Symmetries Definition and 187 Threads

Symmetry (from Greek συμμετρία symmetria "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definition, and is usually used to refer to an object that is invariant under some transformations; including translation, reflection, rotation or scaling. Although these two meanings of "symmetry" can sometimes be told apart, they are intricately related, and hence are discussed together in this article.
Mathematical symmetry may be observed with respect to the passage of time; as a spatial relationship; through geometric transformations; through other kinds of functional transformations; and as an aspect of abstract objects, including theoretic models, language, and music.This article describes symmetry from three perspectives: in mathematics, including geometry, the most familiar type of symmetry for many people; in science and nature; and in the arts, covering architecture, art and music.
The opposite of symmetry is asymmetry, which refers to the absence or a violation of symmetry.

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  1. P

    What are symmetries in a Lagrangian?

    Homework Statement Consider the Lagrangian of a particle moving in a potential field L = m/2( \dot{x}2 + \dot{y}2 + \dot{z}2) - U(r), r = sqrt(x^2 + y^2) (a) Introduce the cylindrical coordinates and derive an expression for the Lagrangian in terms of the coordinates. (b) Identify the...
  2. mnb96

    Symmetries and shift of coordinates in 3D

    Hello, given a vector x=(a,b) in 2D, and considering another vector obtained by shifting cyclically the coordinates of x, we get x'=(b,a). It is straightforward to prove that x and x' are simply the reflection of each other on the line k(1,1). Now let's suppose we are in 3D space. Given a...
  3. A

    Commutation Relations and Symmetries for SU(2)

    Homework Statement I'm working through a bit of group theory (specifically SU(2) commutation relations). I have a question a bout symmetries in the SU(2) group. It's something I'm trying to work through in my lecture notes for particle physics, but it's a bit of a mathsy question so I thought...
  4. T

    Interpretation of symmetries in physics

    Hi all. I was wondering about symmetries in physics, and have a couple of questions. The symmetries of space/time translation and rotation are often explained in terms of doing an experiment e.g. 10m away instead of right here and getting the same result, as long as we are in the same...
  5. T

    What is the Integral with Symmetries in a Minimization Problem?

    Homework Statement Hi, i need to integrate a function which has a very nice structure. however, in case i cannot do the same, i would need a bound on the value of the integral. The problem is basically a minmization problem with n parameters. Please follow the link to acquaint your self with...
  6. R

    QFT: Gauge Invariance, Ghosts, Symmetry & Lorentz Invariance

    When quantizing boson fields, ghosts and gauge-fixing terms seem to break gauge invariance. The unitary gauge (where there are no ghosts or gauge-fixing terms) respects gauge invariance however. So which is correct - is the Standard Model a gauge theory or not? Sometimes I hear people speak...
  7. F

    Is the Conservation of Angular Momentum in Quantum Mechanics Only Probabilistic?

    I have a symmetry question in QM that's been puzzling me. To state the puzzle, I use a simplified example of a neutral pion in an otherwise empty universe. Since the pion is spinless, and since it possesses no electric or magnetic dipole (or higher) moments, this little set-up is spherically...
  8. F

    Is the Group of Symmetries of a Pentagram Isomorphic to the Dihedral Group?

    Homework Statement The group of symmetries of a regular pentagram is isomorphic to the dihedral group of order 10. Show that this is true.The Attempt at a Solution It seems to me that the group shown by the "star" has order 5, since, by following the lines from one point, it takes 5 total...
  9. B

    How Do Standard Model Symmetries Influence Theoretical Physics?

    1. The Standard model is an SU(3)xSU(2)xU(1) symmetric theory. To me this means that if you choose any 3 members of the groups and act on the Lagrangian, it is invariant. However, not all terms in the Lagrangian have something for a group member to act on, for example terms that don't involve...
  10. O

    Symmetries and Conversation Laws

    According to Noether Theorem, Every symmetry corresponds to a conversation Law in the nature. For example: If rotational symmetry exits, Angular momentum is conserved. How can I be sure that which symmetry corresponds to which conversation Law? Can you tell me the conversation Laws under...
  11. E

    Understanding Weinberg's Symmetries and Rays

    Hello, I am reading Weinberg's book and in the part on symmetries he speaks about rays, and says basically that 2 vectors U,V which are on the same ray can only differ by a phase factor \phi, so that U=e^{i\phi}V. Is "ray" meaning "direction" here ? Can I rephrase it and say that 2...
  12. C

    Exploring the Symmetries of Nambu-Goto Action in String Theory?

    Hi. First let me recall that there are two equivalent classical bosonic string actions, the Nambu-Goto action S_\mathrm{NG} = - T \iint \mathrm d\sigma \, \mathrm d\tau \sqrt{ (\dot X X')^2 - \dot X^2 {X'}^2} and the Polyakov action S_\mathrm{Pol} = - \frac{T}{2} \iint \mathrm d\sigma...
  13. K

    Symmetries and Transformation Groups of Equilateral Triangle & Icosahedron

    How many symmetries (and what symmetries) and how many elements do the transformation groups of the equilateral triangle and the icosahedron have? thanks
  14. J

    Spontaneous symmetry breaking of gauge symmetries

    hello all gauge symmetries are redundencies of the description of a situation. Therefore they are not real symmetries. So in what sense does it mean to spontaneously break a gauge symmetry? ian
  15. F

    Broken Symmetries (Weinberg p215)

    Hi... A group G is proken to a subgroup H. Let t_{\alpha} the generator of G and t_i the generator of H. The t_i form a subalgebra. Take the x_a to be the other indipendent generator of G. Why any finite element of G may be expressed in the form g=exp[i\xi_ax_a]exp[i\theta_i t_i] even if...
  16. M

    Symmetry in Differential Equations: Benefits & Consequences

    Hello everybody! I have a general question concerning DEs :0 Can one use the symmetry of the equation to somehow get the solution faster? What does such symmetry tell us? e.g.: \dot x=y \dot y=x is the symmetrical system to the second order DE \ddot x-x=0 Now we can easily see the...
  17. C

    Group theory and Spacetime symmetries

    "A manifold (with a metric tensor) is said to be spherically symmetric iff the Lie algebra of its Killing vector fields has a sub-algebra that is the Lie algebra of SO(3)." Why? The statement is paraphrased from texts such as Schutz or D'Inverno, where it is always expressed like a definition...
  18. T

    Obtaining Dirac equation from symmetries

    If we consider nonrelativistic QM, we will find Galilean group under the hood. Thanks to this, group theory enables us to find equations of motion directly from the symmetry principles. For example, if we take only geometric symmetries, we will get that the state space is broken into irreducible...
  19. F

    Solving Maths Problems: Numbers, Symmetries & Groups

    http://img180.imageshack.us/img180/9589/simplell9.jpg Is 1. c) as simple as i think it is? I have gone through my notes and can't find anything to do with it, the module for it is Numbers, symmetries and groups, any ideas or do i simple just wack in 13/7 on my calculator and write down...
  20. S

    Gauge fixing and residual symmetries

    This question comes from reading Schwarz' string theory book, which is why I put it in this section. But it seems like a general QFT question, so maybe this isn't the right forum for it. Starting with the sigma model action, reparametrization and Weyl invariance allow us to "gauge fix" the...
  21. P

    Symmetries and conserved quantities

    I know that if a particle is in a spherically symetric potential its angular momentum will be conserved, but what about if somehow we manage to produce say an elliptically symmetric potential? Will the particle then have a momentum along the curve of the ellipse conserved? Thanks
  22. K

    Symmetries of Silicon: M3M Point Group

    As I'm interested in the simplifications of property tensors due to crystal symmetry, I have been trying to find the symmetries of silicon (i.e. the diamond structure). As silicon belongs to the m3m point group I would e.g. expect to find a mirror plane perpendicular to the [100], [010] and...
  23. H

    Is Calculus a Prerequisite for Abstract Algebra?

    Hi, I want to take this course next term. One reason is because I think it will help me with mechanics, classical and quantum, which are taken next year at advanced level. The problem is I'm taking calc2 atm, and its a listed prereq for this group course. I got all the other prereq's...
  24. S

    Group of symmetries on a regular polygon

    So i began reading up on some group theory and I came across an interesting question, what is the order of the group of symmetries on of a n-sided regular polygon? with a square it's 8, triangle it's 4. I feel like I'm missing something with the pentagon because I'm only finding these: the 5...
  25. H

    What will I learn in a course on Groups and Symmetries?

    Hello. As some of you know I'm a chemistry student, but I plan to take some math for the hell of it next summer. I've come across a course called "Groups and Symmetries" and intend to take it, mainly because it is one of the few upper maths avaialbe in the summer. I've never heard of this...
  26. O

    Does motion break existing symmetries?

    Does motion break existing symmetries? Observations suggest that the observable universe is spatially flat and, on the largest “cosmic”scale, highly symmetric. On this scale it is modeled as always isotropic and homogeneous. In this situation Birkhoff’s theorem tells us that “exterior” matter...
  27. Fra

    What Are the Symmetries of Evolution?

    It seems evolutionary thinking is growing in popularity, even among those that really doesn't seem to be take the is principles as first principles, and developt a theory from scratch. I have a question for those working on some of the "big" approaches: Do you ask yourselves what the...
  28. V

    Symmetries and lagrangian formulation

    I recently noticed there's something that escaped me in Lagrangian mechanics. I recently browsed though the first volume of Landau and L., where it is explained that two systems have identical dynamics if their lagrangians differ only by a total differential to time of a function (because the...
  29. Loren Booda

    DNA and RNA symmetries per the origin(s) of life

    Does the single-handedness of DNA and RNA indicate that life originated few times, if more than once? What might the sameness between early sequences (in bacteria, say) indicate in this regard?
  30. M

    Classical and nonclassical symmetries for Helmholtz Equation

    " Classical and nonclassical symmetries for Helmholtz Equation " solitions help. Thank you.
  31. Loren Booda

    Exploring Symmetries in the Cosmological Realm: From Quantum to Macroscopic

    In the cosmological realm there seem to be no exact symmetries as with the quantum realm. Microscopic axes of symmetry appear to randomize on the large scale. Might even quanta be shown of complex structure? Can you think of any examples where outer space exhibits what would be considered...
  32. marcus

    Baez links Standard Model symmetries to Calabi-Yaus

    new preprint http://arxiv.org/abs/hep-th/0511086 Calabi-Yau Manifolds and the Standard Model John C. Baez 4 pages "For any subgroup G of O(n), define a "G-manifold" to be an n-dimensional Riemannian manifold whose holonomy group is contained in G. Then a G-manifold where G is the Standard Model...
  33. S

    Proving Matrix Symmetries of A with Inverse Matrix

    I have a matrix A which satisfies: A_ij(+B) = A_ji (-B) (the matrix is a 4x4 matrix) EDIT: i also know that each row and line summes to 1. i want to prove that the inverse matrix of A sattisfies the same symmetry property. (But, with no success) Do you have an idea how to do that...
  34. J

    Symmetries in Particle Physicsconservation laws

    Assign specific lepton generation numbers and distincguish between antineutrinos and neutrinos for the following reactions. Make appropriate fixes to show that you know about all the conservation laws not just lepton number conservation \mu -> e^{-}\nu \nu \nu n -> e^{-}p \tau^{-} ->...
  35. T

    Symmetries of a Regular Quindecagon: Order & Classification

    How do i show that a group of symmetries of a regular quindecagon has order |G| = 30 ? And how do i describe the elements of G and classify them by their order? Thanks
  36. V

    Searching for Gauge Symmetries and Their Application in Physics

    I'm searching informations about the Gauge simmetries and their application in physics; where can i search in internet and where on books? thanks for answers
  37. Loren Booda

    Maximal identical configuration symmetries

    Quantum mechanics: classical indistinguishability? What is the quantum number correspondent to the maximal symmetry shared by any two real physical configurations in observable spacetime? For instance, any two protons have a high probability of sharing an identical set of quantum numbers...
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