Line of Symmetry
A line of symmetry divides a figure into two mirror-image halves that are identical when folded along the line.
Definition
A line of symmetry is a line that divides a shape into two halves that are mirror images of each other, so folding the shape along the line makes the two halves match up perfectly. Formally, a line of symmetry (axis of symmetry) of a figure is a line $l$ such that the reflection of the figure across $l$ maps the figure onto itself; every point $P$ on the figure has a corresponding point $P'$ also on the figure, with $l$ the perpendicular bisector of $PP'$. A regular n-gon has exactly $n$ lines of symmetry, since each line either passes through two opposite vertices (even $n$) or through a vertex and an opposite edge midpoint (odd $n$). In group-theoretic terms, $l$-reflection is an element of the symmetry group $\text{Sym}(F)$; for a regular n-gon, $\text{Sym}(F) = D_n$ (the dihedral group of order $2n$), containing $n$ reflections and $n$ rotations, and the lines of symmetry generate the reflection subgroup of $D_n$.
Example
A rectangle has $2$ lines of symmetry (one horizontal, one vertical), an equilateral triangle has $3$, a circle has infinitely many, and the letter "A" has $1$ vertical line of symmetry. An isosceles triangle has exactly $1$ line of symmetry (the altitude from the apex to the base), a square has $4$, a non-regular rectangle has $2$, and a non-rectangular parallelogram has $0$. The symmetry group of an equilateral triangle is $D_3 = S_3$, with $3$ reflections and $2$ non-identity rotations ($120$ and $240$ degrees), and its $3$ lines of symmetry generate the full dihedral group $D_3$.
Key Insight
Fold a piece of paper in half: the fold line is a line of symmetry if both halves match, as seen in a butterfly, a heart shape, or the letter "H." The number of lines of symmetry of a regular polygon always equals its number of sides. Dihedral groups $D_n$ arise in crystallography as symmetry groups of 2-D crystal cross-sections, in chemistry (molecular symmetry), and in art (tilings and rosette patterns); the classification of all 2-D symmetry groups ($17$ wallpaper groups) is a landmark result in group theory with physical applications.