Rotational Symmetry

Rotational Symmetry

A shape has Rotational Symmetry when it still looks exactly the same after some rotation less than one full turn.

As we go all the way around once (a full turn), we can count how many times it matches itself. This number is called the Order of rotational symmetry.

images/symmetry-rotate.js?mode=2

Rotational Symmetry
Order 2
images/symmetry-rotate.js?mode=3

Rotational Symmetry
Order 3

Examples of Different Rotational Symmetry Order

Think of propeller blades (like below). When we turn the propeller around its center, it keeps looking the same again and again.

Order Example Shape Artwork
using Symmetry Artist
The number 2 shown with its rotational symmetry path An oval-like shape with order 2 rotational symmetry Symmetric abstract pattern with order 2 rotational symmetry
The number 3 shown with its rotational symmetry path A three-pointed star-like shape with order 3 rotational symmetry Symmetric abstract pattern with order 3 rotational symmetry
The number 4 shown with its rotational symmetry path A square-like shape with order 4 rotational symmetry Symmetric abstract pattern with order 4 rotational symmetry
... and there's also Order 5, 6, 7, and ...
The number 8 shown with its rotational symmetry path An eight-pointed star-like shape with order 8 rotational symmetry Symmetric abstract pattern with order 8 rotational symmetry
... and Orders 9, 10, and so on.

Try rotational symmetry yourself with Symmetry Artist ... !

An asymmetric shape demonstrating order 1 (no rotational symmetry)

Is there Rotational Symmetry of Order 1 ?

Not really! If a shape only matches itself once as we go around (ie it matches itself after one full rotation) there's really no symmetry at all ...

... because the word "Symmetry" comes from syn- together and metron measure, and there can't be "together" if there's just one thing.

Angles

We can calculate the angle of each turn! Take the full rotation of 360° and divide by the order n:

Angle of Each Turn = 360°n

Symmetric abstract pattern with order 4 rotational symmetry

Example: Order 4

Angle of Each Turn = 360°4 = 90°

Real World Examples

A standard dartboard with alternating black and white sectors
The dartboard pattern has rotational symmetry of Order 10
The US Bronze Star Medal, a five-pointed star
The US Bronze Star Medal has Order 5
The London Eye giant Ferris wheel with multiple passenger capsules
The London Eye has Order ... oops, I lost count!
849, 850, 2137, 2138, 3360, 3361, 3362, 3363, 5033, 5034
Activity: Make a Mandala
Symmetry Symmetry Artist Reflection Symmetry Point Symmetry Transformations Rotation Geometry Index