In mathematics and geometry, a discrete symmetry is a

symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...

that describes non-continuous changes in a system. For example, a square possesses discrete

rotational symmetry Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which ...

, as only rotations by multiples of right angles will preserve the square's original appearance. Discrete symmetries sometimes involve some type of 'swapping', these swaps usually being called ''reflections'' or ''interchanges''. In mathematics and

theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...

, a discrete symmetry is a

under the transformations of a

discrete group In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and o ...

—e.g. a

topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...

with a

discrete topology In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest top ...

whose elements form a finite or a

countable set In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numb ...

. One of the most prominent discrete symmetries in physics is

parity symmetry In physics, a parity transformation (also called parity inversion) is the flip in the sign of ''one'' spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a point refle ...

. It manifests itself in various elementary physical quantum systems, such as quantum harmonic oscillator, electron orbitals of Hydrogen-like atoms by forcing wavefunctions to be even or odd. This in turn gives rise to

selection rules In physics and chemistry, a selection rule, or transition rule, formally constrains the possible transitions of a system from one quantum state to another. Selection rules have been derived for electromagnetic transitions in molecules, in atoms, i ...

that determine which transition lines are visible in atomic

absorption spectra Absorption spectroscopy refers to spectroscopic techniques that measure the absorption of radiation, as a function of frequency or wavelength, due to its interaction with a sample. The sample absorbs energy, i.e., photons, from the radiating ...

References

* Slavik V. Jablan, ''Symmetry, Ornament and Modularity'', Volume 30 of K & E Series on Knots and Everything, World Scientific, 2002. Group theory Theoretical physics Symmetry {{geometry-stub