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)