# Symmetry in mathematics

1. Mar 3, 2012

### spaghetti3451

This is in the introduction of the lecture notes I am using to study group theory:

"An object is symmetric or has symmetry if there is an operation (e.g. a rotation, reﬂection or translation) s.t. the object looks the same after the operation as it did originally. An equilateral triangle is indistinguishable after rotations by 1/3 π and 2/3 π around its geometric centre/ symmetry axis. A square is indistinguishable after rotations by 1/2 π, π, and 3/2 π around its centre. A circle is indistinguishable after any such rotation. The objects are invariant under these symmetry transformations (i.e. the operations) The more symmetry transformations an object admits, the more “symmetric” it is. So, the circle is more symmetric than the square, which is more symmetric than the triangle. But rotations are not the only operations that leave the objects in invariant: there are also mirror planes. The totality of distinct symmetry transformations of an object is the central construction in the mathematical theory of symmetry."

The final two sentences are what are confusing me.

1. What exactly are mirror planes and how do they leave a physical object invariant?

2. Distinct symmetry transformations of an object??? Central construction in the mathematical theory of symmetry??? What do they mean???

Any help would be greatly appreciated.

2. Mar 3, 2012

### tiny-tim

hi failexam!
eg the four mirror planes for a square are the diagonals, and lines bisecting the sides

(a non-square rectangle only has two mirror planes)
they just mean that symmetry is based on symmetry transformations