One-dimensional Symmetry Group
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. A one-dimensional symmetry group is a mathematical group that describe symmetries in one dimension. 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). We first consider patterns for which the group is discrete, i.e. for which the positive values...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. A one-dimensional symmetry group is a mathematical group that describe symmetries in one dimension. 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). We first consider patterns for which the group is discrete, i.e. for which the positive values in the group have a minimum. By rescaling we make this minimum value 1. Such patterns fall in two categories, the two 1D space groups.
