# Definition:Symmetry (Geometry)

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## Definition

A **symmetry** of a geometric figure $P$ is a bijection from $P$ to itself which preserves the distance between points.

In other words, it is a self-congruence.

Intuitively and informally, a **symmetry** is a movement of the figure so that it looks exactly the same after it has been moved.

## Also known as

A **symmetry** of a geometric figure $P$ is also known as:

- a
**symmetry operation on $P$** - a
**symmetry mapping on $P$** - a
**rigid motion**(that is, an**isometry**)**of $P$ onto itself**.

## Examples

### Rotations of Square through $90 \degrees$

Let $S$ be a square embedded in the plane centered at the origin $O$.

A rotation of the plane through an angle of $90 \degrees$ either clockwise or anticlockwise is a symmetry of $S$.

## Also see

## Linguistic Note

The word **symmetry** comes from Greek **συμμετρεῖν** (**symmetría**) meaning **measure together**.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Example $2.5$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: The Definition of Group Structure: $\S 26 \eta$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 34$. Examples of groups: $(5)$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**symmetry** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**symmetry** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**symmetry**(of a geometric figure)