What is Symmetric: Definition and 563 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. E

    Symmetric and Anti-Symmetric Wavefunctions

    I am not sure if my title to this thread is appropriate for the question I am about to ask, but it is what we are currently studying in my Quantum Mechanics class so here it goes. Two non-interacting particles with mass m, are in 1-d potential which is zero along a length 2a and infinite...
  2. P

    Prove R^n is Symmetric for All Positive Integers n

    Question: Let R be a symmetric relation on set A. Show that R^n is symetric for all positive integers n. My "solution": Suppose R is symmetric, \exists a,b \in A ((a,b) \in R \wedge (b,a) \in R) For n=1, R^1=R. Next, assume that (a,b) and (b,a) \in R^k, for k a possitive...
  3. M

    Generating Set for the Symmetric Group - Question

    Explain why the permutations (1 2) and (1 2 ... n) generate all of Sn, the symmetric group (the group of all permutations of the numbers {1,2,...,n}? Perhaps something to do with the fact that (1 2 ... n) = (1 2) (1 3) ... (1 n)? Other than that I haven't got a clue - help! (please!) Thanks
  4. P

    Theorem about symmetric matrices?

    If you have a symmetric, nonsingular matrix A, is it always possible to find a matrix B such that B^T A B = 1, where 1 is the identity?
  5. I

    Twins/triplets again, but *truly* symmetric

    Guys, I'm trying (just for fun) to map out quantitatively from each traveller's perspective what happens in the following situation. Imagine the classic twins paradox, with triplets instead of twins, but not for the purposes of avoiding the turn-around. In my question, Triplet A stays on Earth...
  6. B

    Where Can I Find the Ricci Tensor/Scalar for a Circularly Symmetric Spacetime?

    Does anyone know of a reference (or website) where I can find the Ricci tensor/scalar for a static, circularly symmetric spacetime (2 + 1) ? Thanks:)
  7. Q

    Gre Problem # 64 symmetric fission

    Dear forum contributer, The binding energy of a heavy nucleus is about 7 Mev per nucleon, whereas the binding energy of a medium-weight nucleus is about 8 Mev per nucleon. Therefore, the total kinetic energy liberated when a heavy nucleus undergoes symmetric fission is most nearly (A) 1876...
  8. Y

    Please scrutinize my symmetric concept.

    1.) SYMMETRY (think a arm or leg extention) + REACTION (think a cupboard) = proportion 2.) Question: Water (action) + a Cup's Rim (reaction) = what Proportion ? Answer: A plural format. 3.) Symmetry is a case of action and reaction = proportion. 4.) Merriam-Webster Online...
  9. Galileo

    Spherically Symmetric Potentials

    I have a question concerning the stationary states of a spherically symmetric potential (V=V(r), no angular dependence) By separation of variables the eigenfunctions of the angular part of the Shrödinger equation are the spherical harmonics. However, (apart from Y^0_0) these are not...
  10. S

    Why are s-orbital spherically symmetric?

    these are some review questions for an exam: 1.why are s-orbital spherically symmetric? 2.What is the probability of finding an electron at or very near to the nucleus? (1s, 2s, 2p... 3.Why does the curve for 1s go to zero for r-> 0? (the curve of the probability density associated...
  11. S

    Symmetric, antisymmetric and parity

    Let me see if I can make it clearer. Problem 5.5 In David Griffiths “Introduction to Quantum Mechanics” says: Imagine two non interacting particles, each of mass m, in the infinite square well. If one is in the state psin and the other in state psim orthogonal to psin, calculate < (x1 -...
  12. S

    Symmetric, antisymmetric and parity

    Problem 5.5 In David Griffiths “Introduction to Quantum Mechanics” says: Imagine two non interacting particles, each of mass m, in the infinite square well. If one is in the state psin and the other in state psim orthogonal to psin, calculate < (x1 - x2) 2 >, assuming that (a) they are...
  13. F

    Show that if A is nonsingular symmetric matrix

    Hello everyone! Can anyone help me here in this theorems (prove)? (Or solve) 1. Suppose that A and B are square matrices and AB = 0. (as in zero matrix) If B is nonsingular, find A. 2. Show that if A is nonsingular symmetric matrix, then A^-1 is symmetric. I hope these won't...
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