Symmetries in geology and geophysics

Symmetries have played an important role in a variety of problems in geology and geophysics. A large fraction of studies in mineralogy are devoted to the symmetry properties of crystals. In this paper, however, the emphasis will be on scale-invariant (fractal) symmetries. The earth’s topography is an example of both statistically self-similar and self-affine fractals. Landforms are also associated with drainage networks, which are statistical fractal trees. A universal feature of drainage networks and other growth networks is side branching. Deterministic space-filling networks with side-branching symmetries are illustrated. It is shown that naturally occurring drainage networks have symmetries similar to diffusion-limited aggregation clusters.

A Carboniferous insect gall: insight into early ecologic history of the Holometabola.

Although the prevalence or even occurrence of insect herbivory during the Late Carboniferous (Pennsylvanian) has been questioned, we present the earliest-known ecologic evidence showing that by Late Pennsylvanian times (302 million years ago) a larva of the Holometabola was galling the internal tissue of Psaronius tree-fern fronds. Several diagnostic cellular and histological features of these petiole galls have been preserved in exquisite detail, including an excavated axial lumen filled with fecal pellets and comminuted frass, plant-produced response tissue surrounding the lumen, and specificity by the larval herbivore for a particular host species and tissue type. Whereas most suggestions over-whelmingly support the evolution of such intimate and reciprocal plant-insect interactions 175 million years later, we provide documentation that before the demise of Pennsylvanian age coal-swamp forests, a highly stereotyped life cycle was already established between an insect that was consuming internal plant tissue and a vascular plant host responding to that herbivory. This and related discoveries of insect herbivore consumption of Psaronius tissues indicate that modern-style herbivores were established in Late Pennsylvanian coal-swamp forests.

Scaling in geology: landforms and earthquakes.

Landforms and earthquakes appear to be extremely complex; yet, there is order in the complexity. Both satisfy fractal statistics in a variety of ways. A basic question is whether the fractal behavior is due to scale invariance or is the signature of a broadly applicable class of physical processes. Both landscape evolution and regional seismicity appear to be examples of self-organized critical phenomena. A variety of statistical models have been proposed to model landforms, including diffusion-limited aggregation, self-avoiding percolation, and cellular automata. Many authors have studied the behavior of multiple slider-block models, both in terms of the rupture of a fault to generate an earthquake and in terms of the interactions between faults associated with regional seismicity. The slider-block models exhibit a remarkably rich spectrum of behavior; two slider blocks can exhibit low-order chaotic behavior. Large numbers of slider blocks clearly exhibit self-organized critical behavior.