Group Generators
For the set of standard, symmetrical shapes, the symbols of the edges of the polygon may be understood to be the generators of a group. Then, the polygon, written in terms of group elements, becomes a constraint on the free group generated by the edges, giving a group presentation with one constraint.
Thus, for example, given the Euclidean plane, let the group element act on the plane as while . Then generate the lattice, and the torus is given by the quotient space (a homogeneous space) . More generally, the two generators can be taken to generate a parallelogram tiling, of fundamental parallelograms.
For the torus, the constraint on the free group in two letters is given by . This constraint is trivially embodied in the action on the plane given above. Alternately, the plane can be tiled by hexagons, and the centers of the hexagons form a hexagonal lattice. Identifying opposite edges of the hexagon again leads to the torus, this time, with the constraint describing the action of the hexagonal lattice generators on the plane.
In practice, most of the interesting cases are surfaces with negative curvature, and are thus realized by a discrete lattice in the group acting on the upper half-plane. Such lattices are known as Fuchsian groups.
Read more about this topic: Fundamental Polygon
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