One-dimensional Symmetry Group

A one-dimensional symmetry group is a mathematical group that describes symmetries in one dimension (1D).

A pattern in 1D can be represented as a function f(x) for, say, the color at position x.

The 1D isometries map x to x + a and to ax. Isometries which leave the function unchanged are translations x + a with a such that f(x + a) = f(x) and reflections ax with a such that f(ax) = f(x).

Read more about One-dimensional Symmetry GroupTranslational Symmetry, Patterns Without Translational Symmetry, 1D-symmetry of A Function Vs. 2D-symmetry of Its Graph, Group Action, Orbits and Stabilizers

Other articles related to "group, symmetry group":

One-dimensional Symmetry Group - Orbits and Stabilizers
... Consider a group G acting on a set X ... The orbit of x is denoted by Gx Case that the group action is on R For the trivial group, all orbits contain only one element for a group of translations, an orbit is e.g ... {2,4}, and for the symmetry group with translations and reflections, e.g ...

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