as orbits, which formalizes the notion of congruence: objects which can be taken to each other by a group action. For example, under the group of rigid motions of the plane, the perimeter of a triangle is an invariant, while the set of triangles congruent to a given triangle is a coinvariant. These are connected as follows: invariants are constant on coinvariants (for example, congruent triangles have the same perimeter), while two objects which agree in the value of one invariant may or may not be congruent (for example, two triangles with the same perimeter need not be congruent). In classification problems, one might seek to find a complete set of invariants, such that if two objects have the same values for this set of invariants, then they are congruent. For example, triangles such that all three sides are equal are congruent under rigid motions, via SSS congruence, and thus the lengths of all three sides form a complete set of invariants for triangles. The three angle measures of a triangle are also invariant under rigid motions, but do not form a complete set as incongruent triangles can share the same angle measures. However, if one allows scaling in addition to rigid motions, then the AAA similarity criterion shows that this is a complete set of invariants. Independent of presentation Secondly, a function may be defined in terms of some presentation or decomposition of a mathematical object; for instance, the Euler characteristic of a cell complex is defined as the alternating sum of the number of cells in each dimension. One may forget the cell complex structure and look only at the underlying topological space (the manifold) – as different cell complexes give the same underlying manifold, one may ask if the function is independent of choice of presentation, in which case it is an intrinsically defined invariant. This is the case for the Euler characteristic, and a general method for defining and computing invariants is to define them for a given presentation, and then show that they are independent of the choice of presentation. Note that there is no notion of a group action in this sense. The most common examples are: The presentation of a manifold in terms of coordinate charts – invariants must be unchanged under change of coordinates. Various manifold decompositions, as discussed for Euler characteristic. Invariants of a presentation of a group. Unchanged under perturbation