Is it correct to discuss "Symmetry" and "Breaking of symmetry" on equal footing? My Background: A senior Undergraduate Student With no Idea of Field Theory Whatsoever.
I am looking for a very general and natural answer to my question.
I noticed that Symmetry is not something nature abides by. In layman's terms, the planet revolves in an elliptical orbit and not circular orbit. And It's hard to explain this fact until you do Back calculations with equations in classical mechanics.
when I am saying the word "symmetry" I am using this word in a very general sense and not very accurate and rigorous.

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*What Importance Does "Breaking of Symmetry" plays in our daily observable universe? or What Importance Does "Breaking of Symmetry" play for a physicist in the context of the quest for fundamental theory? If you are a theoretical physics, this will be obvious, But not for an Undergraduate.


*Is breaking of symmetry only important in field theory (Particle Physics, Higgs Field)? Or It is more general that we can find in nature?
I have found that certain optimization problems or Human evolution do not obey symmetry. Otherwise, we would have eyes possible to see in 360 degrees.


*Is finding "Breaking of Symmetry" as normal and as important as finding symmetry in nature? Is the breaking of symmetry as fundamental and natural as finding symmetry?


*With lots many examples of breaking of symmetry (in layman terms), Why do we care about Symmetry.
you are working on theory, if you find symmetry, then good the task becomes easy. if you don't then become stoic.
one can write a good essay or give a talk on symmetry in nature and its uses, similarly, can you intuitively explain the "Breaking of symmetry " How Breaking of symmetry is important in nature, evolution, and fundamental physics?
when one searches for "symmetry breaking" Google only provides you field theory and it seems that you have searched for something wrong which you shouldn't do until you are a graduate student.
I don't know Is it correct to discuss "Symmetry" and "Breaking of symmetry" on equal footing?
 A: Symmetry breaking is, in fact a topic within field theory. To understand it means understanding the mathematics of field theory. Certainly we can handwave and speak in analogies, but this does not convey accurately what the topic is really about or what it does for us.
I will say this though. There is a difference between what we call symmetry and what might colloquially be described as symmetry. Sure there are similarities and overlaps between the two, but they are not generically the same time. When a physicist says 'symmetry" they mean a transformation which leaves the action invariant. The reason to define the word symmetry in this way is because it's precisely transformations which obey this condition that can be used with Noether's theorem, and that's the source of essentially all the power in symmetry.
This definition of symmetry is a in terms of the action, or (almost) equivalently, in terms of the equations that solutions satisfy. This is not a statement about the solutions themselves. In other words, symmetries as physicists define them map solutions to solutions, but do not necessarily map a given solution back to itself.
A: In the most formal sense (contrasting with the informal usage), a symmetry means that if you know something about one state, you can infer information about another state.  The most trivial example is that if you've seen that gravity works in the past, you may assume it will work in the future.  This is an example of a time-translation symmetry.  The laws of physics at $t=t_0$ are the same as the laws of physics at $t=t_0+\Delta T$.  Likewise, if a system has $C_6$ symmetry, hexagonal symmetry without rotation, if you know something about one point, you know something about five other points.
Where there is symmetry, there is always a structure that we can talk to.  In the most common usages in physics, we find that (continuous) symmetries in what is called the Lagrangian correspond to conserved values like energy and momentum.  This particular variant of symmetry breaking is very specific to physics.  You wont see it much elsewhere.
The breaking of symmetry occurs when a very nuanced exacting symmetry "breaks down" into a system with multiple simpler symmetries.  This is a very big deal for particle physics, because highly symmetric answers to "life the universe and everything" are preferred in an anthropomorphic sense.  We like the idea that the rules that are followed are "simple," and "simple" tends to get translated into having a very profound symmetry.  As an intuitive example, we find it "simpler" to have laws of physics which are the same at every point in space, rather than having special rules which apply only to me, and do not apply to everyone else.
Symmetry breaking is a very essential thing in daily life, although our brains tend to handle it for us such that we often don't see it occurring.  Where there are symmetries, there are typically comparisons which have a derivative of 0.  These are situations that leave us with very few opportunities to react quickly.  For a biological example, one can look at looming.  In this situation, we recognize that if you're looking at an object that is approaching you, it doesn't seem to change very much.  There is a projective symmetry associated with an object getting closer to you.  Dragonflies leverage this.  When they attack a prey insect, they approach along a constant-bearing decending-range path.  This means that the dragon fly appears to stay in the same part of the target's field of view and simply gets larger.  As it turns out, this results in very minimal changes in the way the prey insect perceives the threat because there is such a strong symmetry between the scene at one moment and the scene at the next, so the prey doesn't succeed at moving out of the way until it is too later.
As a result, prey flies may choose to fly on a trajectory which breaks as many of these symmetries as possible, giving it the highest liklihood of detecting the dragon fly.  They can try to give the dragonfly as few approaches as possible which can achieve this constant-bearing decreasing-range approach.
In social settings we can recognize this in situations like "you are with us or you are against us."  In these settings, we are aware that there are symmetries to the problem which leave us naked and unprepared to deal with some adversary because we cannot see them acting along a particular path until it is too late to stop them.  We choose to expend energy to break those symmetries, forcing people into camp A or camp B, such that there is no such path that can blindside us.
So these kinds of symmetry breaking operations do appear everywhere.  However, they are typically handled at a very intuitive and informal level.  Symmetry breaking in the physics sense takes it to a whole new level in the amount of raw math they are able to apply to the topic.
A: I think the questions in this  post are  "philosophy of science" material.

What Importance Does "Breaking of Symmetry" plays in our daily observable universe

A great importance in vehicles : think of an asymmetric in mass car. When the weight is not symmetric to the direction of motion one continually compensates. It is important that the car itself is symmetric about its long axis.
The same with constructing buildings, symmetry should not be broken in the various directions so as to have strong structures.etc
A: Symmetry is a much larger concept than symmetry breaking. The latter is specific to the field theory and the condensed matter physics (in the context of phase transitions, which can often be characterized as states with different symmetries).
Symmetry is however a much larger subject: the other applications in physics that readily come to mind are the applications of group theory to molecules (particularly to molecular spectra) and crystallography.
A: There are many examples of symmetry breaking outside of field theory and particle physics. Crystallisation involves symmetry breaking in the transition from a highly symmetric liquid phase to crystals with a smaller discrete set of symmetries. The formation of magnetic domains in a ferromagnetic substance is another example. In biology a symmetric single-celled zygote must develop through a process of regional differentiation to ultimately become a non-symmetric organism. Every time you toss a coin or throw a dice you are breaking a symmetric set of expectations to produce a non-symmetric outcome.
At a theoretical level, stable and unstable equilibrium states are studied in the theory of dynamical systems and the transition from a stable equilibrium state to an unstable state (with subsequent symmetry breaking) is studied in bifurcation theory and catastrophe theory. These topics can certainly be covered at undergraduate level.
