Automorphic is an adjective used in mathematics to describe a kind of property or symmetry possessed by certain mathematical objects, such as numbers, shapes, or functions. Specifically, it refers to the property of remaining unchanged under a certain transformation or operation. More precisely, an automorphic object retains its shape, structure, or essential characteristics when subjected to a specific transformation or action.
In the context of number theory, automorphic numbers are integers that retain the same final digits when squared or raised to a certain power. For example, 25 is an automorphic number because its square ends with the same digits "25." In general, these numbers have the property that their kth power ends with the same digits as the original number, where k is a positive integer.
Automorphic functions, often encountered in complex analysis, refer to functions that are invariant under certain transformations, such as inversion or rotation. These functions remain unchanged or display specific symmetries when subjected to these transformations.
Automorphic symmetry is a type of self-symmetry exhibited by particular geometric shapes, often used in the field of crystallography. These shapes possess translational symmetry, meaning that they can be shifted by a particular distance and still retain their overall structure.
Overall, automorphic is a term used to describe mathematical objects that possess a special property of remaining unchanged or retaining specific symmetries under particular operations or transformations.
