What is a linear symmetry?
It is called line symmetry when we can actually draw a line down the middle of a figure that divides the figure into two mirror images.
If you're talking about convex polygons with equal sides (eg. equilateral triangles, squares, pentagons, hexagons, etc.), then the relationship is a very direct one. In those cases, there are as many lines of symmetry as there are points in the polygons. A triangle has three lines of symmetry, a square has four, a pentagon five, etc.
In molecular orbital theory, MO theory, molecular orbitals are "built" from atomic orbitals. A common approach is to take a linear combination of atomic orbitals (LCAO), specifically symmetry adapted linear combinations (SALC) using group theory. The formation of a bond is essentially down to the overlap of the orbitals, the orbitals being of similar energy and the atomic orbital wave functions having the correct symmetry.
For a polygon with an even number of vertices join each vertex to the opposite vertex. Test for symmetry. join the mid-point of each side to the mid-point of the opposite side. Test for symmetry. For a polygon with an odd number of vertices join each vertex to the mid-point of the opposite side. Test for symmetry.
When a shape is rotated about its centre, if it comes to rest in a position and looks exactly like the original, then it has rotational symmetry. A shape like an equilateral triangle would therefore have an order of rotational symmetry of 3. The general rule for a regular polygon (shapes such as pentagons, heptagons, octagons etc. is, that the number of sides is the same as the number of lines of symmetry, which is…