The idea of symmetry in physics and math can be described as equality under some sort of transformation.
Our usual idea of mathematical equality is an equation, which is a test for numerical equality and involves finding the values of the variable that make the equality true, if such values can be found.
Or, in higher math, where symbols and not just numbers are the currency of the realm, when you "solve an equation" you have a new equation for the dependent variable (the strength of an electric field, or the temperature distribution on the surface of a frying pan, for instance) in terms of the independents (time and place, for instance), and an equation is a test of whether you can accomplish that or not. Boundary conditions and/or initial conditions are also relevant to solving an actual physical "problem".
In contrast to numerical equality, a "transformation" can be considered as a test for geometric equality: what value of the variable (the rotation angle, for instance) produces an arrangement that is identical or equal to the original arrangement? A simple example is the rotation of a square about its center. There are four angles of rotation where exact symmetry or geometric equality is observed: 90, 180, 270 and 360 degrees.
The word used more commonly than "equality" when it comes to a geometric figure is "invariance." In physics, symmetry and invariance are really synonyms. What sort of symmetry or invariance is involved is the next question that naturally arises, as in 4-fold symmetry (for example the square), 8-fold symmetry (octagon), and continuous symmetry (circle). And that's just looking at a few cases in only two dimensions--there are other geometric figures besides the square that have 4-fold rotational symmetry (a plus sign is one of them--you can try to think of others). In general, the type of symmetry or invariance is labeled or categorized by the symmetry group or transformation group, to be talked about later.
Or, in higher math, where symbols and not just numbers are the currency of the realm, when you "solve an equation" you have a new equation for the dependent variable (the strength of an electric field, or the temperature distribution on the surface of a frying pan, for instance) in terms of the independents (time and place, for instance), and an equation is a test of whether you can accomplish that or not. Boundary conditions and/or initial conditions are also relevant to solving an actual physical "problem".
In contrast to numerical equality, a "transformation" can be considered as a test for geometric equality: what value of the variable (the rotation angle, for instance) produces an arrangement that is identical or equal to the original arrangement? A simple example is the rotation of a square about its center. There are four angles of rotation where exact symmetry or geometric equality is observed: 90, 180, 270 and 360 degrees.
The word used more commonly than "equality" when it comes to a geometric figure is "invariance." In physics, symmetry and invariance are really synonyms. What sort of symmetry or invariance is involved is the next question that naturally arises, as in 4-fold symmetry (for example the square), 8-fold symmetry (octagon), and continuous symmetry (circle). And that's just looking at a few cases in only two dimensions--there are other geometric figures besides the square that have 4-fold rotational symmetry (a plus sign is one of them--you can try to think of others). In general, the type of symmetry or invariance is labeled or categorized by the symmetry group or transformation group, to be talked about later.
