The invariant cubic rod (cylinder) packings: symmetries and coordinates

O’Keeffe, Michael; Plévert, Jacques; Teshima, Yoshinori; Watanabe, Yoshinori; Ogama, T.

doi:10.1107/s010876730001151x

Cited by 37 publications

(40 citation statements)

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“…Intuition suggests that straight fibres will minimize the fibre length per unit cell; hence, ideal embeddings should, where possible, correspond to rod packings. For the three-dimensional weaving G 123 (2), 1 a generic version of cubic rod packing G [5], this intuition is well placed: the ideal embedding contains straight rods, giving a fibre-packing fraction of 0.680. The weaving is illustrated in figure 1.…”

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confidence: 99%

“…The constituent fibres form space curves that lie just to one side of the gyroid, within one of the two labyrinths. If all the filaments are straightened, the weaving is the chiral cubic rod packing known as the P þ structure [5].…”

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confidence: 99%

“…The relation of this weaving with the gyroid is likely to be more than coincidental. In pioneering structural studies of the stratum corneum, Norlén and Al-Amoudi [11] noted the possible presence of lipid bilayers folded onto the gyroid surface within individual corneocytes and suggested that this geometry effectively templates an ordered arrangement of keratin fibres, corresponding to another weaving whose entanglements are those of the cubic G rod packing [5]. In contrast to the G 129 weaving, this pattern is achiral and relatively dense.…”

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From three-dimensional weavings to swollen corneocytes

Evans

Hyde

2011

J. R. Soc. Interface.

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A novel technique to generate three-dimensional Euclidean weavings, composed of closepacked, periodic arrays of one-dimensional fibres, is described. Some of these weavings are shown to dilate by simple shape changes of the constituent fibres (such as fibre straightening). The free volume within a chiral cubic example of a dilatant weaving, the ideal conformation of the G 129 weaving related to the S þ rod packing, expands more than fivefold on filament straightening. This remarkable three-dimensional weaving, therefore, allows an unprecedented variation of packing density without loss of structural rigidity and is an attractive design target for materials. We propose that the G 129 weaving (ideal S þ weaving) is formed by keratin fibres in the outermost layer of mammalian skin, probably templated by a folded membrane. Keywords: intermediate filaments; entangled weavings; dilatant materialsWe have developed a technique to generate a variety of crystalline close-packed arrays of one-dimensional filaments, via projection of line arrays in two-dimensional hyperbolic space into conventional three-dimensional space. The procedure relies on mapping of the hyperbolic patterns onto three-periodic minimal surfaces (TPMS), generalizing a technique used to enumerate systematically crystalline nets [1]. A rich catalogue of three-dimensional weavings can be constructed using this technique, with varying degrees of entanglement between filaments. Indeed, the entanglement can be tuned by adjusting the orientation of the line arrays in the hyperbolic plane, thereby varying the pitch of filament windings around channels of the TPMS. Generic examples are woven from curved and twisted filaments, whose detailed form depends on their trajectories on the TPMS [2]. Since these weavings contain close-packed fibres that can unwind at constant fibre volume without unjamming, they offer novel geometries for dilatant structures. An intriguing natural material that can sustain significant variations in fibre-packing densities is corneocytes, which make up the bulk of the outermost layer of mammalian skin, the stratum corneum. Corneocytes contain predominantly packed keratin fibrils, whose arrangement remains uncertain. We have found a dilatant chiral weaving with cubic symmetry that admits a sevenfold increase in the free volume on straightening the fibres cooperatively. The porosity variations exhibited by this weaving match closely the observed variations in human stratum corneum water contents. Further, the geometry of keratin intermediate filaments (IFs) deduced from X-ray crystallography matches well with data for fibres within this weaving in its canonical configuration, corresponding to unhydrated corneocytes.Since the specific curvilinear form of component filaments within a weaving can be varied without changing its entanglement, it is helpful to formulate a canonical geometry for each weaving. We have adapted algorithms developed to form canonical 'ideal' or 'tight' embeddings of knots [3,4] to arrive at these canonical embeddings...

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From three-dimensional weavings to swollen corneocytes

Evans

Hyde

2011

J. R. Soc. Interface.

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“…1 gezeigt [4]. Diese kubische Helix-Packung ist dreifach-periodisch, d. h. invariant unter drei zueinander orthogonalen Translationen, und eng verwandt mit einer Packung unendlich langer Stäbe, die in der Strukturchemie wohlbekannt ist [24]. Diese Filamentstruktur entspricht einem Zopf, der nicht wie gewöhnlich in einer Richtung geflochten ist, sondern in den vier Richtungen, die den Raumdiagonalen der kubischen Symmetriegruppe entsprechen.…”

Section: Die Geometrie Von Schrumpelfingernunclassified

In a Material World. Hyperbolische Geometrie in biologischen Materialien

Evans¹,

Schröder-Turk²

2014

Mitteilungen Der Deutschen Mathematiker-Vereinigung

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“…The difference in occupied voxels between such a helix Gyroid and the single Gyroid PNS approximation adds up to as few as 0.7 % (3.5 % relative to the solid volume fraction), when both models have been adjusted to the same volume fraction φ = 0.205 (PNS iso-surface threshold t = −0.9). This helix Gyroid is closely related to cubic rod packings [51] with the same I4 1 32 space group: Rods either oriented in 1 0 0 ('octagon' 4 1 screw axes with ± Π packings) or in 1 1 1 ('triangle' 3 1 axes with ± Σ packings) directions, correspond to Gyroid helix approximations by replacing the self-avoiding cylinders with helices of maximum overlap. It is a extended special case of [26]'s bi-chiral schema, in which this construction would be called "LH corner-RH helices", with explicitly added BCC symmetry and a subsequent bend radius constraint.…”

Section: Appendix 1: Single Gyroid Built From Solid Helicesmentioning

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Harmonic distortions enhance circular dichroism of dielectric single gyroids

et al. 2017

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Abstract:The departure from strict periodic order in two-phase dielectric materials can offer properties that are otherwise inaccessible to perfectly ordered photonic crystals. Herewith, we investigate the circular dichroism of the single gyroid photonic crystal in the presence of spatial distortions. FDTD simulations and microwave transmission measurements on 3D-printed replicas show that certain harmonic long-wavelength spatial distortions ("sinusoidal chirp") nearly doubles the imbalance of the circular polarisation reflectances, as well as significantly strengthens polarisation-incoherent reflectance. The observed changes are partially rationalised by comparison with simpler distortion models (linear chirp and tetragonal deformation) of the Gyroid. References and links1. S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486Lett. 58, -2489Lett. 58, (1987. 2. E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059Lett. 58, -2062Lett. 58, (1987. 3. J. W. S. Rayleigh, "On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure," Phil.

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The invariant cubic rod (cylinder) packings: symmetries and coordinates

Cited by 37 publications

References 11 publications

From three-dimensional weavings to swollen corneocytes

From three-dimensional weavings to swollen corneocytes

In a Material World. Hyperbolische Geometrie in biologischen Materialien

Harmonic distortions enhance circular dichroism of dielectric single gyroids

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