Symmetry: Line, Point and Rotational
Symmetry means that a figure looks the same after a transformation: line symmetry after a reflection, point symmetry after a half-turn and rotational symmetry after a turn by a smaller angle. In brief: a rectangle has 2 lines of symmetry and point symmetry, a square has 4 lines of symmetry and rotational symmetry of order 4, and the letter S has point symmetry but no line of symmetry.
Symmetry is a way to describe shapes in nature, art and design, and it helps in geometry because it reduces work: if a figure is symmetric, you only need to study half of it. This guide explains the three main kinds of symmetry, how to find the symmetries of polygons and letters, and how symmetry appears in graphs of functions.
What is line symmetry?
A figure has line symmetry (also called reflection or mirror symmetry) if there is a line that divides it into two halves that are mirror images of each other. The line is the axis of symmetry. If you fold the figure along the axis, the two halves coincide. Every point P of the figure has a mirror point P′ that is also in the figure, and the axis is the perpendicular bisector of the segment PP′.
Examples: an isosceles triangle has 1 axis, through the apex and the midpoint of the base. A rectangle that is not a square has 2 axes, through the midpoints of opposite sides. A circle has infinitely many, every line through its center. The parabola y = x2 is symmetric about the y-axis.
What is point symmetry?
A figure has point symmetry if there is a point O, the center of symmetry, such that every point P of the figure has a partner P′ in the figure with O as the midpoint of PP′. This is the same as rotating the figure by 180° about O and getting the same figure.
Parallelograms, including rectangles, rhombi and squares, have point symmetry about the intersection of their diagonals. A regular hexagon does as well. An equilateral triangle does not. The letters H, I, N, O, S, X and Z have point symmetry. For functions, an odd function such as y = x3 has point symmetry about the origin.
What is rotational symmetry?
A figure has rotational symmetry if a rotation about a center by some angle smaller than 360° maps it onto itself. The number of times the figure matches itself in a full turn is the order of the symmetry, and the smallest rotation angle is 360° divided by the order.
An equilateral triangle has order 3, with a smallest angle of 120°. A square has order 4 and a smallest angle of 90°. A regular hexagon has order 6 and 60°. A rectangle has order 2 and 180°, which is point symmetry. Every figure matches itself after a full turn of 360°, which does not count as symmetry, so a figure with order 1 has no rotational symmetry. A propeller with three blades has order 3 but no line symmetry if the blades are curved.
How do you find the symmetries of a figure?
- Lines: try folding. Look for a line such that each part has a mirror image on the other side. Candidates are lines through vertices and through midpoints of sides.
- Point: check if rotating by 180° about a center gives the same figure.
- Rotation: test angles 360°/n for n = 2, 3, 4 and so on.
- List all results, including the number of axes and the order.
Example 1: a rhombus (not a square)
It has 2 axes of symmetry, the diagonals, and rotational symmetry of order 2 (point symmetry about the center). The sides are not axes, as the angles are not equal.
Example 2: a regular pentagon
It has 5 axes of symmetry, each through a vertex and the midpoint of the opposite side, and rotational symmetry of order 5 with a smallest angle of 72°. It has no point symmetry, since 180° is not a multiple of 72°.
Example 3: the letter
The letter A has 1 vertical axis. The letter H has 2 axes and point symmetry. The letter F has none.
Symmetries of regular polygons
| Polygon | Axes of symmetry | Order of rotation | Smallest angle | Point symmetry |
|---|---|---|---|---|
| Equilateral triangle | 3 | 3 | 120° | No |
| Square | 4 | 4 | 90° | Yes |
| Regular pentagon | 5 | 5 | 72° | No |
| Regular hexagon | 6 | 6 | 60° | Yes |
| Regular heptagon | 7 | 7 | about 51.4° | No |
| Regular octagon | 8 | 8 | 45° | Yes |
A regular n-gon always has n axes of symmetry and rotational symmetry of order n. It has point symmetry exactly when n is even.
Where does the idea come from?
Symmetry has been used in art and architecture since prehistoric times. The word comes from the Greek symmetria, meaning “common measure”, which originally referred to proportion more than to mirror images. The mathematical study of symmetry grew in the nineteenth century into group theory, with contributions by Évariste Galois and others, and became a central idea in modern mathematics and physics.
Where is symmetry used today?
Architects and designers use it for balance, and logos often have line or rotational symmetry. Crystals are classified by their symmetries. Biology describes animals as bilaterally symmetric, like humans, or radially symmetric, like starfish. In mathematics, symmetry is used to simplify proofs and calculations, and in physics, symmetries correspond to conservation laws.
Which mistakes are common?
Counting diagonals of a rectangle as axes
A rectangle that is not a square has only 2 axes, the lines through midpoints of opposite sides. A fold along a diagonal does not match the halves.
Confusing point and line symmetry
Point symmetry is a half-turn, and line symmetry is a reflection. A parallelogram has point symmetry but, in general, no line symmetry.
Counting the full turn
A rotation by 360° always works and does not count. The order is the number of different rotations including the trivial one, so a figure with order 1 has no rotational symmetry.
Forgetting the center
Rotational symmetry needs a center. A figure can be rotationally symmetric about one point but not about another.