Line of Symmetry
Geometry & MeasurementA 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. If you fold the shape along the line, the two halves match up perfectly.
Example
A rectangle has $2$ lines of symmetry (one horizontal, one vertical). An equilateral triangle has $3$. A circle has infinitely many. The letter "A" has $1$ vertical line of symmetry.
Key Insight
Fold a piece of paper in half: the fold line is a line of symmetry if both halves match. Many everyday objects have lines of symmetry: a butterfly, a heart shape, the letter "H." Symmetry is everywhere in nature and design.
Definition
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, where $l$ is the perpendicular bisector of $PP'$. A regular n-gon has exactly $n$ lines of symmetry.
Example
An isosceles triangle has exactly $1$ line of symmetry (the altitude from the apex to the base). A square has $4$ (two through opposite vertices, two through opposite edge midpoints). A non-regular rectangle has $2$; a parallelogram (that is not a rectangle) has $0$.
Key Insight
The number of lines of symmetry of a regular polygon equals the number of its sides. This is because each line either passes through two opposite vertices (for even $n$) or through a vertex and an opposite edge midpoint (for odd $n$), giving $n$ total.
Definition
A line of symmetry of a figure $F$ is a line $l$ such that the reflection $R_l$ satisfies $R_l(F) = F$, i.e., $l$-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 $2n$), which contains $n$ reflections and $n$ rotations. The lines of symmetry are the mirror lines generating the reflection subgroup of $D_n$.
Example
The symmetry group of an equilateral triangle is $D_3 = S_3$ (symmetric group on $3$ elements), with $3$ reflections and $2$ non-identity rotations ($120$ degrees and $240$ degrees). The $3$ lines of symmetry generate the full dihedral group $D_3$.
Key Insight
Dihedral groups $D_n$ arise in crystallography as the symmetry groups of 2-D crystal cross-sections, chemistry (molecular symmetry), and 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.