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# Polyhedra and packings from hyperbolic honeycombs

Edited by Robion C. Kirby, University of California, Berkeley, CA, and approved May 25, 2018 (received for review November 29, 2017)

## Significance

The simplest 2D regular honeycombs are familiar patterns, found in an extraordinary range of natural and designed systems. They include tessellations of the plane by squares, hexagons, and equilateral triangles. Regular triangular honeycombs also form on the sphere; they are the triangular Platonic polyhedra: the tetrahedron, octahedron, and icosahedron. Regular hyperbolic honeycombs adopt an infinite variety of topologies; these must be distorted to be situated in 3D space and are thus frustrated. We construct minimally frustrated realizations of the simplest hyperbolic honeycombs.

## Abstract

We derive more than 80 embeddings of 2D hyperbolic honeycombs in Euclidean 3 space, forming 3-periodic infinite polyhedra with cubic symmetry. All embeddings are “minimally frustrated,” formed by removing just enough isometries of the (regular, but unphysical) 2D hyperbolic honeycombs

Triangulations are central constructions in diverse areas of pure and applied sciences, from cartography (1) and signal processing (2) to fundamental mathematics (3). Triangular polyhedra characterize many spatial packings, such as the icosahedral arrangement of discs on a sphere and the penny packing in the flat plane and close packing of equivalent spheres in Euclidean 3 space (4),

Here, we derive a large number of infinite, crystalline patterns, namely nets, triangular infinite polyhedra, and associated disc packings in 3D Euclidean space,

The construction is done in two stages. First, we revisit dense disc packings of 2D hyperbolic space, first explored by and Coxeter (22) and Tóth (23). We deform related hyperbolic nets whose edges link adjacent discs to realize the nets and their associated disc packings in

## Disc Packings and Triangular Patterns

The density of 2D hard disc packing is characterized by the ratio of the total area of the packed objects to the area of the embedding space. Thus, the hexagonal “penny packing” of equal discs realizes the maximal packing density in the plane,

Regular triangulations **1**. So regular arrangements of equal discs realize the densest packings in both

Regular hyperbolic honeycombs have been explored considerably less than their Euclidean relatives, in part due to the “unphysical” nature of

## Mapping from H 2 to E 3 : Commensurate Subgroups

Imagine a pattern in

## A Worked Example for the 3 10 Pattern

The procedure is described in detail in *SI Appendix*; relevant group–subgroup lattices and lowest index commensurate subgroups are described in *SI Appendix*, Figs. S5–S9). To guide the reader, we first briefly describe our constructions of disc packings, nets, and polyhedra derived from one of the regular hyperbolic honeycombs: the

GAP detected six subgroups of the group of the regular honeycomb [orbifold symbol *SI Appendix*, Fig. S8 and Table S5). Of those, only the subgroup with largest index (5, orbifold symbol *SI Appendix*, Fig. S15). The resulting frustrated, irregular, commensurate 3,10 net in *SI Appendix*, Table S7) and curvilinear 3-periodic nets in *N*. The 3-periodic curvilinear nets had space groups induced by the combination of orbifold (*SI Appendix*, Table S11). The net induced by the P embedding relaxed to give strictly equal edges; pairs of edges “collapse” to a common edge, so that nine geometrically distinct edges emerge from each vertex. The (10-valent) nets derived from the D and *SI Appendix*, Fig. S17 and listed in *SI Appendix*, section S4.35. To build the three minimally frustrated *H* and *I*) patterns are open **9**) is a *SI Appendix*, Figs. S5–S7). Second, multiple groups shared some of those orbifolds, leading to multiple *SI Appendix*, Fig. S11). Third, the orbifold 2223 can be embedded in infinitely many ways on the TPMS, leading to a (finite) number of topologically distinct embeddings of the nets in

## Minimally Frustrated Hyperbolic Disc Packings in E 3

Consider the unfrustrated, N-coordinated, regular, dense disc packings in

The unfrustrated, dense, 7-coordinated disc packing in *A*. This regular pattern is characterized by the orbifold *SI Appendix*, Fig. S6). A frustrated dense packing of equal hyperbolic discs, with orbifold *B*. The frustrated packings for 8- and 9-coordinated patterns adopt the commensurate orbifold, *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 packings are drawn in *F–J*. Since all of the minimally frustrated orbifolds were either coincident with the hyperbolic symmetries of the D,

## Minimally Frustrated 3 , N Nets

The deformed

First, the frustrated *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 *A* and *B*. Second, those hyperbolic nets were mapped to *C*. At this stage of our constructions, each parent *SI Appendix*, Table S1). The mapping is dependent on the TPMS, and a pair of maps are possible in general on the *C*, realized on the P TPMS, was processed by Systre to produce a cubic net, of known topology, listed in Reticular Chemistry Structure Resource (RCSR) as **dgp** (39). In general, a hyperbolic orbifold can be embedded in more than one way into the TPMS, so our construction is not unique. (This complication does not arise for the construction of the disc packings above, since their relevant orbifolds have unique embeddings.) Among the orbifolds identified here, the 2223 orbifold admits multiple embeddings, on the P, D, and *SI Appendix*, Fig. S4. Due to computational constraints we have constructed a restricted suite of patterns, indexed as 01, 11, 21, 31, and 41 for

We built embeddings from 36 minimally frustrated hyperbolic quotient graphs: 12, 14, 8, 1, and 1 graph(s) for *SI Appendix*, Fig. S17. The nets are 3 periodic, with cubic symmetry, containing 24, 12, 8, 6, and 4 vertices per unit cell for *SI Appendix*, Tables S8–S12.

## Vertex-Transitive Infinite 3 N Polyhedra

The 3 cycles of these minimally frustrated

A selection of infinite simplicial polyhedra are shown in Fig. 3. Many of them resemble infinite cubic arrays of finite convex polyhedra; these examples are, however, single (infinite) polyhedra. In some cases (indicated in *SI Appendix*, Fig. S17), the Systre relaxation gives overlapping edges and vertices; e.g., the P embedding of the **10** in Fig. 4) contains adjacent triangular faces folded over their common edge by a dihedral angle of π, so that the faces coincide. Infinite polyhedra derived from such collapsed edges, like this *SI Appendix*, Fig. S18). Among these minimally frustrated 3-periodic hyperbolic triangulations with cubic symmetry, we have found 10 infinite deltahedra, 6 of which have, to our knowledge, not been reported elsewhere. The remaining 4 were, to our knowledge, described in refs. 42 and 48. These deltahedra are shown in Fig. 4 and described in Table 2. All retain the regular triangular facets of their parent hyperbolic triangulations, resulting in vertex-transitive cubic infinite deltahedra. However, these embeddings are not edge transitive, despite their equal lengths. The most symmetric examples are edge-2 transitive, and others are edge-3 and -4 transitive. Six of these infinite deltahedra are embedded, sponge-like polyhedra, with infinite 3-periodic internal channels. Some of those embedded deltahedra contain fragments of the regular Platonic deltahedra: **2** is an array of face-sharing regular icosahedra and octahedra; **7** can be decomposed into face-sharing octahedra; and **4** and **6** have extended flat facets, containing multiple triangular faces. Three of them (**5**, **9**, and **10**) collapse to lower-degree nets in **1**). These four nonembedded examples contain fragments of the simpler convex Platonic deltahedra: **1** comprises edge-shared regular octahedra and tetrahedra; **5**, edge-shared octahedra; **9**, edge-shared icosahedra; and **10**, edge-shared tetrahedra.

Edge nets of some of these infinite polyhedra have been reported previously (**svu**, **nca**, **pyc**, and **xay** in ref. 2) and are listed in RCSR (39). However, the deltahedra and simplicial polyhedra are structurally distinct from their skeletal nets.

## Conclusion

We have described constructions in

The construction pipeline outlined here can be generalized to arbitrary hyperbolic patterns. Regular examples, whose skeletal nets have topology

## Acknowledgments

We thank Olaf Delgado-Friedrichs, Benedikt Kolbe, Stuart Ramsden, and Vanessa Robins for discussions related to this paper. M.C.P. acknowledges funding from the Carlsberg Foundation.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: martin.pedersen{at}anu.edu.au.

Author contributions: M.C.P. and S.T.H. designed research, performed research, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1720307115/-/DCSupplemental.

Published under the PNAS license.

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