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My girlfriend is studying to become a math teacher, and asked me what I have ever used symmetries for.

I'm a web developer, and do some designing from time to time, so I answered that symmetry have given me a general understanding of aesthetics.

However, that answer didn't quite work for any of us, so now I'm asking you:

What are the reason we are taught symmetry in school?

What are the practical usages of symmetries in our everyday lives?

Preferably I'm looking for documented answers, but any answer will do at the moment.

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There is a cool article on crossword puzzles that just came out; this involves symmetry since there are certain symmetrical rules governing what is admissible in the Times. In particular, you need to be able to turn the unfilled puzzle 180 degrees and have it look the same. From the AMM: Abstract. The maximum number of clues possible for a 15 x 15 daily New York Times crossword puzzle is shown to be 96, and all possible puzzle grids with 96 clues are presented. Moreover, a crossword puzzle with 96 clues is given, in the theme of this result. –  Benjamin Dickman Jun 9 at 21:11

8 Answers 8

Symmetries give profound insight into structure. This is demonstrated by Noether's theorem, proved just under 100 years ago by Emmy Noether (probably the greatest female mathematician in history).

Noether's theorem states that every symmetry of a physical system is there because something is being conserved. Examples:

  1. Translational symmetry -> conservation of momentum
  2. Rotational symmetry -> conservation of angular momentum
  3. Time symmetry -> conservation of energy
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This strikes me as the kind of answer that would be a good answer on math.SE, but it seems (to me, currently) like a poor answer for matheducators.SE, especially under the primary-education tab! Am I confused about this? How would a primary school teacher, talking to 9-year-olds, find this information useful? –  Chris Cunningham May 5 at 3:38
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Well, just because you are teaching 9-year-olds, that hopefully does not mean we should talk to you like you are a nine-year old. The information here is very basic and appropriate for the question, you should be able to either classify it as too hard for 9-year-olds or contract information and present in a form that is suitable for 9-year-olds. That's part of your job, as a teacher. –  Mon Kee Poo May 5 at 6:06
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If I'm not mistaken, this answer was posted when the question originally had the undergraduate-education tag. Now that the question has been edited, this tag has been replaced with the primary-education tag. –  Joel Reyes Noche May 5 at 13:17
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I would advocate for calling Emmy Noether e.g. "one of the greatest mathematician of the twentieth century" rather than ranking her among female mathematicians: do we ever say that Gauss is "probably the greatest male mathematician in history"? –  Benoît Kloeckner May 24 at 11:19

People tend to use the presence of symmetry in a phenomenon to simplify a model of it. For example, in physics, systems with certain symmetries are the easiest to model (e.g., an infinite plane, an infinitely long cylinder, a sphere, etc.). But we have to be careful here.

Ian Stewart has a very nice article on symmetry-breaking. (The article is from his column in Pour La Science and was reprinted as "Curie's Mistake" Chapter 10 of Another Fine Math You've Got Me Into... (New York: W. H. Freeman and Company, 1992).) Aside from presenting examples of symmetry found in nature, he also presents examples in nature where symmetry breaks (like why certain animals have stripes and why others have spots).

Stewart discusses the question "How does the symmetry of a system affect its behavior?"

A famous answer was given by the great physicist Pierre Curie [...]. In 1894 Pierre Curie gave two logically equivalent statements of a general principle from the folklore of mathematical physics.

  1. If certain causes produce certain effects, then the symmetries of the causes reappear in the effects produced.
  2. If certain effects reveal a certain asymmetry, this asymmetry will be reflected in the causes which give rise to them.

Stewart then goes to see if Curie was correct.

For example, in the Kensington Science Museum in London there is an engineering model of a passenger jet, used in a wind tunnel to study the flow of air around the aircraft. Since the aircraft is bilaterally symmetric, the engineers built only half of the model---tacitly assuming that the air flow had to be bilaterally symmetric as well.

It turns out that Curie was not correct. There are many instances in nature where symmetry breaks, instances where a symmetric state becomes unstable.

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See also Fearful Symmetry: Is God a Geometer by Stewart & Golubitsky, reprinted by Dover. –  J W May 3 at 8:02

Some examples:

  • It's much easier to develop a symmetric plane or boat than to calculate all the forces regarding an asymmetric hull.
  • Car keys being symmetric it's great usability feature: saves time and frustration.
  • Pots are symmetric, because they are easier to produce and nicer to use.
  • Wheels wouldn't work if not for symmetry.
  • Games are often symmetric to achieve a form of fairness.
  • Humans and animals in general are symmetric too (from outside, but the asymmetries inside are not so big either), but I have no idea way (somehow such a shape must be more fit).
  • High-level symmetry is recognizable and feels deeply intentional, this can be used to great effects in art (i.e. not only for aesthetics, but also emphasis or symbolism, etc.); examples include
    • direct use like inversion or retrograde in music, e.g. in fugues by J. S. Bach,
    • less direct use like the same scene from various perspectives, e.g. in Another World or Above and Below by M. C. Escher,
    • even less direct like the same situation happening to pair of characters in a novel, but the second time with their roles reversed (taken to the extremes in the 7th Voyage in the Star Diaries by S. Lem).

I hope this helps $\ddot\smile$

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Schiralli (2007) points out that our sense of the aesthetic is part of our ability to engage in formal scientific and mathematical endeavors:

"The identification of pattern is, therefore, a fundamentally aesthetic apprehension that in systematic inquiry soon moves beyond the immediately perceptable towards the more formal conceptual connections with which scientific and mathematical theory is ultimately concerned." (p. 106)

For a practical example, Frank Wiczek notes in the introduction to Weyl's (2009) Philosophy of Mathematics and Natural Science:

Time-reversal symmetry asserts the equivalence of past and future, in the microscopic laws of physics. Of course, both the concrete history of the universe and (at a mundane, but more specific and practical level) the laws of thermodynamics distinguish past and future. Yet the laws of Newtonian mechanics and Maxwellian electrodynamics--and indeed, all the basic laws of the microcosmos known in Weyl's day--do not. (p. xi)

Wiczek explains that symmetry has played a role in theory, and that violations of theoretical symmetry are important in that they have lead to questions that powered major advances in physics.

But maybe you wanted a more geometric example. Patterns in math are so often useful, and symmetry is no different. Lets say my pool is some strange shape. I want to calculate the volume of water in the pool so I know how much chlorine to use, but I don't have a formula for the shape in question. But then I notice that the pool is symmetrical in some way (choose whatever type of symmetry you prefer). The symmetry involves a shape I do have the formula for. So I calculate the volume of that shape, then iterate accordingly (or just multiply).

Perhaps you meant something different, but noticing symmetry in a system always gives you a way to reduce the complexity in how you represent the system. In the reverse, from a software or computer graphics standpoint, symmetry gives you a way to generate forms programatically by transformation of other forms.


Cited

Schiralli, M. (2007). The meaning of pattern. In N. Sinclair, D. Pimm, & W. Higginson (Eds.), Mathematics and the Aesthetic (pp. 105–125). Springer. Retrieved from http://link.springer.com/chapter/10.1007/978-0-387-38145-9_6

Weyl, H. (2009). Philosophy of Mathematics and Natural Science. Princeton University Press.

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Symmetry in polynomials gives ways to find zeroes. I.e., $$ x^n + a x^{n -1} + b x^{n -2} + \ldots + b x^2 + a x + 1 $$ can be reduced in degree by $x = y + 1/y$.

One often looks for symmetries in problems to simplify solving them, or to check solutions. I can't give a concrete example right now, but it is where I start.

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Also, it might be worth noting that the roots of $a_0x^n+\cdots+a_n$ are the reciprocals of the roots of $a_nx^n+\cdots+a_0$. I know no "applications" of that, apart from giving this (or its variants) as an exercise, its internal elegance/coolness, and the "Aha!" moment when you get it. –  mbork May 4 at 13:32

For a more particular example, when we are finding the area (in calculus) under an even function (symmetrical across the y-axis) from x = -a to x = a, we can ask ourselves the easier question of finding the area from x = 0 to x = a, and multiplying that by two.

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+1 Although I find integrals that cancel out more fun and compelling (odd functions, sine/cosine over a period), especially in multivariable calculus. –  Michael E2 May 4 at 16:07

An example of a rather natural and rich question one can start to investigate at lower grades is that of designing a dice: how many ways are there to assign the values from 1 to 6 to the faces of a dice? As a first subquestion, one can ask whether there are even two different ways? One can also give pupils two different dices and ask whether they are "the same"; then the cubical symmetry makes a lot of sense.

Another example is how to sit people on a round table; in many circumstances, one "quotient out" by circular symmetry.

Another one is the first hand the game of go: many "different" hands have in fact just the same effect by symmetry. It is also interesting to think about what happens after a few plays: when one approaches a corner stone lying on a diagonal, there are two version of most possible move which are locally equivalent because of diagonal symmetry, but global symmetry has usually broke at this point. Then the player has to take into account the distant surroundings, which have a very subtle effect. Many beginners spontaneously consider the situation as symmetric (showing how natural the notion is), while it is not quite so.

More geometrically, high symmetry implies lower diversity in shapes. For example, among quadrilaterals squares have the most symmetries and are entirely described (up to isometry) by their side-length; one needs much more quantities to describe a general quadrilateral, or even a parallelogram.

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You can use symmetry to cut snow flakes, stars or ornaments.

I don't know the english names, but here and here are examples of these stars. The use rotational symmetry.

With translational symmetry you can create a long ornament of angel figures for example. (I don't find links, sorry.)

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