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 Group:  Translational Symmetry, Patterns Without Translational Symmetry, 1D-symmetry of A Function Vs. 2D-symmetry of Its Graph, Group Action, Orbits and Stabilizers

Famous quotes containing the words symmetry and/or group:

    What makes a regiment of soldiers a more noble object of view than the same mass of mob? Their arms, their dresses, their banners, and the art and artificial symmetry of their position and movements.
    George Gordon Noel Byron (1788–1824)

    The poet who speaks out of the deepest instincts of man will be heard. The poet who creates a myth beyond the power of man to realize is gagged at the peril of the group that binds him. He is the true revolutionary: he builds a new world.
    Babette Deutsch (1895–1982)