Schwarzite nets: a wealth of 3-valent examples sharing similar topologies and symmetries

Abstract

We enumerate trivalent reticulations of two- and three-periodic hyperbolic surfaces by equal-sided n -gonal faces, ( n , 3), where n = 7, 8, 9, 10, 12. These are the simplest hyperbolic generalizations of the planar graphene net, (6, 3) and dodecahedrane, (5, 3). The enumeration proceeds by deleting isometries of regular reticulations of two-dimensional hyperbolic space until the ( n , 3) nets can be embedded in euclidean three-space via periodic hyperbolic surfaces. Those nets are then symmetrized in euclidean space retaining their net topology, leading to explicit crystallographic net embeddings whose edges are as equal as possible, affording candidtae patterns for graphitic schwarzites. The resulting schwarzites are the most symmetric examples. More than one hundred topologically distinct nets are described, most of which are novel.

Mapping hyperbolic order in curved materials

“…For example, tessellations of H 2 can be realised by a broad variety of finite tiles, whose edges induce crystalline networks, of relevance to chemistry, physics, and mathematics. [32][33][34][35][36][37][38][39][40][41] Further, tessellations of H 2 by infinite ribbon-shaped tiles, related to so-called ''free tilings'' 34,35,42,43 map to complex interwoven networks in E 3 via TPMS, which match patterns found in numerical simulations of selfassembled star polymers. 42,44,45 The latter will be reconsidered later in this paper.…”

Schwarz P‐surface via isolated sp2 carbon heptagons: Design and properties

“…2 Since then, numerous Schwarzites have been proposed in literature and organized in the database SACADA. 12 TPMS designs are known for which only a single size polygon, N = 7, 8, 9, 10, or 12 is used 6,13 as well as a combination of such polygons with hexagons. 2,3 Designs based on isolated heptagons are important for a possible bottom-up synthesis of TPMS.…”

“…Known in mathematics for a long time, triply periodic minimal surfaces (TPMS, examples shown in Figure 1) have received attention in chemical topology with regard to their potential application in materials science. [2][3][4][5][6] Interest in this subject is evidenced by numerous reports on the incorporation of gyroid structure in block copolymers. 7 Gyroid structure made of graphene flakes has been proposed for the construction of materials with extreme mechanical properties.…”