Minimal nets and minimal minimal surfaces

Abstract

Author(s) of this paper may load this reprint on their own web site or institutional repository provided that this cover page is retained. Republication of this article or its storage in electronic databases other than as specified above is not permitted without prior permission in writing from the IUCr.For further information see http://journals.iucr.org/services/authorrights.html Acta Crystallographica Section A: Foundations of Crystallography covers theoretical and fundamental aspects of the structure of matter. The journal is the prime forum for research in diffraction physics and the theory of crystallographic structure determination by diffraction methods using X-rays, neutrons and electrons. The structures include periodic and aperiodic crystals, and non-periodic disordered materials, and the corresponding Bragg, satellite and diffuse scattering, thermal motion and symmetry aspects. Spatial resolutions range from the subatomic domain in charge-density studies to nanodimensional imperfections such as dislocations and twin walls. The chemistry encompasses metals, alloys, and inorganic, organic and biological materials. Structure prediction and properties such as the theory of phase transformations are also covered.Crystallography Journals Online is available from journals.iucr.org Acta Cryst. (2013 The 3-periodic nets of genus 3 ('minimal nets') are reviewed and their symmetries re-examined. Although they are all crystallographic, seven of the 15 only have maximum-symmetry embeddings if some links are allowed to have zero length. The connection between the minimal nets and the genus-3 zeromean-curvature surfaces ('minimal minimal' surfaces) is explored by determining the surface associated with a net that has a self-dual tiling. The fact that there are only five such surfaces but 15 minimal nets is rationalized by showing that all the minimal nets can serve as the labyrinth graph of one of the known minimal minimal surfaces.

Generating carbon schwarzites via zeolite-templating

“…Of the 15 ZTCs we suggest as experimentally accessible, the majority form surfaces that partition space into two disjoint labyrinths that can be described by the nets running through them, as is common in the study of TPMSs ( Fig. 4 ) ( 16 – 18 ). Only ZTC-RWY does not form a surface that bounds two labyrinths, with the structure instead consisting of a body-centered cubic packing of fullerenes ( Fig.…”

“…To illustrate these points, let us attempt to solve the inverse problem for a set of target TPMSs. A natural choice of targets is the “minimal minimal surfaces,” so called because they have genus 3, the smallest possible genus for a TPMS ( 18 ). Only five minimal minimal surfaces are known; these are the Schwarz D, CLP, H, and P TPMSs, and the Schoen G TPMS (deformations of which admit lower symmetry variants like the tD, mDCLP, and rPD TPMSs) ( 16 , 18 , 41 ).…”

“…Amorphous carbons consisting of local regions with negative Gaussian curvature have been observed [e.g., Bourgeois and Bursill ( 13 ) and Barborini et al ( 14 )]. These materials have been noted to be similar to “random schwarzite” models that have local properties similar to schwarzites ( 15 ), but they lack the long-range ordering required for characterization of their topologies by the periodic nets that describe TPMSs ( 16 – 18 ).…”

Coexistence of both gyroid chiralities in individual butterfly wing scales of Callophrys rubi

“…The gyroid geometry, which serves as biophotonic crystals, has cubic symmetry (I4 1 32) and is characterized by the topologically particularly simple srs-net (the label for the chiral degree-three network modeled on SrSi2) (11). As a chiral structure, it can be realized as one of two enantiomers (related by mirror symmetry) that are here called RH and LH.…”

In a Material World. Hyperbolische Geometrie in biologischen Materialien

“…Die komplexe Form symmetrischer bikontinuierlicher Minimalflächen lässt sich durch ein geometrisches Netzwerk aus Stäben und Knoten, das in den Labyrinthen zentriert liegt, veranschaulichen. Dieses GyroidNetzwerk, bekannt unter dem Namen srs [16,23], besteht aus (bis auf Translation und Symmetrie) identischen, dreivalenten Ecken und identischen Kanten und ist eines der fundamentalen periodischen Netzwerke [3]. Schoen fand den Gyroid bei einer Untersuchung der Möglichkei-ten, zwei identische Netzwerke auf symmetrische Weise ineinanderzulegen, als die Minimalfläche zwischen zwei ineinander verwobenen srs-Netzwerken [30].…”