What is Symmetry in Math?

Mar 30, 2022 | Cherry Hill

“Symmetry is a vast subject and has significance in art and nature. Mathematics lies in symmetry’s root, and it would be very hard to find a better one on which to demonstrate the working of the mathematical intellect.”

-Hermann Weyl


Symmetry is a concept that states that when we move one shape, such as turning, flipping, or sliding it, it becomes identical to the other.


“Symmetry” comes from the Greek word, which implies “to measure together.” 

When two parts of anything are identical, they are symmetrical. Drawing a mirror line through the middle of a figure and observing if both parts are similar is how you check to see if it is symmetrical. Some great examples of symmetry in nature are starfish, peacocks, turtles, sunflowers, honeycombs, snowflakes, rainbows, etc. A figure is asymmetrical when two or more identical pieces face each other or revolve around an axis. 


The Four Types of Symmetry


1. Translational Symmetry: Translational symmetry is where a figure or an image is translated at a set distance in the same direction as the original. The spaces between points, angles, sizes, and shapes of the figure will not change. The only thing that changes is its location. You may move it right or left. You may move it up or down. You may move it through a combination of these two, but these are the only possibilities.


2. Rotational Symmetry: Rotational Symmetry, also known as radial symmetry, is where a shape or an image looks precisely similar to the original form or image after some rotation. We count rotational symmetry by the number of turns it takes, also referred to as order, for a shape to look the same. For example, a rectangle has an order of 2, and a five-point star has an order of 5. You can find rotational symmetry naturally in sea stars, jellyfish, and sea anemones. It also appears in human-made objects like airplane propellers, Ferris wheels, dartboards!


3. Reflectional Symmetry: Reflection symmetry is also known as line symmetry or mirror symmetry. One half of the image or picture reflects the other half. The line which a reflection takes place over is known as the line of symmetry. It’s also important to note that some shapes can have multiple lines of symmetry. Take a square, for example - you can draw four lines of symmetry on a square—one horizontally across the middle, one vertically down the middle, and two going diagonally each way.


 4. Glide Reflection Symmetry: Glide Reflection Symmetry is best thought of as a hybrid between reflection and translational symmetry. It is a type of symmetry where the figure or image looks precisely the original when it is reflected over a line and then translated at a given distance in a given direction. The footprints trail is one of the best examples of Glide Reflection Symmetry.

This type of symmetry involves both processes but in a specific order; reflection over a line and translation along the line. A shape must first be reflected and then translated in any direction for glide reflection to have taken place.

As in translational symmetry, glide-reflectional symmetry exists only for infinite patterns.


Symmetry is an exciting topic of study. It involves the integration of math and art, which allows for a beautiful showcase of the visuals that numbers can produce. We find symmetry in objects like our reflection in the mirror, a butterfly’s beautifully patterned wings, and traffic stop signs. Most things that we see daily have some form of symmetry or balance. Asymmetrical objects also exhibit underlying symmetry principles through subtle patterns that are not immediately visible to the naked eye.


So next time you’re outside, see if you can find some numbers hard at work making nature as beautiful as it is!


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