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 a − x. Isometries which leave the function unchanged are translations x + a with a such that f(x + a) = f(x) and reflections a − x with a such that f(a − x) = 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)
“Even in harmonious families there is this double life: the group life, which is the one we can observe in our neighbour’s household, and, underneath, another—secret and passionate and intense—which is the real life that stamps the faces and gives character to the voices of our friends. Always in his mind each member of these social units is escaping, running away, trying to break the net which circumstances and his own affections have woven about him.”
—Willa Cather (1873–1947)