2001
DOI: 10.1107/s010876730001151x
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The invariant cubic rod (cylinder) packings: symmetries and coordinates

Abstract: Six packings of symmetry‐related cylinders, with cylinder axes in invariant positions (coordinates completely determined by symmetry), are described. Two have axes along 〈100〉 and four have axes along 〈111〉. It is shown that there can be no cubic cylinder packing with axes along 〈110〉. Earlier errors concerning the numbers of such packings and their symmetries are corrected.

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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%
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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.…”
mentioning
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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“…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
“…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
confidence: 99%