Five-fold symmetry in crystalline quasicrystal lattices

To demonstrate that crystallographic methods can be applied to index and interpret diffraction patterns from well-ordered quasicrystals that display non-crystallographic 5-fold symmetry, we have characterized the properties of a series of periodic two-dimensional lattices built from pentagons, called Fibonacci pentilings, which resemble aperiodic Penrose tilings. The computed diffraction patterns from periodic pentilings with moderate size unit cells show decagonal symmetry and are virtually indistinguishable from that of the infinite aperiodic pentiling. We identify the vertices and centers of the pentagons forming the pentiling with the positions of transition metal atoms projected on the plane perpendicular to the decagonal axis of quasicrystals whose structure is related to crystalline η phase alloys. The characteristic length scale of the pentiling lattices, evident from the Patterson (autocorrelation) function, is ∼τ2 times the pentagon edge length, where τ is the golden ratio. Within this distance there are a finite number of local atomic motifs whose structure can be crystallographically refined against the experimentally measured diffraction data.

From symmetry to asymmetry: Phylogenetic patterns of asymmetry variation in animals and their evolutionary significance

Phylogenetic analyses of asymmetry variation offer a powerful tool for exploring the interplay between ontogeny and evolution because (i) conspicuous asymmetries exist in many higher metazoans with widely varying modes of development, (ii) patterns of bilateral variation within species may identify genetically and environmentally triggered asymmetries, and (iii) asymmetries arising at different times during development may be more sensitive to internal cytoplasmic inhomogeneities compared to external environmental stimuli. Using four broadly comparable asymmetry states (symmetry, antisymmetry, dextral, and sinistral), and two stages at which asymmetry appears developmentally (larval and postlarval), I evaluated relations between ontogenetic and phylogenetic patterns of asymmetry variation. Among 140 inferred phylogenetic transitions between asymmetry states, recorded from 11 classes in five phyla, directional asymmetry (dextral or sinistral) evolved directly from symmetrical ancestors proportionally more frequently among larval asymmetries. In contrast, antisymmetry, either as an end state or as a transitional stage preceding directional asymmetry, was confined primarily to postlarval asymmetries. The ontogenetic origin of asymmetry thus significantly influences its subsequent evolution. Furthermore, because antisymmetry typically signals an environmentally triggered asymmetry, the phylogenetic transition from antisymmetry to directional asymmetry suggests that many cases of laterally fixed asymmetries evolved via genetic assimilation.

Symmetry in chemistry from the hydrogen atom to proteins

The last 2 decades have seen discoveries in highly excited states of atoms and molecules of phenomena that are qualitatively different from the “planetary” model of the atom, and the near-rigid model of molecules, characteristic of these systems in their low-energy states. A unified view is emerging in terms of approximate dynamical symmetry principles. Highly excited states of two-electron atoms display “molecular” behavior of a nonrigid linear structure undergoing collective rotation and vibration. Highly excited states of molecules described in the “standard molecular model” display normal mode couplings, which induce bifurcations on the route to molecular chaos. New approaches such as rigid–nonrigid correlation, vibrons, and quantum groups suggest a unified view of collective electronic motion in atoms and nuclear motion in molecules.

Independent determination of symmetry and polarity in the Drosophila eye.

In each facet of the Drosophila compound eye, a cluster of photoreceptor cells assumes an asymmetric trapezoidal pattern. These clusters have opposite orientations above and below an equator, showing global dorsoventral mirror symmetry. However, in the mutant spiny legs, the polarization of each cluster appears to be random, so that no equator is evident. The apparent lack of an equator suggests that spiny legs+ may be involved in the establishment of global dorsoventral identity that might be essential for proper polarization of the photoreceptor clusters. Alternatively, a global dorsoventral pattern could be present, but spiny legs+ may be required for local polarization of individual clusters. Using an enhancer trap strain in which white+ gene expression is restricted to the dorsal field, we show that white+ expression in spiny legs correctly respects dorsoventral position even in facets with inappropriate polarizations; the dorsoventral boundary is indeed present, whereas the mechanism for polarization is perturbed. It is suggested that the boundary is established before the action of spiny legs+ by an independent mechanism.