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Polyhedra and Packings from Hyperbolic Honeycombs

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Polyhedra and Packings from Hyperbolic Honeycombs Polyhedra and packings from hyperbolic honeycombs Martin Cramer Pedersena,1 and Stephen T. Hydea aDepartment of Applied Mathematics, Research School of Physics and Engineering, Australian National University, Canberra ACT 2601, Australia Edited by Robion C. Kirby, University of California, Berkeley, CA, and approved May 25, 2018 (received for review November 29, 2017) We derive more than 80 embeddings of 2D hyperbolic honey- Disc Packings and Triangular Patterns combs in Euclidean 3 space, forming 3-periodic infinite polyhedra The density of 2D hard disc packing is characterized by the ratio with cubic symmetry. All embeddings are “minimally frustrated,” of the total area of the packed objects to the area of the embed- formed by removing just enough isometries of the (regular, ding space. Thus, the hexagonal “penny packing” of equal discs but unphysical) 2D hyperbolic honeycombs f3, 7g, f3, 8g, f3, 9g, 2 realizes the maximal packing density in the plane, E (24, 25). f3, 10g, and f3, 12g to allow embeddings in Euclidean 3 space. Dense packings of equal discs on the surface of the 2 sphere, Nearly all of these triangulated “simplicial polyhedra” have sym- 2, are more subtle, since the sphere’s finite area means that metrically identical vertices, and most are chiral. The most sym- S optimal solutions depend on the disc radius. The formal defini- metric examples include 10 infinite “deltahedra,” with equilateral tion of packing density for equal discs in the third homogeneous triangular faces, 6 of which were previously unknown and some 2 of which can be described as packings of Platonic deltahedra. We 2D space—the hyperbolic plane, H —is also complicated by the describe also related cubic crystalline packings of equal hyper- nature of that space. Nevertheless, Toth´ (23) established that bolic discs in 3 space that are frustrated analogues of optimally packing densities in all three homogeneous 2D spaces can be unified via a simple formula, which gives an upper bound for the dense hyperbolic disc packings. The 10-coordinated packings are 2 2 2 the least “loosened” Euclidean embeddings, although frustra- density of any packing in S , E , or H , tion swells all of the hyperbolic disc packings to give less dense 3 csc π − 6 arrays than the flat penny-packing even though their unfrustrated ρ (N )= N , [1] analogues in H2 are denser. N − 6 N > 3 where denotes the coordination number of the disc pack- APPLIED hyperbolic geometry j nets j minimal surfaces j graph embeddings j ρ(6) = ing, equal to 6 for the penny packing, whose density is MATHEMATICS symmetry groups pπ ≈ 0:91 12 . We can associate a “regular” skeletal net with the penny packing, whose vertices align with the penny centers and riangulations are central constructions in diverse areas of whose edges join pennies in mutual tangency. The net has the Tpure and applied sciences, from cartography (1) and signal topology and geometry of the edges of the equilateral triangular 2 6 processing (2) to fundamental mathematics (3). Triangular poly- tiling of the Euclidean plane, E , denoted 3 . Here, “regularity” hedra characterize many spatial packings, such as the icosahedral implies symmetrically identical faces, edges, and vertices. The 36 arrangement of discs on a sphere and the penny packing in the triangulation can be realized by repeated reflections in the edges flat plane and close packing of equivalent spheres in Euclidean of a triangular kaleidoscope with vertex angles π , π , and π : The 3 2 3 6 3 space (4), E . The Platonic triangular polyhedra can be assem- regular pattern has orbifold ∗236 (6). bled into numerous configurations, many relevant to geometries Regular triangulations 3N and their regular skeletal nets of crystalline, quasi-crystalline, and disordered packings (5–8). (denoted f3, N g) are realizable for all integer values of N ≥ 3 Consequently, triangulated structures are common in condensed (26). They are all kaleidoscopic patterns—characterized by tri- atomic and (bio)molecular assemblies. Triangular polyhedra are angular kaleidoscopic orbifolds ∗23N, illustrated in Fig. 1. For found in glasses and random dense sphere packings (9–12). They example, the regular triangulations 3N , N = 3, 4, 5 consist of are essential building blocks in tetrahedrally close-packed struc- 2 2 tures in alloys and soft materials (13–15); Goldberg polyhedra spherical equilateral triangular faces (in S ). If we scale S so that (16), which describe the structures of many viruses (17, 18); and Boerdijk–Coxeter helices (19) in biological fibers (20) and Significance nanowires (21). Here, we derive a large number of infinite, crystalline pat- terns, namely nets, triangular infinite polyhedra, and associated The simplest 2D regular honeycombs are familiar patterns, 3 disc packings in 3D Euclidean space, E , derived from Coxeter’s found in an extraordinary range of natural and designed 2 systems. They include tessellations of the plane by squares, “regular honeycombs” of 2D hyperbolic space (22), H . These 3- periodic crystalline structures minimize the geometric frustration hexagons, and equilateral triangles. Regular triangular hon- 2 3 eycombs also form on the sphere; they are the triangular that results from mapping H to E . Thus, they are important additions to the compendium of regular patterns. Platonic polyhedra: the tetrahedron, octahedron, and icosa- The construction is done in two stages. First, we revisit hedron. Regular hyperbolic honeycombs adopt an infinite dense disc packings of 2D hyperbolic space, first explored by variety of topologies; these must be distorted to be sit- and Coxeter (22) and Toth´ (23). We deform related hyper- uated in 3D space and are thus frustrated. We construct bolic nets whose edges link adjacent discs to realize the nets minimally frustrated realizations of the simplest hyperbolic 3 and their associated disc packings in E . Finally, we relax honeycombs. 3 symmetrized versions of the nets in E , to recover as simi- lar edge lengths as possible. In those cases where equal edges Author contributions: M.C.P. and S.T.H. designed research, performed research, analyzed result, the nets define edges of infinite triangular polyhedra, data, and wrote the paper. “deltahedra.” More commonly, frustration imposes unequal The authors declare no conflict of interest. edges, describing the skeleta of infinite simplicial polyhedra. This article is a PNAS Direct Submission. An extraordinary wealth of disc packings, nets, and polyhe- Published under the PNAS license. dra emerge from just a small number of hyperbolic triangular 1 To whom correspondence should be addressed. Email: [email protected] honeycombs. The presented methods are readily extended to This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. 3 2 realize patterns in E from patterns in H , beyond triangular 1073/pnas.1720307115/-/DCSupplemental. honeycombs. www.pnas.org/cgi/doi/10.1073/pnas.1720307115 PNAS Latest Articles j 1 of 6 Downloaded by guest on October 1, 2021 where the lower bound is realized by the most symmetric hyper- bolic pattern, ∗237. Coincidentally, this is the orbifold for the 2 simplest regular hyperbolic honeycomb in H , f3, 7g. Any 2D hyperbolic pattern must have this or lower symmetry, in the sense introduced above. But ∗237 is incommensurate, since it is impos- 3 sible to embed a hyperbolic surface in E with this orbifold. Likewise, ∗238, ∗239, ∗23(10), and ∗23(12) are incommensurate. Fig. 1. (A) Regular triangulations 3N formed by decorating an orbifold 5 2 In fact, the most symmetric commensurate hyperbolic orbifold is ∗23N by an edge and vertex. (B and C) The 3 triangulation of S (B) and 36 tiling of 2 (C). (D) The analogous triangulation 37 of 2 (drawn in the the ∗246 orbifold (with jχorb j = 1=24) and all other hyperbolic E H 3 Poincare´ disc model). surfaces embedded in E have jχorb j > 1=24. The pattern ∗246 characterizes the 2D isometries of the most symmetric hyper- 3 bolic surfaces in E , the cubic triply periodic minimal surfaces the triangles’ edges are of unit length, related dense disc pack- (TPMS), known as the D, P, and Gyroid surfaces (29, 30). Anal- ings are realized by locating unit discs on the sphere whose ysis of the intrinsic symmetries of these and other TPMS, as well centers coincide with vertices of the triangulation. These pack- as 2-periodic hyperbolic “mesh” surfaces, has led to enumera- ings have densities precisely equal to ρ(N ) in Eq. 1. So regular tion of the most symmetric commensurate hyperbolic orbifolds arrangements of equal discs realize the densest packings in both (28, 31, 32). The degrees of frustration required to render the 2 2 S and E . Similarly, the densest disc packings for N > 6 are regular hyperbolic honeycombs commensurate are dependent on 2 regular packings in H , and their densities are equal to ρ (N ). the degree of symmetry breaking required to build the pattern in 2 3 Their skeletal nets define regular triangulations of H for all E . We ranked the frustration via the index of a commensurate integers N exceeding 6. In summary, densest 2D packings are subgroup relative to its unfrustrated, regular, parent honey- 2 realized by centering equal discs on the vertices of regular tilings comb in H : Increasing index implies more frustration, since that describe triangular honeycombs, f3, N g, regardless of their more isometries are lost. We generated increasingly frustrated 2 2 2 N 2D geometry (S , E , or H ). Note that since all patterns 3 patterns by systematically dropping isometries of the regular 2 honeycombs, using the GAP software (33) to construct a lattice with N > 6 inhabit H , the overwhelming majority of regular honeycombs are hyperbolic. of subgroups of the parent honeycombs. The subgroups were Regular hyperbolic honeycombs have been explored consid- erably less than their Euclidean relatives, in part due to the 2 2 “unphysical” nature of H .
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