Two polytopes (in particular, polygons or polyhedra) A and B are equicomplementable if there exist additions to them nearly disjoint (i.e., disjoint or with an intersection of lower dimension, with an empty interior) polytopes A1,...,An and B1,...,Bn such that for all i=1,...,n, Ai is congruent to Bi, for which

A ∪ (i=1n Ai)
B ∪ (i=1n Bi)
are congruent.

Some variants of this definition are also used. Sometimes the resulting polyhedra are required only to be equidecomposable; sometimes the Ai's and Bi's are required to be triangles or tetrahedra or d-dimensional simplexes. It turns out that this doesn't matter.

Equicomplementable polygons might be called building-block equivalent: given building blocks of the right shapes, we can glue them onto either A or B to get the same thing.

Naturally, equicomplementable polytopes must have equal measure (area, volume, or what-have-you in d dimensions), since their congruent additions have equal measure, and adding them gives congruent polytopes of equal measure. What about the converse?

In two dimensions, any two polygons of equal area are equicomplementable. In three dimensions, there exist even tetrahedra of equal volume which are not equicomplementable -- this is the substance of Hilbert's third problem.

See also the related concept of equidecomposable polytopes.

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