You have raised too many questions under one thread. Dear @Raja, my contribution will be the easiest one, about Natural Geometry!

"What about natural geometry? For 2500 years, the system of geometry that has appeared most natural to human adults is Euclidean plane geometry: a formal system for characterizing two-dimensional shapes in accord with the distance, angle, and directional relationships among their parts. In Euclidean geometry, all forms that differ from one another have parts that contrast on one or more of these properties; conversely, two forms whose points and lines are characterized by the same distance, angle, and directional relationships are congruent. The geometrical equivalence of two forms with the same length, angle, and sense relations is so immediately apparent to the human mind that philosophers and mathematicians believed for millennia that Euclidean geometry constituted the only logically possible geometric system. Although this view was overturned more than two centuries ago, Euclidean geometry continues to be the first system of formal geometry that students learn, and the system whose principles accord best with human intuition...

Research on young human children, on non-human animals, and on human adults in diverse cultures, provides evidence for at least two core systems of geometry that are present and functional early in human development, that predate the evolution of humans as a species, and that remain universally present in human adults. Research on older children provides evidence for the emergence of capacities to relate these systems. At 4 years, children appear to relate the shapes of objects and of the surface layout only with respect to their common distance relations. With development, however, children also relate these representations on the basis of angle, and by adulthood, direction. We have learned little, thus far, about the processes by which this integration occurs. What leads the child to view the navigable surface layout as a large-scale form, with the angular relationships that apply to visible objects? And what leads the child to view small visible forms as an array of paths and places to which one can navigate in one’s mind?"

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2897178/

Dear Raja, symmetry and pattern are something we as humans perceive, and thus pertains to human perception. It is we who, after realizing the recurrence of a form in nature, declare it a "pattern" or a symmetry. Without human perception, there would be no symmetry or patterns. After experiencing a series of phenomena in nature we infer abstract forms, more perfect that the actual natural forms: we call them "circle", "square", etc. Our circle is "perfect", while natural circles are seldom perfect. This is due to our capacity for abstraction. We endow nature with symmetry, with geometrical shapes, with patterns, although all of them are imperfect in reality.

When you ask what is the force behind symmetry and pattern, I would answer that it is the human force and capacity for inference of shapes and patterns, our capacity to perfect and classify them, and to use them according to our will. Nature does not care about this because nature has no will.

What would have happened if our perception had preferred the accidental, the formless? Our civilization would be quite different, I assure you! It is we humans who seek repetition and symmetry, because nature produces the formless in probably greater quantity. We chose to see perfection among nature's haphazard products.

Thank you for this question, although my answer is probably not what you expected. :-)

Best regards, Lilliana

Dear Raja, the main factor that is common in all your questions, is entropy. It can be shown that entropy is a major factor in evolution, and in the existence of shapes at all.

Thus it is also a factor in the nature of those shapes, and in the Symmetry that they exhibit.

 Dear Napoleon, Dear Ljubomir, Dear Lilliana and Dear Graeme Smith, its very diversify and nice input by you, Thanks.

Still the mystery of symmetry and patterns is there, as the actual force behind it is unknown yet. But it can be assumed that there should be some logical design behind it either using the known forces or some unknown force/particle.......here we can make a hypothetical assumption that symmetry and pattern give ground strength to universe (either its volunteer or in-volunteer) so its important in structural aspects of universe........if it get disturbed then we can face unstable universal system, So there should be some force with any possible nature of wave(s) / particle(s) behind the symmetry and patterns even if such symmetries and patterns are the result of balance among atomic forces................!!

it can be called "SYMPAT" class of  wave / particle(s).....but i think/believe its there with core responsibilities of distance and angles maintenance along some higher level of intelligence among all other kind of particles........it should have some thing unique than other particles and waves, might be well equipped with more than two types of charges to produce directional force even in very fast movements.......if so then it should be among the first few particles which got produce with birth of universe...........its magnitude should be much lesser than photons and its not necessary that it can reside only in atoms............it looks like "Eye of particle" ........................!!

Thanks

Graeme,

yes, entropy and please do not forget energy. Entropy have to increase and energy is going to a minimum. That's all. Geometry and symmetry are human inventions.

Wilfried, I am not sure there isn't a natural geometery, actually I think of entropy as the energy/entropy balance, which is why I mentioned just entropy. When energy and entropy are in balance, you get stability when energy is the greater, you get change, and when entropy is the greater you get crystalization into symmetric shapes.

Dear 

Assuming symmetry exist at some level of meassurable reallity, the forces that produces these symetries are the classical ones : electro-magnetic, mechanical, etc. furtanately they are described by laws: newtonian, thermodinamics (from clasius to boltzman and shrodinger, etc).

Thanks, if the theory of "SYMPAT" class of wave / particles exists.......then it should not be alone, rather, there should be acceptable presence of an unknown "Master Family" of waves / particles exists behind-the-known-scenes of forces.

Graeme,

crystals have a high symmetry. There are some different crystal systems you can find in chemistry, crystallography and metallurgy. All these configurations can be understood by arranging spheres with equal or with different diameters. There is no metaphysical force, no mystery nothing dubious but energy and entropy.

Graeme,

" I am not sure there isn't a natural geometery". 

Naturally, we find there symmetry but nature does not know what symmetry is. We can categorize these structures as highly symmetrical but we do not need this expression to understand the composition.

Wilfried, a ratio isn't mysterious or metaphysical, and symmetry does not need to be part of a natural geometery, so it makes no difference whether nature "Knows" about symmetry. The fact that symmetry is rare suggests merely that energy is very available and the energy/entropy ballance usually swings over to favor energy. This could be because entropy works faster than energy.

Graeme,

I think our positions are not far from each other but we use different imaginations. Entropy is a question of statistics and energy is the motor to drive things. Symmetry is not found in any of the two concepts as a variable.

I think of entropy as the inverse of energy, whether it is expressed in a statistical manner or not is really beside the point. Energy is indeed the driver of the whole system, and it is entropy that causes crystalization. Symmetry is just the result of Crystalization or equivalent effects.

Dear All, I find an interesting article today.....it might help us further, as many things are interlinked.

http://www.physics-astronomy.com/2016/03/the-2-most-dangerous-numbers-in.html?m=1#.Vw973TiCbqA

Graeme,

I am sure you know the GIBBS-HELMHOLTZ equation  dG = dH - T*dS.

in this equation entropy and energy (enthalpie) are mixed together for free energy..

I do not understand your metapher of entropy as "nverse of energy,". Entropy causes also not in every case crystallization, it depends of the temperature T.