Self-intersecting three-periodic minimal surfaces forming one-periodic tubes or finite polyhedra

Koch, Elke

doi:10.1524/zkri.216.8.430.20356

Cited by 5 publications

(2 citation statements)

References 31 publications

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“…Koch, 2000) or into one-periodic tubes or finite polyhedra (cf. Koch, 2001) shows such a generating polygon.…”

Section: Minimal Surfaces With Parallel Flat Labyrinthsmentioning

confidence: 99%

Self-intersecting three-periodic minimal surfaces forming two-periodic (flat) labyrinths

Koch¹

2000

Zeitschrift Für Kristallographie - Crystalline Materials

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14 families of minimal surfaces with straight self-intersections have been derived which subdivide R 3 into infinitely many congruent, two-periodic`flat labyrinths'. For eleven families, all flat labyrinths are parallel to each other. Two sets of mutual perpendicular flat labyrinths have been found three times. All these minimal surfaces are non-orientable. Their Euler characteristics vary between À3 and À13.

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“…Koch, 2000) or into one-periodic tubes or finite polyhedra (cf. Koch, 2001) shows such a generating polygon.…”

Section: Minimal Surfaces With Parallel Flat Labyrinthsmentioning

confidence: 99%

Self-intersecting three-periodic minimal surfaces forming two-periodic (flat) labyrinths

Koch¹

2000

Zeitschrift Für Kristallographie - Crystalline Materials

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“…This is not just an unnecessary word repetition because three-periodic surfaces may have layer-like, rod-like or even polyhedral 'labyrinths'(Koch, 2000b(Koch, , 2001.…”

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confidence: 99%

On the group-theoretical approach to the study of interpenetrating nets

Baburin

2016

Acta Crystallogr A Found Adv

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Using group-subgroup and group-supergroup relations, a general theoretical framework is developed to describe and derive interpenetrating 3-periodic nets. The generation of interpenetration patterns is readily accomplished by replicating a single net with a supergroup G of its space group H under the condition that site symmetries of vertices and edges are the same in both H and G. It is shown that interpenetrating nets cannot be mapped onto each other by mirror reflections because otherwise edge crossings would necessarily occur in the embedding. For the same reason any other rotation or roto-inversion axes from G \ H are not allowed to intersect vertices or edges of the nets. This property significantly narrows the set of supergroups to be included in the derivation of interpenetrating nets. A procedure is described based on the automorphism group of a Hopf ring net [Alexandrov et al. (2012). Acta Cryst. A68, 484-493] to determine maximal symmetries compatible with interpenetration patterns. The proposed approach is illustrated by examples of twofold interpenetrated utp, dia and pcu nets, as well as multiple copies of enantiomorphic quartz (qtz) networks. Some applications to polycatenated 2-periodic layers are also discussed.

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Modelling of the α (clathrate VIII) ⇌ β (clathrate I) phase transition in Eu8Ga16Ge30

Leoni

Carrillo‐Cabrera

Grin

2003

Journal of Alloys and Compounds

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Self-intersecting three-periodic minimal surfaces forming one-periodic tubes or finite polyhedra

Abstract: Abstract17 families of three-periodic minimal surfaces with straight self-intersections have been derived that subdivide

Cited by 5 publications

References 31 publications

Self-intersecting three-periodic minimal surfaces forming two-periodic (flat) labyrinths

Self-intersecting three-periodic minimal surfaces forming two-periodic (flat) labyrinths

On the group-theoretical approach to the study of interpenetrating nets

Modelling of the α (clathrate VIII) ⇌ β (clathrate I) phase transition in Eu8Ga16Ge30

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