Turn it any way
Rotational symmetry of a circle
A turning wheel
A wheel is turning. You see a point of the wheel touching the ground. After some time you look again, and a point of the wheel is touching the ground again.
Can you tell whether it is the same point of the wheel as before?
What this lesson covers
The idea
A circle has complete rotational symmetry: rotated about its centre by any angle, it looks exactly the same.
A turning wheel
A wheel is turning. You see a point of the wheel touching the ground. After some time you look again, and a point of the wheel is touching the ground again.
Can you tell whether it is the same point of the wheel as before?
- Yes, I can tell by looking
- No, there is no way to tell
- Only if the wheel turns slowly
Turn the shapes
The dashed outline shows where each shape started. Turn the shape about its centre and watch when it looks the same as before.
Any angle
A circle has complete rotational symmetry: rotate it about its centre by any angle and it looks exactly the same.
A square looks the same only after turns of 90°, 180°, 270°. An equilateral triangle only after 120° and 240°. A circle looks the same at every angle. That is why you cannot tell which point of a turning wheel touches the ground.
Notes
A circle has complete rotational symmetry: rotated about its centre by any angle, it looks exactly the same.
Check yourself
A circle is turned by 37° about its centre. How does it look?
An equilateral triangle looks the same after a turn of 120°. After what next turn, in degrees, does it look the same again?
Answer: 240 °
120° + 120° = 240°. A triangle does not fit at 180°.
Which shape looks the same after a turn of any angle about its centre?
Why can you not tell which point of a turning wheel touches the ground?
- Different, because 37° is not a quarter turn. A square needs special turns like 90°. A circle does not: any angle works.
- Exactly the same as before — correct. Yes. A circle looks the same after a turn of any angle.
- The same only if it is turned back to 0°. It looks the same after every turn, not only a full one.
- A square. A square looks the same only after 90°, 180° and 270° turns.
- An equilateral triangle. A triangle looks the same only after 120° and 240° turns.
- A circle — correct. Yes. It has complete rotational symmetry.
- Because a circle looks the same after any rotation — correct. Yes. Any two turned positions of the circle look exactly alike.
- Because the wheel turns too fast to see. Even a wheel turned slowly, or in a still photo, looks the same. The reason is the circle's symmetry.
- Because the wheel has no centre. Every circle has a centre. It is the circle's rotational symmetry that hides the turn.