Nature provides impressive examples of chiral photonic crystals, with the notable example of the cubic so-called srs network (the label for the chiral degree-three network modeled on SrSi2) or gyroid structure realized in wing scales of several butterfly species. By a circular polarization analysis of the band structure of such networks, we demonstrate strong circular dichroism effects: The butterfly srs microstructure, of cubic I4(1)32 symmetry, shows significant circular dichroism for blue to ultraviolet light, that warrants a search for biological receptors sensitive to circular polarization. A derived synthetic structure based on four like-handed silicon srs nets exhibits a large circular polarization stop band of a width exceeding 30%. These findings offer design principles for chiral photonic devices.

We construct the entire three-parameter family of embedded constant mean curvature surfaces with three ends and genus zero. They are classified by triples of points on the sphere whose distances are the necksizes of the three ends. Because our surfaces are transcendental, and are not described by any ordinary di¤erential equation, it is remarkable to obtain such an explicit determination of their moduli space.

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Inverse bicontinuous cubic phases with two aqueous network domains separated by a smooth bilayer are firmly established as equilibrium phases in lipid/water systems. The purpose of this article is to highlight the generalisations of these bicontinuous geometries to polycontinuous geometries, which could be realised as lipid mesophases with three or more network-like aqueous domains separated by a branched bilayer. An analysis of structural homogeneity in terms of bilayer width variations reveals that ordered polycontinuous geometries are likely candidates for lipid mesophase structures, with similar chain packing characteristics to the inverse micellar phases (that once were believed not to exist due to high packing frustration). The average molecular shape required by global geometry to form these multi-network phases is quantified by the surfactant shape parameter, v/(al); we find that it adopts values close to those of the known lipid phases. We specifically analyse the 3etc(187 193) structure of hexagonal symmetry P6 3 /mcm with three aqueous domains, the 3dia(24 220) structure of cubic symmetry I 43d composed of three distorted diamond networks, the cubic chiral 4srs(24 208) with cubic symmetry P4 2 32 and the achiral 4srs(5 133) structure of symmetry P4 2 /nbc, each consisting of four intergrown undistorted copies of the srs net (the same net as in the Q G II gyroid phase). Structural homogeneity is analysed by a medial surface approach assuming that the headgroup interfaces are constant mean curvature surfaces. To facilitate future experimental identification, we provide simulated SAXS scattering patterns that, for the 4srs(24 208) and 3dia(24 220) structures, bear remarkable similarity to those of bicontinuous Q G II -gyroid and Q D II -diamond phases, with comparable lattice parameters and only a single peak that cannot be indexed to the wellestablished structures. While polycontinuous lipid phases have, to date, not been reported, the likelihood of their formation is further indicated by the reported observation of a solid tricontinuous mesoporous silicate structure, termed IBN-9, which formed in the presence of surfactants [Han et al