Polyhedra and Packings from Hyperbolic Honeycombs

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APPLIED MATHEMATICS where the lower bound is realized by the most symmetric hyperbolic pattern, ∗237. Coincidentally, this is the orbifold for the 2 simplest regular hyperbolic honeycomb in H , {3, 7}. 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 |χorb | = 1/24) and all other hyperbolic E H 3 Poincare´ disc model). surfaces embedded in E have |χorb | > 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, {3, N }, 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 . H cannot be smoothly embedded 3 2 2 in E , in contrast to S and E , both of which can be embed- 3 ded in E isometrically. (The exact result, due to Hilbert and later refined by Efimov, is discussed in ref. 27.) Just as any 2 2 map of the globe (S ) onto a flat sheet of paper (E ) is sub- 2 3 ject to distortion, H cannot be mapped into E without some “frustration.” Geographers have found ways around the former 2 problem, and here we describe a number of embeddings of H 3 into E that minimize, but cannot completely remove, the frus- 2 3 tration imposed by the embedding from H into E . We explore regular patterns for N = 7, 8, 9, 10, 12 whose distortions are just 3 sufficient to embed them in E , and no more. These “minimally frustrated” structures include a number of symmetric hyperbolic triangular polyhedra 3N , their associated nets [3, N ], and 3 related disc and sphere packings in E . We denote the resulting Euclidean nets [3, N ], in contrast to their regular hyperbolic antecedents, {3, N }.

2 3 Mapping from H to E : Commensurate Subgroups 2 Imagine a pattern in H , analogous to symmetric patterns in 2 the plane or on the sphere. Since all embeddings of H into 3 E are frustrated, the pattern is necessarily distorted in mov- 2 3 ing from H to E . However, some hyperbolic patterns may be 3 moved into E without losing any of their isometries, provided the frustration itself adopts those isometries. Hyperbolic pat- 3 terns are “commensurate” if they can be embedded in E such 3 that all of their 2D (in-surface) isometries are retained in E . 2 2 The 2D isometries of generic patterns, whether in S , E , or 2 N H , are encoded by orbifolds (6). For example, the regular 3 triangulations of the sphere (N = 3, 4, 5) are all commensurate, since patterned spheres with tetrahedral, octahedral, and icosahedral point groups have orbifolds ∗23N (28). We can order patterns from most to least symmetric by the orbifold charac-

teristic of their associated orbifold, χorb , readily calculated from 2 Fig. 2. (A–E) Dense N-coordinated disc packings in H (drawn in the the Conway symbol of the orbifold (6). For example, the more 2 Poincare´ disc model of H ) for N = 7, 8, 9, 10, and 12, together with reg- symmetric the spherical pattern the smaller the area of an asym- ular {3, N} net. (F–J) Embeddings of minimally frustrated disc packings in metric domain, equal to 2πχorb , normalized to spheres of unit 2 H , whose symmetries produce embeddings with commensurate subgroups 2 3 curvature. Likewise, we can order patterns in H by decreasing of E , together with related irregular [3, N] nets. (K–O) Embeddings of the 2 2 1 3 symmetry (28). In contrast to S , orbifolds in H have |χorb | ≥ 84 , frustrated disc packings in E , as coverings of 3-periodic minimal surfaces.

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D e mednsin embeddings net 2 N Gyroid Fg 3 (Fig. .Tid h orbifold the Third, S11 ). Fig. Appendix, (SI Gyroid where }, E 3 and Like . H Gyroid P hs eswr tagtndin straightened were nets Those . atrshv aibeeg lengths. edge variable have patterns I obidtetremnmlyfrus- minimally three the build To . 3 ) and ssoni i.2N Fig. in shown is N N 10 i.S17 Fig. Appendix, SI 3 codntd eua,dnedisc dense regular, -coordinated, ,8 ,1,12 10, 9, 8, 7, = 10 oyer nte3 the on polyhedra mednsaerd re,bu,and blue, green, red, are embeddings I E atrsaeopen are patterns ) the , 3 rdcdb eaigtriangulated relaxing by produced , 3 P E E 12 3 NSLts Articles Latest PNAS 3 3 10 oyern(i.4, (Fig. polyhedron n nec TPMS. each on one , Pm oecm aejust gave honeycomb eut nirregular— in results 01 ae o he rea- three for case, riod (E orbifold. . 3, h 3-periodic The . ∗ 3 P 2obfl.(J orbifold. 22 7 and 23, , E n itdin listed and 3 [3, 3 10 3 8 and , sponges, N and | ] I nets, f6 of 3 F 2 3 E 3 ) A ) 1 3 9) 10 3 3 9 8 ,

APPLIED MATHEMATICS deformed [3, 7] net has orbifold symbol 2223, has coordination 4, as shown in Fig. 2B. The frustrated packings for 8- and 9- coordinated patterns adopt the commensurate orbifold, ∗246, and the 12-coordinated packing forms a minimally frustrated embedding in 3 space with orbifold ∗266. The minimally frustrated [3, N ]nets have orbifolds 2 ∗ 23, 2 ∗ 23, and 2 ∗ 33, for N = 8, 9, and 12, respectively (SI Appendix, Figs. S6, S7, and S9). (The 10-coordinated pattern is discussed in the previous section.) The structural data for all packings are listed in SI Appendix, Table S7. Details of the frustrated disc packings are listed in Table 1. Some examples of the associated minimally frustrated disc 2 packings are drawn in H in Fig. 2 F–J. Since all of the minimally frustrated orbifolds were either coincident with the hyperbolic symmetries of the D, Gyroid, and P cubic TPMS (∗246) or subgroups thereof, they can be mapped onto any one of those cubic 3 TPMS, giving three alternative embeddings in E of the regular packings for each coordination number, N . A selection of those patterns is illustrated in Fig. 2. Minimally Frustrated [3, N]Nets The deformed [3, N ] nets induced by these commensurate disc packings lie in the TPMS, and their net geometry was fixed by their orbifold symmetry and had maximal 2D density. We relaxed 3 those nets in E to build embeddings that both were maximally 3 symmetric in E and had—as far as possible—equal edge lengths. 2 First, the frustrated [3, N ] nets were constructed in H with commensurate symmetries. Fundamental domains of the nets were inferred from decorations of the unfrustrated parent ∗23N orbifolds. That process, illustrated in SI Appendix, Fig. S3 C and D, led to one or more motifs for all hyperbolic crystallographic orbifolds (SI Appendix, Figs. S10–S16). The extended [3, N ]nets 2 in H are the orbits of the relevant decoration by the action of the groups associated with these crystallographic hyperbolic orbifolds. This process is illustrated in Fig. 5 A and B. Second, 3 those hyperbolic nets were mapped to E , via the TPMS, as fol- lows. The 3-periodic P, D, and Gyroid surfaces are covers of a common genus-3 tritorus, built of 96 ∗246 domains. The tri- 2 torus is built from a hyperbolic dodecagon in H , compactified (or “glued”) via six hyperbolic translations (36). The edges of the Fig. 4. Deltahedra, color coded by their TPMS surface embeddings. infinite hyperbolic net contained within the dodecagon therefore also glue up in pairs to form a finite quotient net, modulo those six hyperbolic translations corresponding to the gluing vectors. with a disc centered on each vertex. The Euler characteristic The hyperbolic quotient graphs can be embedded as a reticula- χ = 1 − N + N = 6−N per vertex, V 2 3 6 , since the total number of tion of this tritorus, and we label edges traversing the boundaries N N edges and faces per vertex is 2 and 3 , respectively. The sym- of the dodecagon by their associated hyperbolic translation. We metries of the unfrustrated nets were broken, without changing determine those labeled quotient graphs from the net fragment their topologies [3, N ], to form a commensurate subgroup, whose (plus dangling edges) contained within this tritorus, as sketched orbifold had characteristic χorb . Since each net vertex represents in Fig. 5C. At this stage of our constructions, each parent [3, N ] 3 a (fractional) disc, the fractional number of discs per orbifold is net was mapped into a 3-periodic net in E , whose topology was 6χorb 3 given by 6−N . Quasi-dense packings in E were built by locating determined by the arrangement of the hyperbolic dodecagon rel- 6χorb discs within a single commensurate orbifold whose equal ative to the underlying commensurate hyperbolic [3, N ]net, and 6−N the surface map. The final step was to embed the nets of fixed radii were maximal and then embedding the orbifold, and its 3 topology into E and build triangulations and packings from copies, on the TPMS. The resulting frustrated disc packings are 3 2 those skeleta. The 3-periodic nets in E were constructed by necessarily less dense than their ideal counterparts in H , since 3 3 mapping the hyperbolic translations to lattice vectors in , some symmetry breaking was required to embed them in E . As E a result, they were no longer N coordinated. 2 The unfrustrated, dense, 7-coordinated disc packing in H is Table 1. Dense ideal N0-coordinated hyperbolic disc packings in shown in Fig. 2A. This regular pattern is characterized by the H2 and their minimally frustrated N-connected embeddings in E3 orbifold ∗237; its minimally frustrated commensurate counterparts are described by two groups with common orbifold 23× N0 Fig. ρ0 N Fig. ρ ρ/ρ0 and one with orbifold 2223 (SI Appendix, Fig. S6). A frus- 7 2A 0.914 4 2F 0.774 0.846 trated dense packing of equal hyperbolic discs, with orbifold 8 2B 0.920 4 2G 0.674 0.733 2 ∗ 23, is commensurate with all three of these commensurate 9 2C 0.924 6 2H 0.828 0.897 groups, since the (group associated with) orbifold 2 ∗ 23 is an 10 2D 0.927 8 2I 0.872 0.940 23× 2223 index-2 supergroup of both and . We tuned the disc 12 2E 0.932 6 2J 0.732 0.796 radius and location to give maximum coverage of the 2 ∗ 23 orbifold, consistent with the requisite disc fraction per orbifold. The The regular and frustrated 2D packing densities are denoted ρ0 and ρ, resulting quasi-dense, frustrated disc packing, whose underlying respectively (where ρ0 is given by Eq. 1).

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Table 2. Table in described and deltahedra These 4 to 48. knowl- Fig. were, and our in 42 4 to shown refs. remaining have, in are The described which elsewhere. knowledge, of reported our been 6 not deltahedra, 3- edge, infinite frustrated have we 10 minimally symmetry, these cubic found with Among triangulations triangles hyperbolic S18). by periodic linked Fig. tetrahedra Appendix, tetra- regular irregular (SI and of octahedra packings and include hedra These “immersed.” are they (48) (42) (48) (48) Ebde”plhda(akdwt a with (marked polyhedra “Embedded” Fig. in shown are polyhedra simplicial infinite of selection A N medn fthe of embedding ] esw aecntutdafr elho hyper- of wealth a afford constructed have we nets . N 2 3 (×) 9 (×) 9 12 10 G 2 (X) 8 8 2 (X) 9 9 2223 (X) 8 8 23× (X) 7 7 2223 23 × (×) 7 (X) 7 8 7 23 × (X) 7 2223 7 (×) 9 7 H 1 2 π and ota h ae onie nnt polyhe- Infinite coincide. faces the that so , N G 3 2 E E 12 3 eoetetoebdig nthe on embeddings two the denote 3 hs oyer r eiicn of reminiscent are polyhedra These . 3 oyern(10 polyhedron N riodTM ymtyNet Symmetry TPMS Orbifold E Polyhedra 3 ∗ ∗ ∗ ∗ u oterself-intersections; their to due , 23 33 22 23 (8) (8) (7) 01 23 21 ,the S17 ), Fig. Appendix, SI 3 aea ottofaces two most at have X) N NSLts Articles Latest PNAS G G Fd D Fd D F F D D F D P Pm P P I P oyer nei the inherit polyhedra 2 1 nFg )contains 4) Fig. in [3, 3 N [3, 12 I I ] 4 4 4 43m ¯ IAppendix, SI N 23 1 1 escnbe can nets polyhedron, d d ¯ ¯ 1 3m 3m ¯ ¯ 32 32 32 3 3 3 ¯ ] N topology Gyroid with | f6 of 5 svm xay nca pyc shy svu uty tes . — —

APPLIED MATHEMATICS 3 regular triangular facets of their parent hyperbolic triangu- be realized in E . The five honeycombs analyzed here have min- lations, resulting in vertex-transitive cubic infinite deltahedra. imally frustrated embeddings on the cubic 3-periodic minimal However, these embeddings are not edge transitive, despite surfaces: the D, Gyroid, and P. Consequently, all of the patterns 3 their equal lengths. The most symmetric examples are edge- are realized in E with cubic symmetry. The 3-periodic crystalline 2 transitive, and others are edge-3 and -4 transitive. Six of structures, including arrays of regular Platonic polyhedra and these infinite deltahedra are embedded, sponge-like polyhe- sphere packings, emerge therefore as least frustrated embed- 3 2 dra, with infinite 3-periodic internal channels. Some of those dings in E of regular patterns in H . The regular hyperbolic embedded deltahedra contain fragments of the regular Platonic 3 honeycombs map into E to give multiple minimally frustrated deltahedra: 2 is an array of face-sharing regular icosahedra patterns. We find 44 topologically distinct cubic nets, all with 2D and octahedra; 7 can be decomposed into face-sharing octa- topology [3, 7]; 24 [3, 8] nets; 12 [3, 9] nets; 3 [3, 10] nets; and hedra; and 4 and 6 have extended flat facets, containing mul- 3 [3, 12]nets. All of the minimally frustrated disc packings on the tiple triangular faces. Three of them (5, 9, and 10) collapse 3 P and D surfaces adopt achiral symmetries, while their Gyroid to lower-degree nets in E , forming nonembedded deltahe- analogues are chiral. In contrast, the overwhelming majority of 8 10 12 dra of types 3 , 3 , and 3 . A fourth nonembedded example the nets and infinite polyhedra reported here are chiral, since contains coincident vertices and edges (1). These four nonem- they are induced by hyperbolic subgroups whose isometries do bedded examples contain fragments of the simpler convex Pla- not include 2D reflections or glides. We found just 1, 5, 1, 2, and tonic deltahedra: 1 comprises edge-shared regular octahedra and 1 achiral [3, 7], [3, 8], [3, 9], [3, 10], and [3, 12]nets. tetrahedra; 5, edge-shared octahedra; 9, edge-shared icosahedra; The construction pipeline outlined here can be generalized to and 10, edge-shared tetrahedra. arbitrary hyperbolic patterns. Regular examples, whose skeletal 2 Edge nets of some of these infinite polyhedra have been nets have topology {p, q} in H , are the simplest generaliza- reported previously (svu, nca, pyc, and xay in ref. 2) and are listed tions. Further, irregular hyperbolic z-valent nets with a range in RCSR (39). However, the deltahedra and simplicial polyhedra of ring topologies, such as examples with 2D vertex symbols are structurally distinct from their skeletal nets. 3 (n1.n2 ... nz ), can be embedded into E using this process. Such structures can require generalization of the definition of Conclusion 3 a polyhedron, discussed in ref. 49. We have described constructions in E of regular “hyperbolic 2 honeycombs,” whose natural ambient space is H . The minimally ACKNOWLEDGMENTS. We thank Olaf Delgado-Friedrichs, Benedikt Kolbe, frustrated embeddings of hyperbolic honeycombs were built by Stuart Ramsden, and Vanessa Robins for discussions related to this paper. minimal symmetry breaking of the honeycombs so that they can M.C.P. acknowledges funding from the Carlsberg Foundation.

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