What is Symmetry: Definition and 956 Discussions

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. mnb96

    Symmetry groups on the plane

    Hello, it is known that the symmetry groups on the 2d Euclidean plane are given by the point-groups (n-fold and dihedral symmetries) and the wallpaper groups. However we can create more symmetries on the plane than just those. For example we can stereographically project the 2d plane onto...
  2. tom.stoer

    Photon helicity: Wigner's unitary rep. of Poincare group and gauge symmetry

    1) Since Wigner it is well known that for massless particles of spin s the physical states are labelled by helicity h = ±s; other states are absent. So e.g. for photons the physical states are labelled by |kμ, h> with kμkμ = 0 and h = ±1 and we have two d.o.f. 2) For gauge theories with...
  3. I

    Standard Finite Well Problem (Solve without symmetry)

    Homework Statement Solve the energy eigenvalue problem for the finite square well without using the symmetry assumption and show that the energy eigenstates must be either even or odd. Homework Equations The finite well goes-a to a and has a potential V0 outside the box and a potential...
  4. C

    MHB Find Exact Values of Sin, Cos and Tan with Graph Symmetry

    Hi, I have included a sketch drawing of a graph that I am not entiry sure is correct for what I am trying to do? I have found the exact values of Sin, Cos and Tan of some given values in radians, and am asked to use the symmetry of graphs of sin, cos and tan to find the exact values of some...
  5. H

    Symmetry of E(k) in the first BZ

    Why the curve E(k) in the first brillouin zoon is symmetric? For example why in the first BZ of a one-dimensional lattice we have E(k)=E(-k)?
  6. Y

    Is the Cubic Function Inversely Symmetric about its Point of Inflection?

    As we know, a quadratic function can be expressed in a form of complete square by a method of completing the square. This form enables us to prove that a quadratic equation is symmetric about its stationary point. But for the cubic function, is there a similar way to prove that the cubic curve...
  7. P

    Using symmetry solving Schrödinger equation

    Homework Statement When solving, say, the double delta function potential well, we fix constants using continuity. If the potential is symmetrical about the origin, can we conclude that the wave function, i.e. the solution, will also be symmetric? I found this way made the calculations much...
  8. S

    Noether currents for local gauge symmetry

    hi everyone, I have been trying to understand gauge theory. I am familiar with the Noether's theorem applied in the context of simpler textbook cases like poincare invariant Lagrangians. This is my question: Are there Noether currents corresponding to the local gauge symmetries too and would...
  9. R

    Solving Spherical Symmetry in Hydrogen Atom

    I have a problem; I am trying to show the spherical symmetry in a hydrogen atom, for a sum over the l=1 shell i.e the sum over the quadratics over three angular wave equations in l=1, |Y10|^2 + |Y11|^2 + |Y1-1.|^2 . This should equal up to a constant or a zero to yield no angular dependence...
  10. U

    Charge from field, cylindrical symmetry

    The problem states: From the field with a radial cylindrical component only given by the following equations: E(r)= (ρ0*r3)/(4 * ε0*a2) for r<=a E(r)= (ρ0*a2)/(4*ε0*r2) for r > a obtain the corresponding charge distribution in free space in which the equation is: ρ(r) = ρ0*(r2/a2)...
  11. V

    Crystal Symmetry: Explaining High Temp Effects

    Hi all, Can anyone help explaining how during a phase transition, crystal will be more symmetric at high temperature
  12. T

    Proof of Symmetry for a Geometrical Object

    I was wondering, if there is a general way of finding the axis of symmetry of a geometrical object and also if there is a proof for the fact that the centre of mass must lie on that axis. Let us consider a trivial structure, the cone. We know by intuition that the zz' axis is the axis of...
  13. srfriggen

    Reflective symmetry of a Square

    In my abstract algebra course we learned recently of the symmetries D4. Regarding flips/reflections, of which there are 4, it seems for the 2D object that is a square, you would have to "fold it through the 3rd dimension" to obtain a flip/reflection. Couldn't you just invert the square by...
  14. K

    Symmetry of R: Proof & Solutions

    Homework Statement R is simmetric iff R=R^{-1} Homework Equations ( \forall x \forall y ((x,y) \in R \rightarrow (y,x) \in R)) \leftrightarrow R=R^{-1}The Attempt at a Solution My problem is with my formulation in [2.] of the statement I have to prove. Is that formulation right or the...
  15. G

    Topological Insulators and Inversion Symmetry

    Hi, I was curious if specific symmetries (or lack thereof) in crystal structure are necessary for the formation of topological insulators. Specifically, do we require that inversion symmetry (or inversion asymmetry) be present in the lattice in order to form the TI state? Thanks, Goalie33
  16. grav-universe

    Schwarzschild metric and spherical symmetry

    In deriving the Schwarzschild metric, the first assumption is that the transformation of r^2 (dθ^2 + sin^2 θ dψ) remains unchanged due to the spherical symmetry. What does that mean exactly? What is the logic behind it? Please apply any math involved in algebraic form. Thanks.
  17. TrickyDicky

    Is the Higgs mechanism really a spontaneous symmetry breaking?

    The Higgs mechanism is often explained (both here at PF and in many physics sites including wikipedia) as an example of spontaneous symmetry breaking, but the Nobel winner physicist 't Hooft says in his "for laymen" book about particle physics, "In search of the ultimate building blocks", that...
  18. V

    Symmetry and symmetry breaking

    I am quite new to the branch of quantum physics and therefore am quite inexperienced with certain terminology and definitions. I have looked these topics up time and time again, but still cannot get a grasp on what they mean. Could someone please describe to me what the concept of "symmetry" in...
  19. S

    Mixed symmetry property and degrees of freedom

    How can I calculate degrees of freedom of a rank (o,3) tensor, Aabc, that is mixed symmetry and antisymmetric in the first 2 indices? By mixed symmetry I mean this: Aabc+Acab+Abca=0.
  20. F

    Exploring Spin, Symmetry, and SUSY in Particle Physics

    I'm wondering how the spin of a particle, whether a particle is a fermion or a boson... how does this relate to the symmetry of a particle, U(1) or SU(2) or SU(3)? I'm trying to understand SUSY in relation to the other internal symmetries? Is there spin 1/2 and spin 1 particles associated with...
  21. atyy

    Spontaneously broken gauge symmetry

    I have read 2 arguments that a gauge symmetry cannot be spontaneously broken. 1. Wen's textbook says a gauge symmetry is a by definition a "do nothing" transformation, so it cannot be broken. 2. Elitzur's theorem, eg.http://arxiv.org/abs/hep-ph/9810302v1 The first argument seems sound...
  22. alemsalem

    What happens to conserved currents after spontaneos symmetry breaking?

    should the current still be conserved? since it stills commutes with the Hamiltonian and symmetry is just hidden. but I just read that the linear-σ model was invented to demonstrate how the axial current could be partially conserved? Thanks!
  23. B

    Bi Linear Functionals and Symmetry

    Homework Statement Show that ## \displaystyle B_1(u,v)=\int_a^b (p(x) u \cdot v + q(x) \frac{du}{dx} \cdot v)dx## is a bilinear functional and is NOT symmetric Homework Statement Bilinear relation ##B(\alpha u_1+\beta u_2,v)=\alpha B(u_1,v) +\beta B(u_2,v)## (1) ##B(u, \alpha v_1+...
  24. J

    Eigenvalues, projection and symmetry. Help please.

    Homework Statement In each case describe the eigenvalues of the linear operator and a base in R^3 that consist of eigenvectors of the given linear operator. Write the matrix of the operator with respect to the given base. The Orthogonal Projection on the plane 2x + y = 0 and...
  25. lpetrich

    Analogies for Symmetry Breaking and the Higgs Particle

    Any good ones? I like this one: For full symmetry, imagine a marble and a bowl with rotational symmetry. Drop the marble into the bowl. It will oscillate back and forth and settle down in the center. The bowl+marble system still has rotational symmetry. If you push the marble out of the...
  26. C

    Bose Symmetry Explained: Definition & Meaning

    Ive tried to search around for what Bose-symmetry is, but I can't seem to find any definition. Can someone here provide me with a definition of bose symmetry?
  27. J

    Symmetry of N-Channel MOSFET: VGD & VGS Thresholds

    In an N-channel mosfet, for example, can the roles of the Drain and Source be reversed such that the FET turns on with positive VGD? Will this threshold be the same as the specified one for VGS?
  28. T

    Applications of Double Integrals: Centroids and Symmetry

    Homework Statement A lamina occupies the region inside the circle x2+y2=2y but outside the circle x2+y2=1. Find the center of mass if the density at any point is inversely proportional to its distance from the origin. Here is the solution...
  29. M

    Is radio path chanell respone symmetry ?

    Hello assume I sample signal between Transmiter to Reciver T->R Is the chanell respone would be the same to R->T also? especially I wonder about knife difraction effect ( Fresnel zone )
  30. S

    How to See Symmetry about x = -1?

    Hi. Question: Sketch f(x) = \sqrt{1 - x} + \sqrt{3 + x} On this test question from calculus, I got full marks for my answer. But I'm posting in this forum because I'd like to know how to analyze the symmetry of this function (posted below), which I actually didn't notice until I read the...
  31. C

    Infrared spectroscopy and symmetry

    is it true to say that any molecule having a centre of symmetry will have the symmetric stretching IR inactive? thanks!
  32. jfy4

    O(2) symmetry breaking trouble

    Hi, I'm going to quote a lot of a book so that I can get some help, brace yourselves... First, \phi_{a} is my field with a=0,1 as internal components and my lagrangian is L=\frac{1}{2}\partial_{\mu}\phi_a \partial^\mu \phi_a +\frac{1}{2}\mu^2 \phi_a \phi_a +\frac{1}{4}\lambda (\phi_a...
  33. H

    Explaining Expression 7.3 in Nother's Theorem and Invariance Under Translation

    Hello, I working through my notes on Nother's theorem and invariance under translation. I don't understand how they get expression 7.3, from the line before 7.3 (see attachment). Can anyone explain. Thanks.
  34. N

    Symmetry breaking/ degrees of freedom

    Dear PF... Please help me with basic question more I think more I get confused... In O(n) space there are n(n-1)/2 generators... suppose I have symmetric tensor in O(n) space, it will have n(n+1)/2 independent components... and i am building invariant potential from it (quartic polynomial...
  35. M

    MHB Finding the vertex, y-intercept and axis of symmetry

    I am lost and confused. I have been on the same problem for 2 hours. I know all the formulas, but I'm not doing something right...
  36. Z

    Proving no simplex has 2-fold symmetry

    Heck, I don't even know if this is true. Intuition seems to suggest that it is but I know of no way to prove it. In general I don't know of any way to rigorously determine axes of symmetry. And I am having trouble finding anything about this, so any links are appreciated.
  37. J

    Wave function at high symmetry point

    How to prove that wave function at \Gamma point can always be a real function? I know it is not true for general k point, but for \Gamma and other high symmetry point like X, is there a simple proof? Thanks!
  38. L

    The Importance of Symmetry in Stress Tensors

    Why stress tensor must be symmetric?
  39. J

    Negative Energy and Mass Symmetry

    I was wondering about a system, specifically quantum, though classical solutions are still welcome, which was resisting all applications of Noether's Theorem, and related techniques. If a system is invariant under a switch from E→-E AND m→-m, then what are the conserved quantities (in analogy to...
  40. M

    Question about symmetry with potential fields.

    Quick question (I think anyway) I'm currently trying to solve a problem that essentially is asking for me to find the final velocity of an electron that is traveling between a cathode and an anode of potential difference 300. At half the distance to the anode can I assume by symmetry that it...
  41. S

    Yukowa potential and symmetry breaking

    Hello, I am trying to shortly explain how the Yukowa potential breaks symmetry in weak interactions. I would like to use the mexican hat potential as a specific example. Unfortunately Wikipedia does not go very in depth or explain it very well. Link. Any help on understanding the collapse of...
  42. B

    Understanding the Symmetry of the Resistor Cube Problem

    Homework Statement I am looking at the classic resistor cube problem-the one in which there is a cube with resistors all of the same value on each side. Particularily I am having trouble understanding the symmetry argument for the case in which the current enters and exits across the...
  43. C

    General Relativity: Prove symmetry of Einstein tensor

    Homework Statement Show that Gij = Gji using the Riemann tensor identity (below) Homework Equations Gij = Rij - 1/2(gijR) Rabcd + Rbcad + Rcabd = 0 R = gmrRmr Rmr = Rmnrn The Attempt at a Solution I have tried to put the Ricci tensor and Ricci scalar (from the Gij...
  44. Doofy

    P, CP and CPT symmetry in neutrino oscillation? (quick question)

    Just a quick question about notation really here. In neutrino oscillation we can calculate a probability of an oscillation occurring between two flavour eigenstates - invariably denoted P(\nu_{\alpha} \rightarrow \nu_{\beta}) . I've got some confusion about what happens to this when we apply...
  45. Islam Hassan

    Is Physical Symmetry Limited Due To The Classification Theorem?

    Given that the Classification Theorem says that every finite simple group is isomorphic to one of 4 broad categories of (specific) finite simple groups, does this mean that any conceivable symmetric: i) 2D form; or ii) 3D object is also isomorphic to one of these 4 categories. Otherwise...
  46. L

    Deducing potential from symmetry (from Griffiths EM)

    Homework Statement It's example 3.8 in the Griffiths book in case someone has it. Basically the problem involves a uncharged metal sphere in a uniform field in the +z direction. Naturally, there will be induced positive charges on top and negative charges at the bottom. The question asks for...
  47. M

    Time reversal symmetry in Topological insulators of HgTe quantum Wells

    Hi everyone, While reading about the BHZ model used to describe HgTe quantum well topological insulators, I read at many places that the effective Hamiltonian (which is a 4 x 4 matrix) can be written in block diagonal form and the lower 2x2 block can be derived from upper 2x2 block as...
  48. Islam Hassan

    Is Symmetry Limited in Geometry?

    Given the complete classification of finite simple groups, can one say that the number of all conceivable 2D/3D symmetrical geometric objects/arrangement is limited? Is spatial symmetry limited in our 3D world? IH
  49. H

    Matter-antimatter symmetry and particle charge

    If matter composed of quarks of positive charge, and anti-quarks have negative charge, why do we call electrons particles of matter rather than anti-matter? Is it possible that the matter-antimatter symmetry problem exists because we are actually calling something that is a particle of...
  50. J

    Singularity theorms and perturbation from exact symmetry

    The singularity theorems apply to situations away from exact symmetry ... away from Schwarzschild solution or Friedmann solutions for example. There are a number of accounts of the singularity theorems but none addressing the problem of proving a 'trapped set' still persists after slight...
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