Take your two favorite polygons, P1 and P2, such as the regular hexagon and the square. A necessary condition for P1 and P2 to admit the same polygonal decomposition (they are then said to be equidecomposable) is, quite obviously, that they have the same area (they are then said to be equivalent).
It is easy to see that, for P1 and P2, being "equivalent" does indeed define, in the technical sense of the term, an equivalence relation on the set of polygons. An equivalence relation satisfies three properties: reflexivity (P1 does have the same area as itself), symmetry (if P1 has the same area as P2, then P2 has the same area as P1), and transitivity (if P1 and P2 have the same area and P2 and P3 have the same area, then P1 and P3 also have the same area).
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Shared authorship
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Checking an equidecomposition entails moving the pieces that make up the first polygon to form the second. Such motions are the transformations permitted by a geometry. Euclidean geometry, for instance, is governed by the group of similarities, comprising translations, dilations, and rotations. Because the pieces must retain their shape and size, we restrict attention to the group of isometries, comprising translations and rotations. Reflections, though isometries, are excluded because they require a piece to be flipped—that is, lifted out of the plane and turned over in three dimensions.
