Periodic entanglement I: networks from hyperbolic reticulations

Evans, Myfanwy E.; Robins, Vanessa; Hyde, Stephen T.

doi:10.1107/s0108767313001670

Cited by 38 publications

(63 citation statements)

References 56 publications

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“…Consider, for example, structures containing two interpenetrating srs nets, both of equivalent chirality. Two catenations are known theoretically, with distinct entanglements [27]. On tightening (figure 18), we find that the G 124RT (cosh −1 (3)) structure has a significantly lower ropelength than the G 129RT (cosh −1 (5)) structure: L/D values are 28.1472 and 41.0965, respectively.…”

Section: (D) Multiple Catenated Netsmentioning

confidence: 81%

“…Catenations of four and eight srs nets have also been generated theoretically [23,27]. The ropelengths of ideal configurations of these patterns (figure 19) are 8.49 and 28.48, respectively.…”

Section: (D) Multiple Catenated Netsmentioning

confidence: 99%

“…Given those limitations, it is worthwhile exploring ideal tight embeddings of nets via the PB-SONO algorithm. (Ideal configurations of many of these nets are published in [27]. )…”

Section: Ideal Configurations Of Periodic Network Entanglementsmentioning

confidence: 99%

“…A useful additional feature of ideal nets is the possibility of forming ideal embeddings for patterns containing multiple catenated disjoint nets [27] via tightening. Multiple inter-grown nets are frequently encountered in atomic and molecular materials [24,28,31] and are also of interest because of their photonic properties [84].…”

Section: (D) Multiple Catenated Netsmentioning

confidence: 99%

“…Ideal embeddings of the simplest three-periodic nets: (a) srs, (b) dia, (c) pcu, and of two simple two-periodic nets (d) hcb, (e) (4,4,8,8). The edge diameter has been decreased to show the ideal form [27]. (Online version in colour.…”

Section: (A) Rod Packingsmentioning

confidence: 99%

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Ideal geometry of periodic entanglements

2015

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Three-dimensional entanglements, including knots, knotted graphs, periodic arrays of woven filaments and interpenetrating nets, form an integral part of structure analysis because they influence various physical properties. Ideal embeddings of these entanglements give insight into identification and classification of the geometry and physically relevant configurations in vivo. This paper introduces an algorithm for the tightening of finite, periodic and branched entanglements to a least energy form. Our algorithm draws inspiration from the ShrinkOn-No-Overlaps (SONO) (Pieranski 1998 In Ideal knots (eds A Stasiak, V Katritch, LH Kauffman), vol. 19, pp. 20-41.) algorithm for the tightening of knots and links: we call it Periodic-Branched Shrink-On-No-Overlaps (PB-SONO). We reproduce published results for ideal configurations of knots using PB-SONO. We then examine ideal geometry for finite entangled graphs, including θ-graphs and entangled tetrahedron-and cube-graphs. Finally, we compute ideal conformations of periodic weavings and entangled nets. The resulting ideal geometry is intriguing: we see spontaneous symmetrisation in some cases, breaking of symmetry in others, as well as configurations reminiscent of biological and chemical structures in nature.

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Section: (D) Multiple Catenated Netsmentioning

confidence: 81%

Section: (D) Multiple Catenated Netsmentioning

confidence: 99%

“…Given those limitations, it is worthwhile exploring ideal tight embeddings of nets via the PB-SONO algorithm. (Ideal configurations of many of these nets are published in [27]. )…”

Section: Ideal Configurations Of Periodic Network Entanglementsmentioning

confidence: 99%

Section: (D) Multiple Catenated Netsmentioning

confidence: 99%

Section: (A) Rod Packingsmentioning

confidence: 99%

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Ideal geometry of periodic entanglements

2015

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Decoupling Minimal Surface Metamaterial Properties Through Multi‐Material Hyperbolic Tilings

Callens

Arns

Kuliesh

et al. 2021

Adv Funct Materials

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Rapid advances in additive manufacturing have kindled widespread interest in the rational design of metamaterials with unique properties over the past decade. However, many applications require multi-physics metamaterials, where multiple properties are simultaneously optimized. This is challenging since different properties, such as mechanical and mass transport properties, typically impose competing requirements on the nano-/micro-/ meso-architecture of metamaterials. Here, a parametric metamaterial design strategy that enables independent tuning of the effective permeability and elastic properties is proposed. Hyperbolic tiling theory is applied to devise simple templates, based on which triply periodic minimal surfaces (TPMS) are partitioned into hard and soft regions. Through computational analyses, it is demonstrated how the decoration of hard, soft, and void phases within the TPMS substantially enhances their permeability-elasticity property space and offers high tunability in the elastic properties and anisotropy at constant permeability. Also shown is that this permeability-elasticity balance is well captured using simple scaling laws. The proposed concept is demonstrated through multi-material additive manufacturing of representative specimens. The approach, which is generalizable to other designs, offers a route towards multi-physics metamaterials that need to simultaneously carry a load and enable mass transport, such as load-bearing heat exchangers or architected tissue-substituting meta-biomaterials.

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A Theoretical Schema for Building Weavings of Nets via Colored Tilings of Two‐Dimensional Spaces and Some Simple Polyhedral, Planar and Three‐Periodic Examples

Thompson

Hyde

2018

Israel Journal of Chemistry

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We analyse two-component "weavings" made of a pair of dual (p,q) and (q,p) nets that undulate on both sides of the sphere, the plane and the hyperbolic plane. Families of weavings are described, sharing a common parent net. The examples describe zero-, two-and three-periodic weavings in three-space. We derive all edge-2-transitive weavings with (p,q) = (3,3), (4,3), (4,4), (4,6) using Delaney-Dress tiling theory, described in detail. The two-dimensional hyperbolic weaving are mapped into (euclidean) three-space to form a pair of catenated crystalline nets. The examples suggest generalisations to other weavings on surfaces, including weavings of filaments. A simple hyperbolic weaving of filaments is derived, analogous to the common warp-andweft filament weaving in the plane. The resulting threeperiodic pattern is related to the molecular-scale weaving in the synthetic COF-505 material synthesized by Liu et al.

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Periodic entanglement I: networks from hyperbolic reticulations

Cited by 38 publications

References 56 publications

Ideal geometry of periodic entanglements

Ideal geometry of periodic entanglements

Decoupling Minimal Surface Metamaterial Properties Through Multi‐Material Hyperbolic Tilings

A Theoretical Schema for Building Weavings of Nets via Colored Tilings of Two‐Dimensional Spaces and Some Simple Polyhedral, Planar and Three‐Periodic Examples

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