Whereas in crystallography, as for example in the case of symmetry classification of atomic crystal groups there are only 234 possible classes of symmetry space-groups (that are precisely defined mathematically in terms of symmetry operations such as reflections and rotations, etc.), in material science, nanoscience (e.g. molecular crystals and quasi-crystals) things may appear not be as neatly and completely defined even when the local symmetries are precise and are well-defined because mathematical groups become then insufficient to classify the latter structures that possess broken symmetry, or only local symmetries thereof. A certain type of asymmetry or noncommutativity is fundamental in non-Abelian mathematics, such as Non-Abelian Algebraic Topology, Anabelian Geometry and Noncommutative Geometry.
http://hal.archives-ouvertes.fr/docs/00/20/97/39/PDF/ajp-jphys_1984. This botany paper has pretty far reaching and simple mathematical properties. Limitation is necessary for growth and disrupts symmetry, but growth continues and creates symmetry despite disruption. This gets down to system entropy and is often based on fibonacci approximation of golden mean in oscillating limits. It's general but "nature" is a pretty big place to look!
Hi Ion: You must be talking about some high crystallography! I know only 230 space from my International Tables! You can of course increase them largely if you take into acount time reveral symmetry (magnetic black and white space groups)!! I was just joking of course!
You have raised a very important point. The reductionists think that if you go down and down and discover the fundamental laws of nature they you can explain everything. This is of course a fool's philosophy. The physical laws are highly symmetric. But the world we see is not symmetric. The symmetry breaking is the most important ingredient to explain our macroscopic world. A large collection of particle (N-body problem) often beak symmetries of the fundamental physical laws and lo the emergent phenomena: materials science and even biology.
I don't have any more now. I shall come back to you later and discuss further with you about symmetry breaking and emergent phenomena. More is different as P>W> Anderson says!
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