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…mehr

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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 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.