TY - JOUR

T1 - Schwarzite nets

T2 - A wealth of 3-valent examples sharing similar topologies and symmetries

AU - Hyde, Stephen T.

AU - Cramer Pedersen, Martin

N1 - Publisher Copyright:
© 2021 The Author(s).

PY - 2021/2/3

Y1 - 2021/2/3

N2 - We enumerate trivalent reticulations of two- A nd 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.

AB - We enumerate trivalent reticulations of two- A nd 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.

KW - chemical nets

KW - graph embeddings

KW - hyperbolic geometry

KW - symmetry groups

UR - http://www.scopus.com/inward/record.url?scp=85102872701&partnerID=8YFLogxK

U2 - 10.1098/rspa.2020.0372

DO - 10.1098/rspa.2020.0372

M3 - Article

SN - 1364-5021

VL - 477

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

IS - 2246

M1 - 20200372

ER -