There are many structures that mathematicians study because of their intrinsic interest, simplicity and being quick starting. Examples that come to mind are plane polygons and graphs.
Plane polygons have a variety of properties that one can look at: area, perimeter, minimum number of vertex guards, number of reflex angles, number of right angles, etc. An individual graph can have a variety of "invariants" that can be studied: coloring number, clique number, being eulerian, or hamiltonian, etc. For two graphs there is no list of invariants that guarantees that the two graphs are isomorphic. The complexity of checking when two graphs are isomorphic is still a dynamic area to investigate. I do not know any list of properties that guarantee that two polygons are congruent going beyond specifying the lengths of the sides and the measure of the angles. Finding new combinatorial/geometric properties of polygons seems to continue to be very worthwhile.
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