A square is a shape with a high degree of symmetry. When discussing the symmetry of a geometric figure, we typically consider all transformations (like reflections and rotations) that map the figure onto itself.
In total, a square has 8 distinct symmetries.
Symmetries can be broadly categorized into two types for a polygon like a square: reflectional symmetry (lines of symmetry) and rotational symmetry.
Lines of Symmetry (Reflectional Symmetry)
A line of symmetry is a line through the shape such that a reflection across that line leaves the shape unchanged. The provided reference states:
In a square, there are four lines of symmetry, each of which divides it into two identical parts. The symmetry lines of a square are both its diagonals and the lines joining the midpoints of its opposite sides (bisectors).
Based on this information and common geometric principles, the four lines of symmetry in a square are:
- The two diagonals: These lines run from one corner to the opposite corner.
- The two lines joining the midpoints of opposite sides: These lines run horizontally and vertically through the center of the square.
These four lines allow you to fold the square in half perfectly onto itself.
Rotational Symmetry
Rotational symmetry exists when a shape can be rotated about a central point by a certain angle and appear exactly as it did before the rotation. For a square, the center of rotation is the intersection of its diagonals.
A square has rotational symmetry at certain angles:
- 0° (or 360°): Any shape looks the same after a 0° rotation. This is the identity symmetry.
- 90°: Rotating a square by 90° clockwise or counterclockwise about its center makes it coincide with its original position.
- 180°: A rotation of 180° also maps the square onto itself.
- 270°: A rotation of 270° (which is -90°) also maps the square onto itself.
Thus, there are 4 rotational symmetries for a square.
Total Symmetries
The total number of symmetries for a square is the sum of its reflectional symmetries (lines of symmetry) and its rotational symmetries.
| Type of Symmetry | Number | Description |
|---|---|---|
| Reflectional Symmetry | 4 | Lines through diagonals and midpoints of sides |
| Rotational Symmetry | 4 | Rotations of 0°, 90°, 180°, 270° |
| Total Symmetries | 8 |
These symmetries form a mathematical structure known as a group, specifically the dihedral group of order 8, often denoted as D₄ or Dih₄. Each of these 8 transformations is a rigid motion that preserves the square's shape and size.
Therefore, a square possesses a total of 8 symmetries.
