimedean, Catalan) packs densely for particles without central symmetry and is via truncation. The amount of truncation is varied in each a candidate for the solution of the general packing problem family by shifting the truncation planes radially in a manner in situations when lattice packings are not a good solution. respecting a centrosymmetric point-symmetry group. Such Chen et al. [34,51] and Haji-Akbari et al. [38] used symmetry groups are known as (finite) spheric triangle groups Δ Kuperberg pairs in their studies of the densest packings. p;q;r. A spheric triangle group is generated by three reflec- π ; π ; π Exceptions where the densest packings are realized in tionsacrossthesidesofaspherictrianglewithanglesfp q rg. neither the Minkowski lattice nor a Kuperberg pair are The finite irreducible spheric triangle groups are Δ3;2;3 known. For example, the densest packing of ellipsoids (tetrahedral, Schönflies notationTd),Δ4;2;3 (cubic/octahedral, has n ¼ 2 but is not a double-lattice packing because the Oh), and Δ5;2;3 (dodecahedral/icosahedral, Ih) [54]. two ellipsoids in the primitive unit cell are not antiparallel [41]. The densest packing of tetrahedra requires n ¼ 4 B. Truncation planes [34,35,37,38]. In two dimensions, space-filling packings Three two-parameter families of polyhedra are con- (tilings) of pentagons are known with n 2,3,4,6,8[53]. ¼ structed by truncating the vertices and edges of the dodecahedron or icosahedron (523 family), the cube or III. CONSTRUCTION OF TWO-PARAMETER octahedron (423 family), and the tetrahedron (323 family). FAMILIES OF POLYHEDRA The results are three types of equivalent face-normal α; β; γ p; q; r In order to define a packing problem, we need to specify vectors f g that are axes of f g-fold symmetry 2π ; 2π ; 2π the container and the objects. Here, we focus on packing in (rotation by angles f p q r g). The triangle group maps infinite Euclidean space, so there is no container. The other any axis to any other axis of the same type. We define the Ξ parameter we need to specify is the object or particle. polyhedron as the intersection of half-spaces for all a; b; c Because the maximum packing density is a function of the normal vectors, where the parameters f g specify the geometric shape of the particles, it is useful to have an amount of truncation or position of the bounding plane: analytical and continuous way to describe shape, i.e., an 8 > N-parameter family of shapes. An N-parameter family of <> ξ · α ≤ a N 3 three-dimensional shapes is a function F∶R → R . The Ξ ξ · β ≤ b N > parameters X ¼hX1; …;XNi ∈ R then represent specific :> ξ · γ ≤ c: operations on the shape Ξ ¼ FðXÞ. The advantages of such

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FIG. 3 Left column: Representative polyhedra for the families 523, 423, 323þ, and 323−. The two pairs of diagonally opposite corners correspond to dual polyhedra. The 323 family also has reflection symmetry about the diagonal a ¼ c. Middle column: The direction of face-normal vectors fα; β; γg, the parameter range ha; ci, 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 fα; β; γg.

011024-5 CHEN et al. PHYS. REV. X 4, 011024 (2014) 8 The normal vectors and parameter ranges for the three < ζ · α ≤ a þ a families are given by Z Ξ Ξ ζ β ≤ b b ¼ þ : · þ 8 ffiffiffi > p ζ · γ ≤ c þ c; <> α ¼h1; 0;si 1 ≤ a ≤ s 5 523 β 2; 0; 0 2 b > ¼h i ¼ 8 :> ζ α ≤ a c γ ¼hS; S; Si S2 ≤ c ≤ 3; < · þ pffiffiffi pffiffiffi Z ¼ Ξ − Ξ ζ · β ≤ b þ b S ¼ 1ð 5 þ 1Þ;s¼ 1ð 5 − 1Þ; : 2 2 ζ · γ ≤ c þ a: 8 < α ¼h1; 0; 0i 1 ≤ a ≤ 2 IV. METHODS 423 β 1; 1; 0 2 b : ¼h i ¼ For each family, we analyze the densest packings in two γ ¼h1; 1; 1i 2 ≤ c ≤ 3; steps. First, we generate dense packings using Monte Carlo simulations by compressing a small number of n particles ( with periodic boundary conditions. We then use these α ¼ −h1; 1; 1i 1 ≤ a ≤ 3 results as a guide to construct an analytic surface of 323 β ¼h1; 0; 0i 1 ¼ b maximum packing density (as in Ref. [38] for one shape γ ¼þh1; 1; 1i 1 ≤ c ≤ 3. parameter). For polyhedra with central symmetry (523, 423, and We fix b to be constant because the packing density 323− families), we investigate only lattice packings because remains invariant if we rescale all parameters by the same the densest known packings of centrally symmetric shapes scalar. Figure 3 shows representative polyhedra on the are most likely found to be lattice packings [33,36–38]. square ha; ci parameter domain. Note that at the boundaries For polyhedra without central symmetry (323þ family), we of the square domain, the polyhedra have degeneracies study packings with n ¼ 1, 2, 3, 4 particles in the primitive (faces with zero area, edges with zero length, coincident unit cell. In the following, we use the notation 323 · n for a points). If we include the degeneracies, all family members packing of a shape in the 323 family with n particles in the have the same number of faces, edges, and vertices. If we primitive unit cell. exclude the degeneracies, the polyhedra at the corners of the domain are edge transitive. Formulas for the polyhedron A. Monte Carlo simulated compression U α; β; γ volume and face areas f g are given in the Our computational techniques closely follow previous Supplemental Material [52] (p. 5). works [37,38,55,56]. We study small systems of n identical polyhedra in a box of volume V with periodic boundary C. Sum and difference bodies conditions. The particle positions and orientations evolve in Most polyhedra in the pqr family have a point- time according to a Monte Carlo trial-move update scheme, symmetry group that is identical to the triangle group where polyhedra are chosen randomly and then rotated and translated by a random amount. In addition, the simulation Δp;q;r. The only exceptions occur in the 323 family, where the polyhedra with central symmetry are also members of box is updated in the isobaric-isotension ensemble by randomly perturbing the coordinates of the three box the 423 family, so they have the higher Δ4;2;3 symmetry. In fact, 323 is the only family with noncentrally symmetric vectors. Strong elongations of the box vectors are avoided by using a lattice-reduction technique. Trial moves are polyhedra. We derive two subfamilies: (i) the family of sum accepted if the generated configurations are free of overlaps bodies 323þ that is identical to 323 and (ii) the family of and rejected otherwise. Overlap checks are performed difference bodies 323− that corresponds to 323 along the using the Gilbert-Johnson-Keerthi (GJK) distance algo- diagonal a ¼ c and does not change in the orthogonal rithm [57]. In contrast to previous works, here, self- direction a þ c ¼ 0. overlaps are accounted for with periodic copies due to The sum and difference bodies of polyhedra in the 523 the small dimensions of the simulation box. Compared and 423 families with central symmetry are identical: to other compression techniques in the literature 8 [33,36,58,59], our scheme consistently finds equivalent < ζ · α ≤ 2a or denser packings; see Figs. S2 and S3 in the Z 2Ξ ζ β ≤ 2b ¼ : · Supplemental Material [52] for a comparison. ζ · γ ≤ 2c: For each of the three families, we choose polyhedra from a fine 101 × 101-parameter grid in the ha; ci-parameter For the 323 family without central symmetry, the sum domain. In some parts of the domain, where the packings (323þ) and difference (323−) bodies are different: change rapidly with parameters ha; ci, we apply a finer grid

011024-6 COMPLEXITY IN SURFACES OF DENSEST PACKINGS … PHYS. REV. X 4, 011024 (2014) to achieve a higher resolution. Although there is some Because the particles in Kuperberg pairs are related to overlap between the shapes of the 423 and 323 families, the one another by simple inversion, all intersection equations total number of unique shapes simulated for this study is in the n ¼ 1, 2, 4 cases are linear and can be solved more than 55 000. The simulation is initialized at low analytically. However, for arbitrary shapes (e.g., truncated density and then slowly compressed by gradually increas- triangular bipyramids [38]) and n ≥ 3, rotations between ing the pressure using an exponential protocol over 7 × 105 neighboring particles can result in a system of quadratic steps. Because our approach resembles the simulated intersection equations. For example, in the 323 · 3 family annealing technique replacing temperature with pressure, the nonlinear intersection equations must be solved we call it “simulated compression.” Each compression run numerically. is repeated 10 times, and the densest packing is recorded for The intersection equations of all analyzed families are each parameter choice. The result of the algorithm is a shown in the Supplemental Material [52] (pp. 6–42). numerical candidate for the densest packing function over Numeric versus analytic maximum packing densities are the two shape parameters. compared in the Appendix, Figs. 12–17. In all cases, we verify that (i) the packing-density data B. Analytic optimization from simulated compression are always a lower bound to the analytic results and (ii) adjacent regions match up We use the densest packings from simulated compres- correctly. sion as a guide to analytically construct small primitive unit-cell packings that are locally optimal under rotations and translations of the particles. C. Classification of packings For a lattice packing (n ¼ 1), we perform the following. We classify the densest packings based on the types of (1) We analyze the neighbor contacts in the densest contacts between neighboring particles in the primitive packings obtained with simulated compression. unit cell. There are six topological types of contact: (2) We write the (abstract) intersection equations in vertex-vertex, vertex-edge, vertex-face, edge-edge, edge- terms of the lattice basis vectors fχ; ψ; ωg and face, and face-face. Each contact is mathematically difference-body faces (where the neighbors’ centers expressed in the form of an intersection equation. We live), which are functions of the shape parame- refer to packings with the same topological types of ters ha; ci. contacts and intersection equations as topologically (3) We reduce the parameters fχ; ψ; ω;a;cg to a min- equivalent. Note that the contacts map to each other imal set of free parameters. via isometry between lattice vectors and/or isomorphism (4) If the (abstract) lattice volume V ¼ det½χ; ψ; ω has between basis vectors. The equivalence relation partitions free parameters, we find the values of the free the domain of each N-parameter family of packings into parameters that minimize V and therefore maximize equivalence classes that we call packing regions and the packing density ϕ ¼ nU=V. denote by the symbol ρi. Within each region, the packing Ingeneral,for agivenlattice,therearemultiplewaystochoose density varies continuously and smoothly with shape a set of basis vectors that generates the lattice. For centrally parameters X because the intersection equations and symmetric shapes, we choose basis vectors to satisfy one therefore the lattice vectors are smooth (algebraic) func- of the three Minkowski types. This choice simplifies the tions of the shape parameters X. At the boundaries optimization procedure and maintains consistency. between adjacent regions, the packing density might be For the double-lattice packings (n ¼ 2)in323 · 2,we nonsmooth (discontinuous derivative). The packing den- repeat the same process as for the lattice packing with sity ϕðXÞ is necessarily continuous everywhere on the the offset vector ς as an additional parameter and we use domain. the difference body for parallel neighbors and the sum When crossing between adjacent regions, a minimal body for antiparallel neighbors. By coincidence, the set of intersection equations changes. This minimal set n ¼ 4 case (323 · 4 family) reduces to double-lattice pack- depends on the symmetries of adjacent packings and the ings. The only exception is an area near the tetrahedra number of particles in the primitive unit cell. We use the (ha; ci¼h1; 3i and h3; 1i) that is a double-lattice packing discontinuities of the first derivative of ϕðXÞ to distinguish of dimers only slightly rotated (face to face, almost edge three boundary types, as depicted in Fig. 4. We define the to edge). Note that we use a packing of two monomers, following. whereas the Kuperbergs [49] use a packing of one dimer (1) A valley ½ϕi∨ϕj is a “soft” boundary. Extending (Kuperberg pair), and thus there are additional degrees of beyond a valley is allowed but gives suboptimal freedom in our packing. The packing of the 323 · 2 family packings. A valley is a generic boundary; inter- (and most of the 323 · 4 family) is a combination of the section equations of adjacent regions need not be sum-body 323þ and difference-body 323− packings, in related, so the lattice vectors can change discon- that the neighbors in the densest packing are either parallel tinuously. We can write the optimal density for a or antiparallel. valley as

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FIG. 4 Boundaries between adjacent regions of topologically distinct packings are classified into three boundary types: valley (left), ridge (middle), and tangent (right).

  ϕ ϕ ≥ ϕ ϕ ∂ϕ ≤ ∂ϕ i i j ϕ ϕ ∽ϕ i i j ϕi∨j ¼½ϕi∨ϕj¼maxfϕi;ϕjg¼ (1) i∽j ¼½ i j¼ ϕ ∂ϕ ≥ ∂ϕ : (4) ϕj ϕi ≤ ϕj. j i j

(2) A ridge ½ϕi∧ϕj is a “hard” boundary. Extending beyond a ridge introduces overlaps into the packing, V. RESULTS which is not allowed. The lattice volume and lattice The results from simulated compression and analytic vectors vary continuously but not smoothly across a optimization are presented in Figs. 5–10. In each figure, the ridge. On the ridge, the set of contacts is the union of top 3 × 3 grid of images shows the surface of maximal the sets of contacts on both sides. Thus, the set 18 packing density ϕha; ci, 49 ≤ ϕ ≤ 1, in 3D space ha; c; ϕi. of intersection equations is the union of the sets of The normal vector of the surface of maximum packing intersection equations on both sides. In other words density maps to the color sphere: white along the north pole that there are more contacts on the ridge than on (ϕ direction), bright colors along the equator. The central either side. We can think of a ridge as a lower- cube has perspective from the þϕ direction. The peripheral dimensional region in its own right. For a ridge cubes are rotated by a zenith angle from the central cube boundary, π (3 from the þϕ direction). a; c  The bottom row shows a top view on the h i plane of ϕi ϕi ≤ ϕj the packing regions (separated by solid lines). The colors ϕ ϕ ∧ϕ ϕ ;ϕ (2) i∧j ¼½ i j¼minf i jg¼ for each region interpolate between the three colors that ϕj ϕi ≥ ϕj. correspond to the face types in Fig. 3 {magenta, yellow, and cyan}. On the bottom left, if applicable, the numbers ϕ ∼ ϕ ϕ ≤ ϕ ϕ ∽ϕ ϕ ≥ ϕ (3) A tangent ½ i jð i jÞ or ½ i jð i jÞ indicate the Minkowski lattice type. The colors indicate is a hybrid between a valley and a ridge. Extend- which types of faces are in contact between parallel ing from the lower-density side to the higher- neighbors only. On the bottom center, the numbers label density side is allowed but gives suboptimal the regions in order of area size, 0 being the smallest. The packings. Extending from the higher-density side colors indicate the proportion of each type of face over to the lower-density side introduces overlaps. The the shape’s total surface area, thus aiding visualization of lattice volume and lattice vectors are continuous the dominant faces. Finally, if the polyhedra are not and smooth across a tangent. The intersection centrally symmetric, there is a bottom right panel. The equations of the higher-density region are a subset colors indicate which types of faces are in contact between of the intersection equations of the lower-density antiparallel neighbors only. region, and in this case, more constraints result in lower packing density. For a tangent boundary, we take partial derivatives in any direction transverse A. 523 results to the tangent boundary, from the ϕi to the ϕj side: Figure 5 shows the surface of maximum packing density for the 523 family. There are 11 regions fρig, all with  6− ϕ ∂ϕ ≥ ∂ϕ Minkowski lattice type G , which includes fcc. All boun- ϕ ϕ ∼ ϕ i i j (3) i∼j ¼½ i j¼ ϕ ∂ϕ ≤ ∂ϕ ; daries are valleys, so the overall packing-density function is j i j simply the maximum of the functions for each region

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6− 6− 6− 6− 6− 6− 6− ϕ ¼ maxfϕ10 ; ϕ9 ; ϕ8 ; ϕ7 ; ϕ6 ; ϕ5 ; ϕ4 ; exactly on the ridge. That new contact will become an ϕ6−; ϕ6−; ϕ6−; ϕ6− ; overlap if we extend beyond the ridge. There are only three 3 2 1 0 g possible types of lattice packings (G6−, G6þ, G7þ), and two of them (G6− and G6þ) have the same number of contacts. wherewe enumerate the regions (subscripts) and indicate the Thus, the only way that regions can intersect via a ridge is if Minkowski lattice type (superscripts). 6− 6 the packings have type G on one side, type G þ on the Consider the colors in the bottom left panel and the 7 other side, and type G þ at the boundary. An example of the numbers in the bottom center. Along the sequence of three Minkowski lattice types across a ridge is shown regions ρ6; ρ4; ρ5; ρ8; ρ7; ρ2; ρ3 , contacts of yellow β f g f g in Fig. 2. faces between neighbors (which are perpendicular to the twofold axes) are gradually replaced by contacts of cyan fγg faces (threefold axes). The same is true for the C. 323 family sequence fρ6; ρ9; ρ10g, where contacts of yellow faces are Figure 7 shows the surface of maximum packing density gradually replaced by contacts of magenta fαg faces for the 323⋅1 family. There are eight regions of all three (fivefold axes). There is a close similarity between the Minkowski lattice types, symmetric about the diagonal transitions in colors characterizing the face contact a ¼ c. All boundaries are valleys except for two ridges 6− 6þ 6− 6þ (bottom left panel) and the colors characterizing the face ½ϕ6 ∧ϕ1 and ½ϕ4 ∧ϕ0 . By combining regions area (bottom center panel). This similarity in the tran- separated by ridges, we can write down the overall sition in color demonstrates that the face type with the packing-density function simply as the maximum of the largest area also has the maximum number of neighbor packing-density function for each region: contacts. 7þ 7þ 7þ 7þ ϕ ¼ maxfϕ6∧1; ϕ4∧0; ϕ7 ; ϕ5 ; ϕ3 ; ϕ2 g: B. 423 results As required, ridges separate regions of Minkowski lattice Figure 6 shows the surface of maximum packing density types G6− and G6þ and have type G7þ at the boundary; for the 423 family. We find 18 regions enumerated in the see Fig. 2. bottom center panel. Figure 6 shows a complex surface, Figure 8 shows the surface of maximum packing density 6 12 where seven regions meet at the point ha; ci¼h5 ; 5 i.All for the 323 · 3 family. We have not constructed the analytic 6− 6þ boundaries are valleys except for three ridges ½ϕ15 ∧ϕ11 , surface because there are variable rotations among the 6þ 6− 6− 6þ 7þ 6þ three particles in the primitive unit cell that give quadratic ½ϕ13 ∧ϕ7 , and ½ϕ12 ∧ϕ3 and one tangent ½ϕ2 ∼ ϕ0 . We can combine regions so that they all intersect at valleys; (nonlinear) intersection equations. The figure gives an thus, the overall packing-density function is simply the impression of the noise in the raw numerical data and thus maximum of the packing-density functions for each region: demonstrates the significance of the analytic optimization for obtaining a clean surface of maximum packing density. 6− 6− 7þ The comparison of the maximum densities for 323 · 1 and ϕ ¼ maxfϕ15∧11; ϕ13∧7; ϕ12∧3; ϕ2∼0; ϕ ; ϕ ; ϕ ; 17 16 14 323 · 3 shows that three particles in the primitive unit cell 7þ 7þ 7þ 7þ 7þ 7þ 6− ϕ10 ; ϕ9 ; ϕ8 ; ϕ6 ; ϕ5 ; ϕ4 ; ϕ1 g: pack denser than lattice packings. Figure 9 shows the surface of maximum packing density Unlike the 523 family, here the colors in the bottom left for the 323 · 2 family. The surface is remarkably complex panel (characterizing the face contact) do not always follow and has reflection symmetry about the diagonal a ¼ c. the colors in the bottom center panel (characterizing the Since all packings are double-lattice packings, we can face area). The two panels transition in color in the same solve for the intersection equations analytically. There are direction across valleys and in the opposite direction across 8 × 1 þ 62 × 2 ¼ 132 regions, taking into account reflec- ridges. An example is the sequence fρ14; ρ11; ρ15g. While tion symmetry. Eight regions straddle the diagonal, and 62 the boundary between ρ14 and ρ11 is a valley and the color mirror pairs are reflected across the diagonal. All bounda- changes from cyan toward purple in both panels, ρ15 has ries are valleys except 13 ridges and 11 tangents. We find more cyan than ρ11 in the bottom left panel. This change in multiple junctions where several ridges or several tangents color means that parts of the magenta faces that were in meet at a point. contact in ρ11 are no longer in contact in ρ15, despite the Figure 10 shows the surface of maximum packing magenta faces being larger in ρ15. density for the 323 · 4 family. There are 8 × 1 þ 61 × 2 ¼ Furthermore, we observe that at a ridge, the Minkowski 130 regions, eight that straddle the diagonal, and 61 mirror lattice type always changes from G6− to G6þ. We can pairs reflected across the diagonal. All boundaries are rationalize this behavior in the following way. By defi- valleys except 13 ridges and 11 tangents, which are the nition, a ridge is the boundary where if we extend beyond same as for the 323 · 2 above. from either side, the packing has overlaps. The implication On the symmetry axis a ¼ c, which is identical to the is that as the shape is deformed, a new contact is introduced subfamily of 323− polyhedra with central symmetry, all

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FIG. 5 523-family optimal density surface (3 × 3 grid of cubes), parallel contacts (bottom left), face areas, and region size (bottom center).

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FIG. 6 423-family optimal density surface (3 × 3 grid of cubes), parallel contacts (bottom left), face areas, and region size (bottom center).

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FIG. 7 323 · 1-family optimal density surface (3 × 3 grid of cubes), parallel contacts (bottom left), face areas, and region size (bottom center).

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FIG. 8 323 · 3-family optimal density surface (3 × 3 grid of cubes) and face areas (bottom center).

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FIG. 9 323 · 2-family optimal density surface (3 × 3 grid of cubes), parallel contacts (bottom left), antiparallel contacts (bottom right), face areas, and region size (bottom center).

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FIG. 10 323 · 4-family optimal density surface (3 × 3 grid of cubes), parallel contacts (bottom left), antiparallel contacts (bottom right), face areas, and region size (bottom center).

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323 families have the same surface of maximum packing VII. CONCLUSIONS density and the same packing, which is the Minkowski We studied three two-parameter families of symmetric lattice G7þ. Near the symmetry axis, in the regions polyhedra, where the shape is continuously deformed via straddling it, all 323 families have the same density vertex and edge truncations. The surface of maximum function but possibly different packings. The densest packing density was determined as a function of shape packings for 323 · 2 and 323 · 4 are identical except for a parameters ha; ci. We defined an equivalence relation of small area near the tetrahedron corners ha; ci¼h3; 1i packings based on the topological type of contacts and and its mirror image h1; 3i, where the regions ρ60 and intersection equations, which allowed a classification of ρ34 consist of a Kuperberg pair of dimers. packings into regions, as well as a definition of three types of boundaries between adjacent regions: valleys, ridges, VI. Comparison with other packing studies and tangents. We note that ridges are special boundaries; Packing studies of one-parameter families have observed the lattice deforms continuously across a ridge, but the set some of the maximum-density surface topography pre- of neighbors in contact changes. We have shown that for sented here [33,36–38,41–43,46]. In this section, we centrally symmetric shapes, a ridge always separates a 6− apply our classification of regions of packings and boun- region of Minkowski type G from a region of Minkowski 6 daries to earlier works and compare their analyses of the type G þ. On the ridge, the packing has Minkowski 7 maximum-density graphs to our surfaces of maximum type G þ. density. A similar study of maximum-density surfaces can be In Fig. 3 of Ref. [43] on puffy tetrahedra, Kallus and performed for general shapes (not necessarily polyhedra). If Elser describe four regions and three transitions, two of we define a continuous family of shapes, then the maxi- them as abrupt (between D1 and S1, and S1 and D0) and one mum-density surface will also be continuous. Thus, we expect the same classification of boundaries between as continuous (between S0 and D1). From their data, we packing regions (valleys, ridges, and tangents) for general would define five regions and four transitions (two ridges S shapes, too. For example, the smoothness of the shape itself and two valleys). Specifically, there is a ridge between 0 (e.g., curvature versus sharp vertices or edges) does not D and 1, where we expect the lattice vectors to change affect our boundary classification as long as the shape continuously but not smoothly. A second local maximum parameter varies continuously. within the S1 region is not identified as a transition but The holy grail of packing studies is to predict structures according to our theory is a ridge, which means the two based solely on shape. It is tempting from the large data set sides must have different intersection equations. presented here to hope that a simple relation between shape The one-parameter family of truncated tetrahedra in and packing might be discerned. However, our results Ref. [37] [see their Fig. 2(a)] corresponds to the left edge demonstrate that a complete appreciation of the way (and by symmetry also to the bottom edge) of the 323 · 4 polyhedra pack at high density must be achieved through family. The authors find eight regions and seven transitions, exploration of not only the specific shape of immediate with which we are in agreement: fρ60; ρ66; ρ68; ρ31; ρ56; ρ62; interest but also shapes near it in shape space, as achieved ρ41; ρ70g. We note that their vector-length curves are by small deformations. Here, we investigated deformation continuous across ridges, as expected. The ridges are via truncations, but other shape-anisotropy dimensions may ½ρ66∧ρ68 and ½ρ31∧ρ56. be explored in the same manner [60–62]. This knowledge The one-parameter family of Gantapara et al. [46] should be especially helpful in the design and synthesis of interpolating from the cube to the octahedron via the particles intended to pack into target structures and/or cuboctahedron corresponds to the symmetry diagonal in densities, as we demonstrate below. the 323 family. While they report 14 regions (see their Our results suggest the following general principles Fig. 1 and Supplemental Material [52]), we find that some relating packing and shape that we hope will both serve of these regions are identical and instead we identify only as useful guidelines to experiments and be tested by eight: fρ7; ρ2; ρ5; ρ0; ρ4; ρ6; ρ1; ρ3g; see Fig. 7. Our region theorists. ρ4 corresponds to their regions VI, VII, and VIII, and our (i) A single shape descriptor is generally insufficient for region ρ6 corresponds to their IX, X, XI, and XII. The predicting the densest packing fractions or structures. discontinuities in the lattice basis-vector lengths and angles Instead, one needs to consider higher-dimensional that appear due to ordering them by magnitude were surfaces of optimal packing densities and/or structures interpreted by the authors as the boundary X–XI. Other of nearby shapes (in shape space) to anticipate how discontinuities were interpreted as boundaries VI–VII, well or poorly a given shape might pack. VII–VIII, IX–X, and XI–XII, but we think they appear (ii) In experiments, particles at all scales (nano, colloidal, due to the specific choice of basis vectors. In addition, our and even 3D printed) have shape imperfections and region ρ2 corresponds to their regions II and III, but we find some degree of polydispersity. The densest packings no boundary in between. are more likely to be achieved experimentally and

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more tolerant to small differences if the packing lies in latticesare the same.For example,in Fig. 6 going from a smooth region of the maximum-density surface. In the octahedron to the cuboctahedron via vertex trun- contrast, it may be very difficult to achieve and less cations (moving to the left along the lower edge), there tolerant to imperfections if the packing lies in a is a ridge between regions 11 and 15; thus, although complex region of the surface (i.e., adjacent to many the contacts are different, the lattice is the same. different packing regions or on boundaries). For Furthermore, this feature might allow the fabrication example, for the truncated cube (Fig. 6, region 17), of reconfigurable structures whose interparticle ori- we foresee that small imperfections in the shape will entations (contacts) can be toggled back and forth make little difference in the packing structure. On the while the crystal superlattice remains the same. other hand, for the edge-and-vertex truncated tetrahe- An important caveat in the general principles outlined dron (Fig. 10, region 35), small imperfections in the above is that they hold for the two shape parameters particle shape may give completely different packing explored here. Of course, there are many other shape structures than expected. parameters that may similarly affect the densest packing (iii) The topography of the maximum-density surface as a landscape. Thus, a surface that appears smooth for the two function of shape is useful if we are interested in parameters considered here may be rough when other obtaining a target density. If the slope of the region is parameters are also considered. Moreover, here we have steep, small imperfections in shape will achieve very not considered surfaces of suboptimal packings, which may different packing densities. Knowledge of the under- affect the practical achievability of a given densest packing. lying surface can help to anticipate how sensitive the These two points further emphasize the value in thinking packing density is to small shape imperfections and of shape not as a fixed and static property but rather as even suggest alternative shapes that yield the desired part of a continuum of shape-anisotropy characteristics packing fraction. (dimensions). Thus, if one is interested in dense packings, (iv) Given the landscape topography in the neighborhood the relevant question to ask is not “what is the densest of a specific shape, one can deduce a posteriori what packing of a particular shape” but rather “what is the kind of imperfection the sample of shapes may have, maximum-packing-density surface in the neighborhood of in the absence of other information. For example, that shape?” consider a shape that lies on a region of the landscape where the packing density changes rapidly with edge ACKNOWLEDGMENTS truncations but not much with vertex truncations, as in region 10 of Fig. 5. The deformation from the E. R. C. acknowledges the National Science Foundation dodecahedron to the icosidodecahedron (going down MSPRF Grant No. DMS-1204686. D. K. acknowledges along the left edge) changes the packing density only the FP7 Marie Curie Actions of the European Commission, slightly. We can thus be certain that imperfections on Grant Agreement No. PIOF-GA-2011-302490 Actsa. The the shape’s vertices will make little difference to the numerical studies of the densest packings were supported packing density. However, if we observe that the by the U.S. Department of Energy, Office of Basic Energy packing density in an experiment is much lower than Sciences, Division of Materials Sciences and Engineering, expected for a given perfect shape, we may deduce Biomolecular Materials Program, under Award that our polyhedron has imperfections on its edges, No. DEFG02-02ER46000. S. C. G., P. F. D., and M. E. rather than its vertices. We expect such information to were supported by the DOD/ASD(R&E) under Grant be useful in the synthesis of nanoparticles for target No. N00244-09-1-0062. Any opinions, findings, and con- applications that exploit dense packings. clusions or recommendations expressed in this publication (v) For centrally symmetric shapes, we showed that if the are those of the authors and do not necessarily reflect shape lies on a surface that is near a ridge, the contacts the views of the DOD/ASD(R&E). E. R. C. and D. K. on either side of the ridge are different, while the contributed equally to this work.

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APPENDIX

FIG. 11 Sum body 323þ (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).

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FIG. 12 Contours of optimal packing density ϕ for the 523 family, numeric (left) versus analytic (right). The color gradient {blue, 3 2 1 1 magenta, red, yellow, green, cyan, and blue} represents the ϕ gradient fϕ − 60 ; ϕ − 60 ; ϕ − 60 ; ϕ; ϕ þ 60 ; 2 3 8 9 ϕ þ 60 ; ϕ þ 60g; yellow contours occur at ϕ ¼f10 ; 10g.

FIG. 13 Contours of optimal packing density ϕ for the 423 family, numeric (left) versus analytic (right). The color gradient {blue, 3 2 1 1 magenta, red, yellow, green, cyan, and blue} represents the ϕ gradient fϕ − 60 ; ϕ − 60 ; ϕ − 60 ; ϕ; ϕ þ 60 ; 2 3 8 9 ϕ þ 60 ; ϕ þ 60g; yellow contours occur at ϕ ¼f10 ; 10 ; 1g.

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FIG. 14 Contours of optimal packing density ϕ for the 323 · 1 family, numeric (left) versus analytic (right). The color gradient {blue, 3 2 1 1 2 magenta, red, yellow, green, cyan, and blue} represents the ϕ gradient fϕ − 60 ; ϕ − 60 ; ϕ − 60 ; ϕ; ϕ þ 60 ; ϕ þ 60 ; 3 4 5 6 7 8 9 ϕ þ 60g; yellow contours occur at ϕ ¼f10 ; 10 ; 10 ; 10 ; 10 ; 10 ; 1g.

FIG. 15 Contours of optimal packing density ϕ 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 ϕ gradient 3 2 1 1 2 3 fϕ − 60 ; ϕ − 60 ; ϕ − 60 ; ϕ; ϕ þ 60 ; ϕ þ 60 ; ϕ þ 60g; yellow contours occur at 7 8 9 ϕ ¼f10 ; 10 ; 10 ; 1g.

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FIG. 16 Contours of optimal packing density ϕ for the 323 · 2 family, numeric (left) versus analytic (right). The color gradient {blue, 3 2 1 1 2 magenta, red, yellow, green, cyan, and blue} represents the ϕ gradient fϕ − 60 ; ϕ − 60 ; ϕ − 60 ; ϕ; ϕ þ 60 ; ϕ þ 60 ; 3 8 9 ϕ þ 60g; yellow contours occur at ϕ ¼f10 ; 10 ; 1g.

FIG. 17 Contours of optimal packing density ϕ for the 323 · 4 family, numeric (left) versus analytic (right). The color gradient {blue, 3 2 1 1 magenta, red, yellow, green, cyan, and blue} represents the ϕ gradient fϕ − 60 ; ϕ − 60 ; ϕ − 60 ; ϕ; ϕ þ 60 ; 2 3 9 ϕ þ 60 ; ϕ þ 60g; yellow contours occur at ϕ ¼f10 ; 1g.

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