Geometry of interfaces: topological complexity in biology and materials

Hyde, Stephen T.; Schröder‐Turk, Gerd E.

doi:10.1098/rsfs.2012.0035

Cited by 51 publications

(47 citation statements)

References 65 publications

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“…[ 20,21 ] Indeed, by joining the centrelines of the gyroid to create a periodic graph ("srs-net") one can readily identify local helices along the various screw axes of the gyroid. its mirror image cannot be transformed to coincide with itself by translation and rotation).…”

Section: Gyroid Geometry and Mathematical Representationmentioning

confidence: 99%

“…[ 13,15,21,56,60,68 ] Whereas 3D photonic crystals still present various challenges to artifi cial synthesis, nature provides a relative abundance evolved over millions of years. [ 13,15,21,56,60,68 ] Whereas 3D photonic crystals still present various challenges to artifi cial synthesis, nature provides a relative abundance evolved over millions of years.…”

Section: Biological Gyroid Structuresmentioning

confidence: 99%

See 1 more Smart Citation

Optical Properties of Gyroid Structured Materials: From Photonic Crystals to Metamaterials

Dolan

Wilts

Vignolini

et al. 2014

Advanced Optical Materials

234

224

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The gyroid is a continuous and triply periodic cubic morphology which possesses a constant mean curvature surface across a range of volumetric fi ll fractions. Found in a variety of natural and synthetic systems which form through self-assembly, from butterfl y wing scales to block copolymers, the gyroid also exhibits an inherent chirality not observed in any other similar morphologies. These unique geometrical properties impart to gyroid structured materials a host of interesting optical properties. Depending on the length scale on which the constituent materials are organised, these properties arise from starkly different physical mechanisms (such as a complete photonic bandgap for photonic crystals and a greatly depressed plasma frequency for optical metamaterials). This article reviews the theoretical predictions and experimental observations of the optical properties of two fundamental classes of gyroid structured materials: photonic crystals (wavelength scale) and metamaterials (sub-wavelength scale).

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Section: Gyroid Geometry and Mathematical Representationmentioning

confidence: 99%

Section: Biological Gyroid Structuresmentioning

confidence: 99%

Optical Properties of Gyroid Structured Materials: From Photonic Crystals to Metamaterials

Dolan

Wilts

Vignolini

et al. 2014

Advanced Optical Materials

234

224

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“…Consider for example an elastic curve on a minimal surface. Locally, such surfaces minimize area; under appropriate conditions, they also minimize the surface bending energy of a symmetric fluid membrane; as such, they feature frequently in the morphology of biological membranes (see, for example, [14]). Because its mean curvature vanishes, the surface assumes a symmetric saddle shape almost everywhere.…”

Section: Introductionmentioning

confidence: 99%

Environmental bias and elastic curves on surfaces

Guven¹,

Valencia²,

Vázquez-Montejo³

2014

J. Phys. A: Math. Theor.

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The behavior of an elastic curve bound to a surface will reflect the geometry of its environment. This may occur in an obvious way: the curve may deform freely along directions tangent to the surface, but not along the surface normal. However, even if the energy itself is symmetric in the curve's geodesic and normal curvatures, which control these modes, very distinct roles are played by the two. If the elastic curve binds preferentially on one side, or is itself assembled on the surface, not only would one expect the bending moduli associated with the two modes to differ, binding along specific directions, reflected in spontaneous values of these curvatures, may be favored. The shape equations describing the equilibrium states of a surface curve described by an elastic energy accommodating environmental factors will be identified by adapting the method of Lagrange multipliers to the Darboux frame associated with the curve. The forces transmitted to the surface along the surface normal will be determined. Features associated with a number of different energies, both of physical relevance and of mathematical interest, are described. The conservation laws associated with trajectories on surface geometries exhibiting continuous symmetries are also examined.

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“…In many cases, the color of bird feathers, butterfly wings, and the carapaces or some fruits and leaves are not produced by pigments or dyes. For a detailed analysis and classification of crystalline structures in biological photonic crystals, see the work of Hyde and Schroder-Turk [231]. In fact, these nanopatterns are nothing other than what is known nowadays as photonic structures, a kind of novel structure of great technological interest that life started to create about 600 million years ago [228][229][230].…”

Section: Crystals To Manipulate Lightmentioning

confidence: 99%