Description / Table of Contents:
INTRODUCTION Theoretical modelling and the use of mathematical methods are presently gaining
in importance since progress in both geology and mathematics offers new possibilities
to combine both fields. Most geological problems are inherently geometrical and morphological,
and, therefore, amenable to a classification of forms from a "Gestalt point of
view". Geometrical objects have to possess an inherent stability in order to preserve
their essential quality under slight deformations. Otherwise, we could hardly conceive
of them or describe them, and today's observation would not reproduce yesterday's result
(DANGELMAYR & GÜTTINGER, 1982). This principle has become known as "structural
stability" (THOM, 1975), i.e. the persistence of a phenomenon under all allowed perturbations.
Stability is also, of course, an assumption of classical Newtonian physics, which
is essentially the theory of various kinds of smooth behavior (POSTON &STEWART, 1978).
However, things sometimes "jump". A new species with a different morphology appears
suddenly in the paleontological record (EI.DREDGE & GOULD, 1972), a fault develops,
a landslide moves, a computer program becomes unstable with a certain data configuration,
etc. It is, surprisingly, the topological approach which permits the study of a broad
range of such phenomena in a coherent manner (POSTON & STEWART, 1978; LU, 1976;
STEWART, 1982). The universal singularities and bifurcation processes derived from the
concept of structural stabiIity determine the spontaneous formation of qualitatively similar
spatio-temporal structures in systems of various geneses exhibiting critical behavior
(DANGELMAYR & GÜTTINGER, 1982; THOM, 1975; POSTON & STEWART, 1978; GÜTTINGER & EIKEMEIER, t979; STEWART, 1981). In addition, this return to a "geometrization
of phenomena"-- after decades of algorithmization-- comes much closer to the
geologist's intuitive geometric reasoning. It is the aim of this study to elucidate, by
examples, how the qualitative geometrical approach allows one to classify forms and
to control the behavior of complex computer algorithms...
Pages:
Online-Ressource (229 Seiten)
ISBN:
9783540139836
URL:
https://doi.org/10.1007/BFb0010505
Language:
English
