What do we mean by symmetry?
Symmetry, arguably, is an aesthetic; it measures how nice something looks. But there is a more quantitative aspect to it. Namely, something symmetric is something that ''looks the same'' from different angle. For instance, if your face possesses left-right symmetry, then that means that it looks exactly the same as the face you see in the mirror. If, on the other hand, it
didn' t possess left-right symmetry (say, if you had a mole on your left cheek) then your mirror image looks ''different'' from your own image. In the mirror image, the mole is on the right cheek.
A circle is an example of a ''lot'' of symmetry
. You could rotate the paper any amount around the center of the circle, and you still get a circle. You can also reflect the circle about any of its diameters and get back the circle: the circle enjoys a ''mirror symmetry''. [[Image:Circlegraffiti.png|thumb|400px|right|A symmetry-rich circle is rendered symmetry-free by graffiti]]
In other words, symmetry is the fact that if you ''make some change'' (either in the object itself or in your perspective) the object looks exactly the same. The ''extent'' of symmetry can now be described by the number of such different perspectives you can use. For instance, an equilateral triangle possesses ''some'' symmetry: if you rotate by certain angles, it doesn't change. But rotating by an arbitrary angle does ''not'' send the equilateral triangle to itself. So the equilateral triangle isn't quite as symmetric as the circle.
The extent of symmetry of an object is measured by the set of transformations that map the object to itself.
Object can be replaced by a structure, or rule. We're used to thinking of symmetries of concrete objects (like mice, clocks, and historical monuments). But symmetry is a deeper concept.
===Symmetry from a fairness perspective===
Let's go a little further with the idea of symmetry in laws, but this time, from the point of view of ''law'' in the sense of laws made by humans. We'd ideally like our laws to enjoy a certain kind of symmetry: a ''fairness''. For instance
, if a set of people appear in a certain court case in a certain case, the outcome should be independent of race, gender, caste etc. of the people. So we'd like our law to be invariant under the ''symmetries'' of changing the ''people'' involved.
same can be said about providing any service fairly: the level of service provided should be invariant under all the ways of '' permuting'' the people around.
This also relates to another important fact: equality , and fairness , translate in group theory to invariance under a certain group action. The group action is the one that permutes the elements.
===Symmetry from an indistinguishability perspective===
''A priori'' distinguishability is much weaker than ''a posteriori'' distinguishability. The former is like saying: ''I cannot give beforehand a criterion that is satisfied by one and not by the other''. The latter is saying: ''Given the two, I cannot tell whether they are the same''.
For instance, if you believe that time has no natural origin, then any point in time looks like any other. You cannot give any ''a priori'' criterion that is satisfied by one point in time and not by the other. But given two points in time, you can certainly compare them: you can, for instance, say which one came earlier .
The fact that there 's no ''a priori'' way of defining a point in time is related to the assumption that physical laws are invariant under time translation.
In a similar way, you may say that there is no ''a priori'' mathematical way of defining a particular unit of length. In other words, there is no criterion that is satisfied by one unit of length, and not by the other. However, ''given'' two units of length, you can certainly ask which is bigger.
The inability to distinguish between various lengths is the '' dilation-invariance'' of mathematical (particularly geometric) theorems: the fact that the laws of geometry are invariant under rescaling lengths.
==Where groups come in==
This is the fact that function composition is associative. Another way of thinking of it is: if you have instructions (1) and (2) on the first page and instruction (3) on the second page, that has the same net effect as having instruction (1) on the first page and instruction (2) and (3) on the second page. (The instructions just tell us what transformations to do).
So in hindsight, ''associativity'' was really an appropriate condition to have a reasonable notion of a group ''acting'' on something.
===The identity map is in===
Certainly! The identity map preserves everything
you could dream of. So, the identity map is actually inside, and it is the identity element of the group.
There may be a tendency to dismiss the identity element as unimportant or uninteresting. However, this is akin to thinking that vacuum is uninteresting in physics. The identity element is right in the middle of the group, and though it doesn't have much character of its own, it is needed for practically everything.
The existence of inverses is a debatable point, but the rough idea is that if your transformation actually ''
was'' a symmetry, you should be able to undo it. Symmetries that aren't reversible aren't genuine symmetries. The ''reverse'' transformation is precisely the inverse element in the group.
There ''are'' certain kinds of situations where we want to relax the assumption about invertibility. Groups ''without inverses'' (i.e. sets with associative binary operation having an identity element) are termed [[monoid]]s. Monoids also ''act'', just like groups do, but the actions aren't often dubbed symmetries. You can't call a one-way street symmetric.
The bigger the symmetry group, the more the symmetry. However, the ''number of elements'' in the symmetry group isn't always the best measure of how symmetry-rich the structure is. Rather, various measures of complexity of the group are better.
For example the straight line has quite a lot of ''symmetry'', but
I guess not too many people would marvel at its symmetry. But a beautiful lattice picture with a huge symmetry group is worth noticing.
There are two general rules:
* The more structural conditions we impose, i.e. the more things we demand remain invariant, the less the symmetry: If you settled for a less-than-perfect notion of '' invariance'', you' d get more symmetry elements. * The larger the universe of transformations we look in, the more the symmetry
==Groups and the big advances in physics==
===The role of symmetry in the physical world===
A lot of serious work in physics is devoted to finding out the ''group of symmetries'' for physical laws. For instance, should physical laws provide a natural way of distinguishing between left and right, or do they possess mirror-symmetry? Do physical laws provide us with natural units of length, mass and time, or do they possess a
Symmetry may have been the realm of philosophers, who believed that the symmetries of the universe should be things that we can derive by thought. For instance, if we do not see any way of distinguishing one direction from the other in space, then all spatial directions should be equivalent. If we do not see a natural way of distinguishing one moment of time from another, then all moments of time must be equal. These philosophical arguments led to the initial study of symmetry in science and art and led to the belief that certain groups of symmetry described the symmetry of the physical world.
However, our thinking is limited by the specific world we live in, and our intuition may not explain what happens at the microscopic (atomic) scales and the macroscopic (galactic) scales. It may not help understand events that happen in extremely short periods of time or be any help in comprehending cosmic timescales.
===The special theory of relativity===
Intuitively, we expect that, sitting in space, there is no way to distinguish one direction from the other, and no way to distinguish one origin from another, but the notion of distance remains invariant under symmetries. Thus, the symmetry group
the physicists believed to capture spatial symmetry , was the group of those transformations of <math>\R^3</math> that preserve distance. Thus, these symmetries preserve everything geometric but do not give a distinguished origin or direction (in technical parlance, we say that space is ''homogeneous'' for having no distinguished origin, and ''isotropic'' for having no distinguished direction). In other words, a physical law should have the same prediction if we translate or rotate.
Independently, there is a symmetry group for time: we expect that time has no natural origin, but an interval of time is an intrinsic quantity. Thus, the group of symmetries of time was the group of time translations. In other words, a physical law should have the same predictions regardless of the point in time when they are applied.
The special theory of relativity
challenged the assumption that we have two independent symmetry groups for space and time. Rather, in the special theory of relativity, a single symmetry group was discovered that governed both space and time, ''together''. This symmetry group roughly treats time as a ''fourth'' dimension along with space, but as a fourth dimension that is not directly interchangeable with the other three dimensions. In other words, the four-dimensional spacetime doesn't look the same in all four dimensions: the three spatial dimensions look similar, but the time dimension behaves differently. The group of symmetries proposed by Einstein contains the earlier groups of symmetries, but offers new symmetries that allow for '' exchanging'' space and time, albeit in a convoluted way.
In summary, the special theory of relativity proposed a new group of symmetries of the physical universe.