Construction of weavings in the plane

Miro, Eden Delight; Zambrano, Aliw Iw; Garciano, Agnes

doi:10.1107/s205327331701422x

Cited by 3 publications

(3 citation statements)

References 9 publications

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“…From here on, we analyse a special class of weavings built of a pair of dual nets, i.e., the parent net is the union of a net and its two‐dimensional dual, with appended polychromatic vertices on the mid‐edges of both nets. The constructions of two‐dimensional weavings developed here complements earlier work ,,…”

Section: Weavings and The Parent Netmentioning

confidence: 78%

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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Section: Weavings and The Parent Netmentioning

confidence: 78%

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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“…Of particular interest are perfect colorings of tilings, that is, colorings where every symmetry of the uncolored tiling sends all tiles of a given color to tiles of the same color (Frettlo ¨h, 2008;Junio & Walo, 2019). In addition, colorings of patterns in the hyperbolic plane have garnered attention because of their connection with quasicrystals and structural chemistry (De Las Pen ˜as et al, 2006;Bugarin et al, 2013;Miro et al, 2018;Thompson & Hyde, 2018).…”

Section: Introductionmentioning

confidence: 99%

Perfect precise colorings of plane semiregular tilings

Loquias,

Santos

2023

Acta Cryst Sect A

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A coloring of a planar semiregular tiling {\cal T} is an assignment of a unique color to each tile of {\cal T}. If G is the symmetry group of {\cal T}, the coloring is said to be perfect if every element of G induces a permutation on the finite set of colors. If {\cal T} is k-valent, then a coloring of {\cal T} with k colors is said to be precise if no two tiles of {\cal T} sharing the same vertex have the same color. In this work, perfect precise colorings are obtained for some families of k-valent semiregular tilings in the plane, where k ≤ 6.

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“…Hyde et al [2,3,4,10,18], as well as E.D. Miro et al [14,15], or through knot theory as considered by S. Grishanov et al [6,7,8], but the community has not agreed to a formal definition of weavings yet, as well as a way to distinguish them from other entangled networks. This paper is the second in a series of articles which aim to contribute to this objective by proposing a new systematic approach, based on low-dimensional topology, combinatorics and geometric principles, to describe, construct, and classify weaves.…”

Section: Introductionmentioning

confidence: 99%

Construction and Classification of Combinatorial Weaving Diagrams

Fukuda¹,

Kotani²,

Mahmoudi³

2021

Preprint

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This paper introduces a new systematic algorithm for constructing periodic Euclidean weaving diagrams with combinatorial arguments. It is shown that such a weaving diagram can be considered as a specific type of four-regular periodic planar tiling with over or under information at each vertex. Therefore, a weaving diagram can be constructed using two sets of cycles, one to build a tiling, and a second to define the crossing information. However, this construction method does not guarantee the uniqueness of the diagram, so we define the notion of equivalence classes of weaving diagrams using the concept of crossing-matrices. Finally, we present a classification of our periodic structures according to the minimum number of crossings on a unit cell.

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Construction of weavings in the plane

Cited by 3 publications

References 9 publications

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

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

Perfect precise colorings of plane semiregular tilings

Construction and Classification of Combinatorial Weaving Diagrams

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