What is Rotational Symmetry? - Definition & Examples

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  • 0:01 What is Rotational Symmetry?
  • 0:57 What is the Center of…
  • 1:16 What is Order of Symmetry?
  • 2:03 What is Degree of Rotation?
  • 3:07 Lesson Summary
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Lesson Transcript
David Karsner

David holds a Master of Arts in Education

Expert Contributor
Amanda Robb

Amanda holds a Masters in Science from Tufts Medical School in Cellular and Molecular Physiology. She has taught high school Biology and Physics for 8 years.

If you have ever tried to hang a picture and not been able to tell which end was up, then your picture might have had rotational symmetry. Rotational symmetry is the characteristic that makes an object look the same even after you've rotated it.

What is Rotational Symmetry?

The recycling icon is a very common symbol, and like most effective icons, the image itself is suggestive of its meaning. The arrows of the image appear to be moving in a circular manner, suggesting the circular concept of recycling. Adding to this perception is that if you were to rotate the image 120 degrees, and another 120 degrees, and a third 120 degrees, it would look the same at all three stops.

This attribute is called rotational symmetry. Many shapes have rotational symmetry, such as rectangles, squares, circles, and all regular polygons. Choose an object and rotate it up to 180 degrees around its center. If at any point the object appears exactly like it did before the rotation, then the object has rotational symmetry. In this lesson, you will be given several image examples, as well as definitions of the relevant concepts center, order, and degree of rotation.

What is the Center of an Object?

The center of a shape or object with rotational symmetry is the point around which the rotation occurs. If one was to spin a basketball on the tip of his finger, the tip of his finger would be the center of the rotational symmetry. If an object has rotational symmetry, its center will also be its center of balance.

Order of Symmetry

The order of symmetry - or for short, order - is the number of times an object or shape can be rotated and still look like it did before rotation began. Let's look at some examples.

Rotational symmetry with 180 degrees of rotation

Here we have a hexagon. It has an order of 2. Let's look at another example.

Rotational symmetry with two rotations of 72 degrees
Rotational symmetry with 72 degrees of rotation

Both the blue and orange shape and starfish have an order of 5 because you can turn them 5 times and they still look the same as they did before they were rotated.

Let's look at one more example.

Rotational symmetry with 120 degrees rotation

This one has an order of 3.

The smallest order would be an order of 2. You cannot have a shape or object that has an order of 1. An order of 1 would mean that you can complete a full rotation without it appearing as it did before the rotation. In other words, an order of 1 would mean that is has no rotational symmetry.

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Additional Activities

Rotational Symmetry and Art

In this creative project, students will create a work of art that has rotational symmetry. Students can use any medium they desire, such as clay, paint, colored pencils, or collage. However, the art that students create should have rotational symmetry and be able to be hung in any direction, as described in the lesson. For example, students might create a mandala like shape from cut outs of colored paper in a color scheme they enjoy. This would be an example of rotational symmetry.

Student Instructions

Now that you understand what rotational symmetry is and how to identify it, it's time to try your own hand at creating something with this type of symmetry. In this creative assignment you will be creating a work of art that has rotational symmetry. You can choose any medium to work with, such as paint, clay, collage, or something else of your choosing. However, your end product should have rotational symmetry. For a list of constraints, see the criteria for success below.

Criteria For Success

  • Art should have rotational symmetry
  • Art should include a color scheme that is attractive
  • The student should be able to discuss their vision for the piece and explain why it has rotational symmetry

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