And in general, it's a parallelgrammatic lattice. (So square and hexagonal lattices are very special rhombic lattices.) If it has a rectangle as a fundamental region, it's a rectangular lattice. (A rhombus is a parallelogram with equal sides.) If a lattice has a rhombus as a fundamental region, it's a rhombic lattice. That's because in that case, the points in the lattice nearest any one point in the lattice are the vertices of a regular hexagon. If it has a 60° rhombus as a fundamental region, it's called a hexagonal lattice. If a lattice has a square fundamental region, it's called a square lattice. We can classify lattices into five different kinds. So the lattice of dots corresponds to the translation symmetries.
Furthermore, you can see what composition you need by seeing where the red dot needs to go.
Thus, every translation is of the form T mU n where m and n are integers. By taking a composition of these two translations and their inverses, you can construct every other translation symmetry of the pattern. A different translation, call it U, is indicated by the rose colored arrows it translates up and a bit right. One translation, call it T, is indicated by the green arrows it translates right and a bit upward. In the example at the left, the translates of the one point colored red are indicated by dark blue dots. For any point, the collection of translates of it by translation symmetries of a pattern forms a lattice. Just by considering the translation symmetries of a pattern we can begin to classify patterns.
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