Symmetry (geometry)

A drawing of a butterfly with bilateral symmetry, with left and right sides as mirror images of each other.

In geometry, an object has symmetry if there is an operation or transformation (such as translation, scaling, rotation or reflection) that maps the figure/object onto itself (i.e., the object has an invariance under the transform).[1] Thus, a symmetry can be thought of as an immunity to change.[2] For instance, a circle rotated about its center will have the same shape and size as the original circle, as all points before and after the transform would be indistinguishable. A circle is thus said to be symmetric under rotation or to have rotational symmetry. If the isometry is the reflection of a plane figure about a line, then the figure is said to have reflectional symmetry or line symmetry;[3] it is also possible for a figure/object to have more than one line of symmetry.[4]

The types of symmetries that are possible for a geometric object depend on the set of geometric transforms available, and on what object properties should remain unchanged after a transformation. Because the composition of two transforms is also a transform and every transform has, by definition, an inverse transform that undoes it, the set of transforms under which an object is symmetric form a mathematical group, the symmetry group of the object.[5]

  1. ^ Martin, G. (1996). Transformation Geometry: An Introduction to Symmetry. Springer. p. 28.
  2. ^ "Symmetry | Thinking about Geometry | Underground Mathematics". undergroundmathematics.org. Retrieved 2019-12-06.
  3. ^ "Symmetry - MathBitsNotebook(Geo - CCSS Math)". mathbitsnotebook.com. Retrieved 2019-12-06.
  4. ^ Freitag, Mark (2013). Mathematics for Elementary School Teachers: A Process Approach. Cengage Learning. p. 721.
  5. ^ Miller, Willard Jr. (1972). Symmetry Groups and Their Applications. New York: Academic Press. OCLC 589081. Archived from the original on 2010-02-17. Retrieved 2009-09-28.