Chapter 9: Symmetry
Symmetry is one of the most pleasing ideas in mathematics , and one of the most visible. A butterfly's wings, the petals of a hibiscus, the Taj Mahal seen from the front, the design of a rangoli, the snowflake on a winter window: all of them are symmetric. Symmetry is what our eyes love about a shape.
There are two main kinds of symmetry we will study. Line symmetry (also called reflection symmetry, mirror symmetry, or bilateral symmetry) is when a shape looks the same on both sides of a line , like a butterfly's two wings. Rotational symmetry is when a shape looks the same after being turned by some angle , like the petals of a daisy.
A shape can have line symmetry, rotational symmetry, both, or neither. The square has four lines of symmetry and rotational symmetry of order , turn it by and it looks unchanged. The letter "Z" has rotational symmetry (turn ) but no lines of symmetry. The letter "T" has one line of symmetry but no rotational symmetry. By the end of the chapter you will be able to look at any figure and read off all its symmetries.
What's inside
- Line symmetry , folding to find mirror lines.
- Multiple lines of symmetry , shapes that fold many ways.
- Rotational symmetry , turning without changing.
- Combining symmetries , and shapes with both.
- Symmetry in nature, art, and design.
Key results / Formula card
| Idea | Quick statement |
|---|---|
| Line of symmetry | A line such that folding along it makes the two halves match |
| Number of lines of symmetry | depending on the shape |
| Rotational symmetry of order | Shape looks the same after turning |
| Equilateral triangle | lines, order |
| Square | lines, order |
| Regular hexagon | lines, order |
| Regular -gon | lines, order |
| Circle | Infinitely many lines, infinite order |
How to read this chapter
Bring a pair of scissors and paper. Folding, cutting, and turning are how symmetry truly sinks in. Many constructions also work brilliantly with a mirror , try one if you have it handy.