Given two equal-area polygons, one can be dissected into a finite number of pieces, then reassembled into the other.
There exists a square that is decomposable into smaller, different, squares.
All polygons are decomposable into triangles by adding only interior edges.
For polygons, all such decompositions have the same number of triangles.
Every polygon has every interior point visible from some vertex.
A 2-D Voronoi diagram's complexity is linear.
Given two convex polygons, there exists an edge of one that separates them.
Rotations commute.
For each edge of a polygon, consider the half-plane of points on the inside side of that edge. Then, the polygon's interior can be expressed as a Boolean expression in those half planes, with each half plane used only once.
but only a finite number of regular polyhedra.
This is usually not true for pairs of equal-volume polyhedra.
There is no cube that is decomposable into smaller, different, cubes.
Not all polyhedra are decomposable into tetrahedra by adding only interior faces.
Some polyhedra can be decomposed different ways into different numbers of tetrahedra.
Some polyhedra have interior points not visible from any vertex.
A 3-D Voronoi diagram's complexity can be quadratic.
Given two polyhedra, it is possible that none of their faces separate them.