Periodic entanglement I: Networks from hyperbolic reticulations

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

doi:10.1107/S0108767313001670

Periodic entanglement I: Networks from hyperbolic reticulations

Myfanwy E. Evans^*, Vanessa Robins, Stephen T. Hyde

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

31 Citations (Scopus)

Abstract

High-symmetry free tilings of the two-dimensional hyperbolic plane (H ²) can be projected to genus-3 3-periodic minimal surfaces (TPMSs). The three-dimensional patterns that arise from this construction typically consist of multiple catenated nets. This paper presents a construction technique and limited catalogue of such entangled structures, that emerge from the simplest examples of regular ribbon tilings of the hyperbolic plane via projection onto four genus-3 TPMSs: the P, D, G(yroid) and H surfaces. The entanglements of these patterns are explored and partially characterized using tools from TOPOS, GAVROG and a new tightening algorithm.

Original language	English
Pages (from-to)	241-261
Number of pages	21
Journal	Acta Crystallographica Section A: Foundations of Crystallography
Volume	69
Issue number	3
DOIs	https://doi.org/10.1107/S0108767313001670
Publication status	Published - May 2013

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title = "Periodic entanglement I: Networks from hyperbolic reticulations",

abstract = "High-symmetry free tilings of the two-dimensional hyperbolic plane (H 2) can be projected to genus-3 3-periodic minimal surfaces (TPMSs). The three-dimensional patterns that arise from this construction typically consist of multiple catenated nets. This paper presents a construction technique and limited catalogue of such entangled structures, that emerge from the simplest examples of regular ribbon tilings of the hyperbolic plane via projection onto four genus-3 TPMSs: the P, D, G(yroid) and H surfaces. The entanglements of these patterns are explored and partially characterized using tools from TOPOS, GAVROG and a new tightening algorithm.",

keywords = "catenation, crystal nets, enumeration, frameworks, graphs, hyperbolic reticulations, ideal knots, networks, periodic entanglement, surfaces, tight knots, tilings",

author = "Evans, {Myfanwy E.} and Vanessa Robins and Hyde, {Stephen T.}",

year = "2013",

month = may,

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language = "English",

volume = "69",

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journal = "Acta Crystallographica Section A: Foundations of Crystallography",

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T1 - Periodic entanglement I

T2 - Networks from hyperbolic reticulations

AU - Evans, Myfanwy E.

AU - Robins, Vanessa

AU - Hyde, Stephen T.

PY - 2013/5

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N2 - High-symmetry free tilings of the two-dimensional hyperbolic plane (H 2) can be projected to genus-3 3-periodic minimal surfaces (TPMSs). The three-dimensional patterns that arise from this construction typically consist of multiple catenated nets. This paper presents a construction technique and limited catalogue of such entangled structures, that emerge from the simplest examples of regular ribbon tilings of the hyperbolic plane via projection onto four genus-3 TPMSs: the P, D, G(yroid) and H surfaces. The entanglements of these patterns are explored and partially characterized using tools from TOPOS, GAVROG and a new tightening algorithm.

AB - High-symmetry free tilings of the two-dimensional hyperbolic plane (H 2) can be projected to genus-3 3-periodic minimal surfaces (TPMSs). The three-dimensional patterns that arise from this construction typically consist of multiple catenated nets. This paper presents a construction technique and limited catalogue of such entangled structures, that emerge from the simplest examples of regular ribbon tilings of the hyperbolic plane via projection onto four genus-3 TPMSs: the P, D, G(yroid) and H surfaces. The entanglements of these patterns are explored and partially characterized using tools from TOPOS, GAVROG and a new tightening algorithm.

KW - catenation

KW - crystal nets

KW - enumeration

KW - frameworks

KW - graphs

KW - hyperbolic reticulations

KW - ideal knots

KW - networks

KW - periodic entanglement

KW - surfaces

KW - tight knots

KW - tilings

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DO - 10.1107/S0108767313001670

M3 - Article

SN - 0108-7673

VL - 69

SP - 241

EP - 261

JO - Acta Crystallographica Section A: Foundations of Crystallography

JF - Acta Crystallographica Section A: Foundations of Crystallography

IS - 3

ER -

Periodic entanglement I: Networks from hyperbolic reticulations

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