We require that in a finite region there are only finitely many distinct points so obtained. Transformation groups that satisfy this requirement are called discontinuous . In summary, we are restricting our investigation to discontinuous group of rotations and translations in the plane. We set out to find all such groups. Note that the study of plane symmety is a study of structure.
Structure is an algebraic notion and has nothing to do with angles or distances.
Do not get confused by elements in our group with points, angles, or distances in the plane. This is why we do talk about distances and angles in discussion of symmetry. We will denote groups using set notation, with curly braces. More often we'll need set-bulding notation. These groups are all isomorphic to the cyclic group of infinite order.