Polyhedra and packings from hyperbolic honeycombs

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 $\left\{\mathrm{3,7}\right\}$, $\left\{\mathrm{3,8}\right\}$, $\left\{\mathrm{3,9}\right\}$, $\left\{\mathrm{3,10}\right\}$, and $\left\{\mathrm{3,12}\right\}$ to allow embeddings in Euclidean 3 space. Nearly all of these triangulated “simplicial polyhedra” have symmetrically identical vertices, and most are chiral. The most symmetric examples include 10 infinite “deltahedra,” with equilateral triangular faces, 6 of which were previously unknown and some of which can be described as packings of Platonic deltahedra. We describe also related cubic crystalline packings of equal hyperbolic discs in 3 space that are frustrated analogues of optimally dense hyperbolic disc packings. The 10-coordinated packings are the least “loosened” Euclidean embeddings, although frustration swells all of the hyperbolic disc packings to give less dense arrays than the flat penny-packing even though their unfrustrated analogues in ${\mathbb{H}}^2$ are denser.

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), ${\mathbb{E}}^3$. The Platonic triangular polyhedra can be assembled into numerous configurations, many relevant to geometries of crystalline, quasi-crystalline, and disordered packings (5–8). Consequently, triangulated structures are common in condensed atomic and (bio)molecular assemblies. Triangular polyhedra are found in glasses and random dense sphere packings (9–12). They are essential building blocks in tetrahedrally close-packed structures in alloys and soft materials (13–15); Goldberg polyhedra (16), which describe the structures of many viruses (17, 18); and Boerdijk–Coxeter helices (19) in biological fibers (20) and nanowires (21).

Here, we derive a large number of infinite, crystalline patterns, namely nets, triangular infinite polyhedra, and associated disc packings in 3D Euclidean space, ${\mathbb{E}}^3$, derived from Coxeter’s “regular honeycombs” of 2D hyperbolic space (22), ${\mathbb{H}}^2$. These 3-periodic crystalline structures minimize the geometric frustration that results from mapping ${\mathbb{H}}^2$ to ${\mathbb{E}}^3$. Thus, they are important additions to the compendium of regular patterns.

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 ${\mathbb{E}}^3$. Finally, we relax symmetrized versions of the nets in ${\mathbb{E}}^3$, to recover as similar edge lengths as possible. In those cases where equal edges result, the nets define edges of infinite triangular polyhedra, “deltahedra.” More commonly, frustration imposes unequal edges, describing the skeleta of infinite simplicial polyhedra. An extraordinary wealth of disc packings, nets, and polyhedra emerge from just a small number of hyperbolic triangular honeycombs. The presented methods are readily extended to realize patterns in ${\mathbb{E}}^3$ from patterns in ${\mathbb{H}}^2$, beyond triangular honeycombs.

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, ${\mathbb{E}}^2$ (24, 25). Dense packings of equal discs on the surface of the 2 sphere, ${\mathbb{S}}^2$, are more subtle, since the sphere’s finite area means that optimal solutions depend on the disc radius. The formal definition of packing density for equal discs in the third homogeneous 2D space—the hyperbolic plane, ${\mathbb{H}}^2$—is also complicated by the nature of that space. Nevertheless, Tóth (23) established that packing densities in all three homogeneous 2D spaces can be unified via a simple formula, which gives an upper bound for the density of any packing in ${\mathbb{S}}^2$, ${\mathbb{E}}^2$, or ${\mathbb{H}}^2$,

$$\rho \left(N\right)=\frac{3\csc \left(\frac{\pi }{N}\right)-6}{N-6},$$ [1]

where $N>3$ denotes the coordination number of the disc packing, equal to 6 for the penny packing, whose density is $\rho \left(6\right)=\frac{\pi }{\sqrt{12}}\approx 0.91$. We can associate a “regular” skeletal net with the penny packing, whose vertices align with the penny centers and whose edges join pennies in mutual tangency. The net has the topology and geometry of the edges of the equilateral triangular tiling of the Euclidean plane, ${\mathbb{E}}^2$, denoted $3^6$. Here, “regularity” implies symmetrically identical faces, edges, and vertices. The $3^6$ triangulation can be realized by repeated reflections in the edges of a triangular kaleidoscope with vertex angles $\frac{\pi }{2},\frac{\pi }{3}$, and $\frac{\pi }{6}$: The regular pattern has orbifold $*$236 (6).

Regular triangulations $3^N$ and their regular skeletal nets (denoted $\left\{3,N\right\}$) are realizable for all integer values of $N\ge 3$ (26). They are all kaleidoscopic patterns—characterized by triangular kaleidoscopic orbifolds $*$23N, illustrated in Fig. 1. For example, the regular triangulations $3^N$, $N=\mathrm{3,4,5}$ consist of spherical equilateral triangular faces (in ${\mathbb{S}}^2$). If we scale ${\mathbb{S}}^2$ so that the triangles’ edges are of unit length, related dense disc packings are realized by locating unit discs on the sphere whose centers coincide with vertices of the triangulation. These packings have densities precisely equal to $\rho \left(N\right)$ in Eq. 1. So regular arrangements of equal discs realize the densest packings in both ${\mathbb{S}}^2$ and ${\mathbb{E}}^2$. Similarly, the densest disc packings for $N>6$ are regular packings in ${\mathbb{H}}^2$, and their densities are equal to $\rho \left(N\right)$. Their skeletal nets define regular triangulations of ${\mathbb{H}}^2$ for all integers $N$ exceeding 6. In summary, densest 2D packings are realized by centering equal discs on the vertices of regular tilings that describe triangular honeycombs, $\left\{3,N\right\}$, regardless of their 2D geometry (${\mathbb{S}}^2$, ${\mathbb{E}}^2$, or ${\mathbb{H}}^2$). Note that since all patterns $3^N$ with $N>6$ inhabit ${\mathbb{H}}^2$, the overwhelming majority of regular honeycombs are hyperbolic.

Fig. 1.

(A) Regular triangulations $3^N$ formed by decorating an orbifold $*$23N by an edge and vertex. (B and C) The $3^5$ triangulation of ${\mathbb{S}}^2$ (B) and $3^6$ tiling of ${\mathbb{E}}^2$ (C). (D) The analogous triangulation $3^7$ of ${\mathbb{H}}^2$ (drawn in the Poincaré disc model).

Regular hyperbolic honeycombs have been explored considerably less than their Euclidean relatives, in part due to the “unphysical” nature of ${\mathbb{H}}^2$. ${\mathbb{H}}^2$ cannot be smoothly embedded in ${\mathbb{E}}^3$, in contrast to ${\mathbb{S}}^2$ and ${\mathbb{E}}^2$, both of which can be embedded in ${\mathbb{E}}^3$ isometrically. (The exact result, due to Hilbert and later refined by Efimov, is discussed in ref. 27.) Just as any map of the globe (${\mathbb{S}}^2$) onto a flat sheet of paper (${\mathbb{E}}^2$) is subject to distortion, ${\mathbb{H}}^2$ cannot be mapped into ${\mathbb{E}}^3$ without some “frustration.” Geographers have found ways around the former problem, and here we describe a number of embeddings of ${\mathbb{H}}^2$ into ${\mathbb{E}}^3$ that minimize, but cannot completely remove, the frustration imposed by the embedding from ${\mathbb{H}}^2$ into ${\mathbb{E}}^3$. We explore regular patterns for $N=\mathrm{7,8,9,10,12}$ whose distortions are just sufficient to embed them in ${\mathbb{E}}^3$, and no more. These “minimally frustrated” structures include a number of symmetric hyperbolic triangular polyhedra $3^N$, their associated nets $\left[3,N\right]$, and related disc and sphere packings in ${\mathbb{E}}^3$. We denote the resulting Euclidean nets $\left[3,N\right]$, in contrast to their regular hyperbolic antecedents, $\left\{3,N\right\}$.

Mapping from ${\mathbb{H}}^2$ to ${\mathbb{E}}^3$: Commensurate Subgroups

Imagine a pattern in ${\mathbb{H}}^2$, analogous to symmetric patterns in the plane or on the sphere. Since all embeddings of ${\mathbb{H}}^2$ into ${\mathbb{E}}^3$ are frustrated, the pattern is necessarily distorted in moving from ${\mathbb{H}}^2$ to ${\mathbb{E}}^3$. However, some hyperbolic patterns may be moved into ${\mathbb{E}}^3$ without losing any of their isometries, provided the frustration itself adopts those isometries. Hyperbolic patterns are “commensurate” if they can be embedded in ${\mathbb{E}}^3$ such that all of their 2D (in-surface) isometries are retained in ${\mathbb{E}}^3$. The 2D isometries of generic patterns, whether in ${\mathbb{S}}^2$, ${\mathbb{E}}^2$, or ${\mathbb{H}}^2$, are encoded by orbifolds (6). For example, the regular $3^N$ triangulations of the sphere ($N=\mathrm{3,4,5}$) 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 characteristic of their associated orbifold, ${\chi }_{orb}$, readily calculated from the Conway symbol of the orbifold (6). For example, the more symmetric the spherical pattern the smaller the area of an asymmetric domain, equal to $2\pi {\chi }_{orb}$, normalized to spheres of unit curvature. Likewise, we can order patterns in ${\mathbb{H}}^2$ by decreasing symmetry (28). In contrast to ${\mathbb{S}}^2$, orbifolds in ${\mathbb{H}}^2$ have $\left|{\chi }_{orb}\right|\ge \frac{1}{84}$, where the lower bound is realized by the most symmetric hyperbolic pattern, $*237$. Coincidentally, this is the orbifold for the simplest regular hyperbolic honeycomb in ${\mathbb{H}}^2$, $\left\{\mathrm{3,7}\right\}$. Any 2D hyperbolic pattern must have this or lower symmetry, in the sense introduced above. But $*237$ is incommensurate, since it is impossible to embed a hyperbolic surface in ${\mathbb{E}}^3$ with this orbifold. Likewise, $*238$, $*239$, $*23\left(10\right)$, and $*23\left(12\right)$ are incommensurate. In fact, the most symmetric commensurate hyperbolic orbifold is the $*246$ orbifold (with $\left|{\chi }_{orb}\right|=1/24$) and all other hyperbolic surfaces embedded in ${\mathbb{E}}^3$ have $\left|{\chi }_{orb}\right|>1/24$. The pattern $*246$ characterizes the 2D isometries of the most symmetric hyperbolic surfaces in ${\mathbb{E}}^3$, the cubic triply periodic minimal surfaces (TPMS), known as the $D$, $P$, and $Gyroid$ surfaces (29, 30). Analysis of the intrinsic symmetries of these and other TPMS, as well as 2-periodic hyperbolic “mesh” surfaces, has led to enumeration of the most symmetric commensurate hyperbolic orbifolds (28, 31, 32). The degrees of frustration required to render the regular hyperbolic honeycombs commensurate are dependent on the degree of symmetry breaking required to build the pattern in ${\mathbb{E}}^3$. We ranked the frustration via the index of a commensurate subgroup relative to its unfrustrated, regular, parent honeycomb in ${\mathbb{H}}^2$: Increasing index implies more frustration, since more isometries are lost. We generated increasingly frustrated patterns by systematically dropping isometries of the regular honeycombs, using the GAP software (33) to construct a lattice of subgroups of the parent honeycombs. The subgroups were identified via their 2D hyperbolic orbifolds. All subgroups with $\chi \ge -\frac{1}{6}$ were generated, since that constraint was found to be sufficiently generous to determine the least frustrated embeddings of hyperbolic triangular honeycombs. Among these, we found seven distinct commensurate orbifolds, including 2, 5, 4, 1, and 1 for $\left\{3,N\right\}$ honeycombs with $N=\mathrm{7,8,9,10,12}$, respectively. Those orbifolds, and their associated subgroups on TPMS, give multiple embeddings in ${\mathbb{E}}^3$. We catalog the associated disc packings, nets, and polyhedra emerging in ${\mathbb{E}}^3$ from the five regular hyperbolic honeycombs below.

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 $3^{10}$ hyperbolic honeycomb.

GAP detected six subgroups of the group of the regular honeycomb [orbifold symbol $*23\left(10\right)$] whose orbifolds had characteristics $\left|{\chi }_{orb}\right|\le \frac{1}{6}$ (listed in SI Appendix, Fig. S8 and Table S5). Of those, only the subgroup with largest index (5, orbifold symbol $3*22$) was commensurate; this is the intrinsic symmetry of minimally frustrated patterns. This orbifold is unusually simple, as it can be cut open to give a unique pentagonal form in ${\mathbb{H}}^2$ that is a union of four $*246$ triangles. By design, this domain was also a union of five $*23\left(10\right)$ triangles, and a suitable decoration by (three partial) edges and a (quarter) vertex was inferred from the decoration of edges and vertices within each $*23\left(10\right)$ triangle in its parent regular $\left\{3,N\right\}$ net (SI Appendix, Fig. S15). The resulting frustrated, irregular, commensurate $\left[\mathrm{3,10}\right]$ net in ${\mathbb{H}}^2$ was vertex transitive and edge-3 transitive, with unequal edges. Disc packings in ${\mathbb{H}}^2$ were constructed by placing hyperbolic discs at the vertices of the regular $\left\{3,N\right\}$ and frustrated irregular $\left[\mathrm{3,10}\right]$ nets. The radius of congruent discs was maximized to form packings. The frustration induced by the commensurate disc packing opens fissures in the disc array, resulting in an 8-coordinated packing, with slightly enhanced symmetry compared with the underlying edge net, namely $*344$. This is the densest commensurate embedding of the regular dense 10-coordinated regular disc packing, filling $87\%$ of ${\mathbb{H}}^2$, compared with the regular dense filling fraction of $93\%$. We mapped that commensurate $\left[\mathrm{3,10}\right]$ net and associated quasi-dense disc packing onto the $P$, $Gyroid$, and $D$ TPMS, guided by the underlying $*246$ triangles and their placement on the TPMS (34) forming three topologically distinct disc packings (SI Appendix, Table S7) and curvilinear 3-periodic nets in ${\mathbb{E}}^3$, one on each TPMS. The disc packing on the $P$ is shown in Fig. 2N. The 3-periodic curvilinear nets had space groups induced by the combination of orbifold ($3*22$) and TPMS (28): namely $Pm\overline{3}$, $P23$, and $I{2}_{1}3$ for the $P$, $D$, and $Gyroid$. Those nets were straightened in ${\mathbb{E}}^3$, conserving all isometries induced by those space groups, to yield nets with unit average edge length and minimal differences in edge lengths (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 $Gyroid$ patterns have variable edge lengths. These net data are shown in SI Appendix, Fig. S17 and listed in SI Appendix, section S4.35. To build the three minimally frustrated $3^{10}$ polyhedra, we inserted triangles in the faces of the relaxed nets. The resulting “simplicial” polyhedra derived from the $Gyroid$ and $D$ (Fig. 3 H and I) patterns are open $3^{10}$ sponges, with both irregular and regular triangular faces. Pairs of faces of the $D$ polyhedron are coplanar: It can be described as a simple cubic array of Archimedean cuboctahedra sharing common square faces, with half of their triangular faces removed, leaving a pair of diamond labyrinths. The $P$ polyhedron (Fig. 4, 9) is a $3^{10}$ “deltahedron” with equilateral triangular faces, forming a simple-cubic array of edge-shared convex Platonic icosahedra. In summary, the essential frustration imposed on the regular $3^{10}$ hyperbolic honeycomb to embed it in ${\mathbb{E}}^3$ results in irregular—although very symmetric—8-coordinated disc packings, degree-9 and -10 nets, and infinite polyhedra with both open and closed cellular morphologies, containing simpler convex Platonic and Archimedean polyhedra. The constructions follow a similar path for all honeycombs. The minimally frustrated $3^7$, $3^8$, and $3^9$ honeycombs are much richer than the $3^{10}$ case, for three reasons. First, more than one maximally symmetric commensurate orbifold was detected (SI Appendix, Figs. S5–S7). Second, multiple groups shared some of those orbifolds, leading to multiple $\left[3,N\right]$ nets in ${\mathbb{H}}^2$ (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 ${\mathbb{E}}^3$. Like $3^{10}$, the $3^{12}$ honeycomb gave just one least frustrated commensurate orbifold, with unique form in ${\mathbb{H}}^2$. We report these constructions derived from all triangular honeycombs, $\left\{3,N\right\}$, where $N=\mathrm{7,8,9,10,12}$.

Fig. 2.

(A–E) Dense $N$-coordinated disc packings in ${\mathbb{H}}^2$ (drawn in the Poincaré disc model of ${\mathbb{H}}^2$) for $N=\mathrm{7,8,9,10},$ and 12, together with regular $\left\{3,N\right\}$ net. (F–J) Embeddings of minimally frustrated disc packings in ${\mathbb{H}}^2$, whose symmetries produce embeddings with commensurate subgroups of ${\mathbb{E}}^3$, together with related irregular $\left[3,N\right]$ nets. (K–O) Embeddings of the frustrated disc packings in ${\mathbb{E}}^3$, as coverings of 3-periodic minimal surfaces.

Fig. 3.

Some cubic, vertex-transitive, infinite, simplicial polyhedra formed from symmetrized $\left[3,N\right]$ net embeddings in ${\mathbb{E}}^3$, produced by relaxing triangulated reticulations of the $P$, $D$, and $Gyroid$ TPMS. Spheres are located at vertices for clarity. $P$, $D$, and (the pair of) $Gyroid$ embeddings are red, green, blue, and yellow (as in SI Appendix, Fig. S17). (A–D) Four distinct $3^7$ polyhedra produced from the same hyperbolic pattern on the $2223_{01}$ orbifold. (E and F) $3^8$ patterns via the $2223_{14}$ orbifold and the $2223_{21}$ orbifold. (G) The $3^9$ polyhedra via the $2223_{31}$ orbifold. (H and I) $3^{10}$ polyhedra on the $3*22$ orbifold. (J) A $3^{12}$ polyhedron formed by decorating the $2*33$ orbifold.

Fig. 4.

Deltahedra, color coded by their TPMS surface embeddings.

Minimally Frustrated Hyperbolic Disc Packings in ${\mathbb{E}}^3$

Consider the unfrustrated, $N$-coordinated, regular, dense disc packings in ${\mathbb{H}}^2$. The skeleta of those packings form $\left[3,N\right]$ nets, with a disc centered on each vertex. The Euler characteristic per vertex, ${\chi }_V=1-\frac{N}{2}+\frac{N}{3}=\frac{6-N}{6}$, since the total number of edges and faces per vertex is $\frac{N}{2}$ and $\frac{N}{3}$, respectively. The symmetries of the unfrustrated nets were broken, without changing their topologies $\left[3,N\right]$, to form a commensurate subgroup, whose orbifold had characteristic ${\chi }_{orb}$. Since each net vertex represents a (fractional) disc, the fractional number of discs per orbifold is given by $\frac{6{\chi }_{orb}}{6-N}$. Quasi-dense packings in ${\mathbb{E}}^3$ were built by locating $\frac{6{\chi }_{orb}}{6-N}$ discs within a single commensurate orbifold whose equal radii were maximal and then embedding the orbifold, and its copies, on the TPMS. The resulting frustrated disc packings are necessarily less dense than their ideal counterparts in ${\mathbb{H}}^2$, since some symmetry breaking was required to embed them in ${\mathbb{E}}^3$. As a result, they were no longer $N$ coordinated.

The unfrustrated, dense, 7-coordinated disc packing in ${\mathbb{H}}^2$ is shown in Fig. 2A. This regular pattern is characterized by the orbifold $*237$; its minimally frustrated commensurate counterparts are described by two groups with common orbifold $23\times$ and one with orbifold $2223$ (SI Appendix, Fig. S6). A frustrated dense packing of equal hyperbolic discs, with orbifold $2*23$, is commensurate with all three of these commensurate groups, since the (group associated with) orbifold $2*23$ is an index-2 supergroup of both $23\times$ and $2223$. We tuned the disc radius and location to give maximum coverage of the $2*23$ orbifold, consistent with the requisite disc fraction per orbifold. The resulting quasi-dense, frustrated disc packing, whose underlying deformed $\left[\mathrm{3,7}\right]$ 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 $\left[3,N\right]$ 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.

Table 1.

Dense ideal $N_0$-coordinated hyperbolic disc packings in ${\mathbb{H}}^2$ and their minimally frustrated $N$-connected embeddings in ${\mathbb{E}}^3$

$N_0$	Fig.	${\rho }_0$	$N$	Fig.	$\rho$	$\rho /{\rho }_0$
7	2A	0.914	4	2F	0.774	0.846
8	2B	0.920	4	2G	0.674	0.733
9	2C	0.924	6	2H	0.828	0.897
10	2D	0.927	8	2I	0.872	0.940
12	2E	0.932	6	2J	0.732	0.796

The regular and frustrated 2D packing densities are denoted ρ₀ and ρ, respectively (where ρ0 is given by Eq. 1).

Some examples of the associated minimally frustrated disc packings are drawn in ${\mathbb{H}}^2$ 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 TPMS, giving three alternative embeddings in ${\mathbb{E}}^3$ of the regular packings for each coordination number, $N$. A selection of those patterns is illustrated in Fig. 2.

Minimally Frustrated $\left[3,N\right]$ Nets

The deformed $\left[3,N\right]$ 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 those nets in ${\mathbb{E}}^3$ to build embeddings that both were maximally symmetric in ${\mathbb{E}}^3$ and had—as far as possible—equal edge lengths.

First, the frustrated $\left[3,N\right]$ nets were constructed in ${\mathbb{H}}^2$ 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 $\left[3,N\right]$ nets in ${\mathbb{H}}^2$ 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, those hyperbolic nets were mapped to ${\mathbb{E}}^3$, via the TPMS, as follows. The 3-periodic $P$, $D$, and $Gyroid$ surfaces are covers of a common genus-3 tritorus, built of 96 $*246$ domains. The tritorus is built from a hyperbolic dodecagon in ${\mathbb{H}}^2$, compactified (or “glued”) via six hyperbolic translations (36). The edges of the 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. The hyperbolic quotient graphs can be embedded as a reticulation of this tritorus, and we label edges traversing the boundaries of the dodecagon by their associated hyperbolic translation. We determine those labeled quotient graphs from the net fragment (plus dangling edges) contained within this tritorus, as sketched in Fig. 5C. At this stage of our constructions, each parent $\left[3,N\right]$ net was mapped into a 3-periodic net in ${\mathbb{E}}^3$, whose topology was determined by the arrangement of the hyperbolic dodecagon relative to the underlying commensurate hyperbolic $\left[3,N\right]$ net, and the surface map. The final step was to embed the nets of fixed topology into ${\mathbb{E}}^3$ and build triangulations and packings from those skeleta. The 3-periodic nets in ${\mathbb{E}}^3$ were constructed by mapping the hyperbolic translations to lattice vectors in ${\mathbb{E}}^3$, described in ref. 34 (listed in SI Appendix, Table S1). The mapping is dependent on the TPMS, and a pair of maps are possible in general on the $Gyroid$ (37). This step therefore produces three or four distinct labeled quotient graphs. Those labeled graphs were input into the Systre package, which relaxes 3-periodic nets in ${\mathbb{E}}^3$ to give explicit embeddings with maximal crystallographic symmetry (38). We imposed a heavy cost on unequal edge lengths during relaxation to produce, where possible, net embeddings with approximately equal edges. The example in Fig. 5C, 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 $Gyroid$ (40). The geometric scheme is outlined in 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 $N=\mathrm{7,8,9}$ as well as 12, 13, 14, 23, and 32 for $N=7$ and $N=8$, using the notation of ref. 40. The source of potential multiplicity from nontrivial automorphisms of the embedded subgroups in the surfaces described in ref. 34 does not lead to any distinct patterns for our subgroups.

Fig. 5.

Construction of minimally frustrated $\left[3,N\right]$ nets from a commensurate subgroup. (A) A $*2224$ fundamental domain in ${\mathbb{H}}^2$ shown in gray, decorated with a motif (in red) derived from subgroup 9 in SI Appendix, Table S3. (B) The orbit of the decorated orbifold is an embedding of the $\left[\mathrm{3,8}\right]$ graph in ${\mathbb{H}}^2$. (C) A dodecagonal domain that builds a single genus-3 unit cell of the $D$, $Gyroid$, and $P$ TPMS. The graph edges within that domain of ${\mathbb{H}}^2$ define the quotient graphs of infinite 3-periodic nets in ${\mathbb{H}}^2$, modulo the six translation vectors joining opposite edges of the dodecagon (SI Appendix, Fig. S2). (D) The Euclidean quotient graphs in C were symmetrized using the Systre algorithm (35), giving embeddings in ${\mathbb{E}}^3$. In this example, we show the embedding via the $P$ surface.

We built embeddings from 36 minimally frustrated hyperbolic quotient graphs: 12, 14, 8, 1, and 1 graph(s) for $N=\mathrm{7,8,9,10,12}$, respectively. The final embeddings generated by Systre—optimized to minimize the variations of edge lengths—retained their maximal symmetry consistent with their topology. We quantified the homogeneity of edge lengths by the ratio of the shortest edges to the average edge length. Most of these polyhedra have unequal edge lengths. In the spirit of nets reported in RCSR (39), we distinguished “good” skeletal nets from others: The shortest vector between polyhedral vertices in a good net is spanned by an edge. These data are summarized graphically in SI Appendix, Fig. S17. The nets are 3 periodic, with cubic symmetry, containing 24, 12, 8, 6, and 4 vertices per unit cell for $N=7$, 8, 9, 10, and 12, respectively. In some cases, the Euclidean nets form multigraphs, with double edges between the vertices. The resulting Systre embeddings in ${\mathbb{E}}^3$ collapse those edges to a single common edge, giving nets of reduced degree compared with their parent patterns. The resulting nets display a range of geometries and symmetries, both chiral and achiral. In some cases, identical nets formed in ${\mathbb{E}}^3$ from hyperbolic nets with different commensurate subgroups: We constructed 136 nets that relaxed to give 86 distinct examples. As they derive from honeycombs, the parent hyperbolic nets are regular. With the exception of 7 nets (with two distinct vertices), their minimally frustrated embeddings in ${\mathbb{E}}^3$ are vertex-1 transitive, yet their edge transitivities vary from 2 to 8. Detailed data for the nets can be found in SI Appendix, Tables S8–S12.

Vertex-Transitive Infinite $3^N$ Polyhedra

The 3 cycles of these minimally frustrated $\left[3,N\right]$ nets can be faceted, giving infinite triangulated polyhedra which in many cases are embedded in ${\mathbb{E}}^3$. These polyhedra are reminiscent of Coxeter’s “skew polyhedra” (41), since renamed “skew apeirohedra.” Other examples of semiregular infinite polyhedra have been found (42). Since our infinite polyhedra are frustrated, many examples—although not all—contain triangular facets with unequal sides. We call such polyhedra simplicial polyhedra, while the cases whose faces are equilateral are named deltahedra (43). The study of deltahedra has a long history. The Platonic tetrahedron, octahedron, and icosahedron are among the eight convex deltahedra (44). An infinite number of nonconvex deltahedra can be constructed, including the celebrated stella octangula (45) and the Boerdijk–Coxeter helices (19). Apart from the tetrahedron, octahedron, and icosahedron, just one further vertex-transitive deltahedron is usually listed: the great icosahedron, one of the four regular Kepler–Poinsot stellated (and nonconvex) polyhedra (46). Higher genus “toroidal” deltahedra are known, although scarce: An example can be constructed from eight regular octahedra (47). The $\left[3,N\right]$ nets we have constructed afford a wealth of hyperbolic equilateral-triangulated infinite polyhedra analogous to finite deltahedra. Since these infinite $3^N$ polyhedra inherit the symmetries of their skeletal nets, they are largely uninodal, as are the regular triangular polyhedra and the great icosahedron. These infinite triangular polyhedra conserve the $\left[3,N\right]$ topology of their parent honeycombs, in contrast to the altered topologies of the frustrated disc packs and edge-collapsed nets. Since their edges are not symmetrically equivalent, they are irregular.

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 $3^{12}$ polyhedron (10 in Fig. 4) contains adjacent triangular faces folded over their common edge by a dihedral angle of $\pi$, so that the faces coincide. Infinite polyhedra derived from such collapsed edges, like this $3^{12}$ polyhedron, are strictly not embedded in ${\mathbb{E}}^3$, due to their self-intersections; they are “immersed.” These include packings of irregular tetrahedra and octahedra and regular tetrahedra linked by triangles (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 ${\mathbb{E}}^3$, forming nonembedded deltahedra of types $3^8$, $3^{10}$, and $3^{12}$. A fourth nonembedded example contains coincident vertices and edges (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.

Table 2.

Vertex-1–transitive cubic infinite deltahedra, $3^N$ with equilateral triangular faces, shown in Fig. 4

Deltahedron	$N_{{\mathbb{H}}^2}$	$N_{{\mathbb{E}}^3}$	Orbifold	TPMS	Symmetry	Net
1	7	9 ($\times$)	$2223_{21}$	$D$	$F{4}_{1}32$	—
2 (48)	7	7 ($✓$)	$23\times$ (7)	$D$	$F\overline{d}3$	svu
3	7	7 ($✓$)	$23\times$ (8)	$D$	$F\overline{d}3$	svm
4	7	7 ($✓$)	$23\times$ (8)	$P$	$I23$	—
5	8	7 ($\times$)	$2223_{23}$	$G_1$	$I{4}_{1}32$	tes
6 (48)	8	8 ($✓$)	$2223_{01}$	$G_2$	$I{4}_{1}32$	nca
7 (48)	8	8 ($✓$)	$2*23$	$D$	$Fd\overline{3}m$	pyc
8 (42)	9	9 ($✓$)	$2*23$	$D$	$Fd\overline{3}m$	uty
9	10	9 ($\times$)	$3*22$	$P$	$Pm\overline{3}$	shy
10	12	9 ($\times$)	$2*33$	$P$	$P\overline{4}3m$	xay

“Embedded” polyhedra (marked with a ✓) have at most two faces sharing any edge. G₁ and G₂ denote the two embeddings on the Gyroid.

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 ${\mathbb{E}}^3$ of regular “hyperbolic honeycombs,” whose natural ambient space is ${\mathbb{H}}^2$. The minimally frustrated embeddings of hyperbolic honeycombs were built by minimal symmetry breaking of the honeycombs so that they can be realized in ${\mathbb{E}}^3$. The five honeycombs analyzed here have minimally frustrated embeddings on the cubic 3-periodic minimal surfaces: the $D$, $Gyroid$, and $P$. Consequently, all of the patterns are realized in ${\mathbb{E}}^3$ with cubic symmetry. The 3-periodic crystalline structures, including arrays of regular Platonic polyhedra and sphere packings, emerge therefore as least frustrated embeddings in ${\mathbb{E}}^3$ of regular patterns in ${\mathbb{H}}^2$. The regular hyperbolic honeycombs map into ${\mathbb{E}}^3$ to give multiple minimally frustrated patterns. We find 44 topologically distinct cubic nets, all with 2D topology $\left[\mathrm{3,7}\right];$ $24\hspace{0.17em}\left[\mathrm{3,8}\right]$ nets; $12\hspace{0.17em}\left[\mathrm{3,9}\right]$ nets; $3\hspace{0.17em}\left[\mathrm{3,10}\right]$ nets; and $3\hspace{0.17em}\left[\mathrm{3,12}\right]$ nets. All of the minimally frustrated disc packings on the $P$ and $D$ surfaces adopt achiral symmetries, while their $Gyroid$ analogues are chiral. In contrast, the overwhelming majority of the nets and infinite polyhedra reported here are chiral, since they are induced by hyperbolic subgroups whose isometries do not include 2D reflections or glides. We found just 1, 5, 1, 2, and 1 achiral $\left[\mathrm{3,7}\right],$ $\left[\mathrm{3,8}\right],$ $\left[\mathrm{3,9}\right],$ $\left[\mathrm{3,10}\right],$ and $\left[\mathrm{3,12}\right]$ nets.

The construction pipeline outlined here can be generalized to arbitrary hyperbolic patterns. Regular examples, whose skeletal nets have topology $\left\{p,q\right\}$ in ${\mathbb{H}}^2$, are the simplest generalizations. Further, irregular hyperbolic $z$-valent nets with a range of ring topologies, such as examples with 2D vertex symbols $\left(n_1.n_2\dots n_z\right)$, can be embedded into ${\mathbb{E}}^3$ using this process. Such structures can require generalization of the definition of a polyhedron, discussed in ref. 49.