Schwarzite nets: a wealth of 3-valent examples sharing similar topologies and symmetries
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Schwarzite nets : a wealth of 3-valent examples sharing similar topologies and symmetries. / Hyde, Stephen T.; Pedersen, Martin Cramer.
In: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 477, No. 2246, 20200372, 03.02.2021.Research output: Contribution to journal › Journal article › Research › peer-review
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TY - JOUR
T1 - Schwarzite nets
T2 - a wealth of 3-valent examples sharing similar topologies and symmetries
AU - Hyde, Stephen T.
AU - Pedersen, Martin Cramer
PY - 2021/2/3
Y1 - 2021/2/3
N2 - 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.
AB - 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.
KW - hyperbolic geometry
KW - chemical nets
KW - graph embeddings
KW - symmetry groups
KW - CARBON ALLOTROPES
KW - PATTERNS
KW - SURFACES
KW - KLEIN
U2 - 10.1098/rspa.2020.0372
DO - 10.1098/rspa.2020.0372
M3 - Journal article
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
SN - 1364-5021
IS - 2246
M1 - 20200372
ER -
ID: 276381601