Definition:Decomposable Set

A set $$S \subset \mathbb{R}^n \ $$ is decomposable in $$m \ $$ sets $$A_1,\dots, A_m \subset \mathbb{R}^n \ $$ if there exist isometries $$\phi_1,\ldots,\phi_m: \mathbb{R}^n \to \mathbb{R}^n \ $$ such that


 * 1) $$S = \bigcup_{k=1}^m \phi_k(A_k) \ $$
 * 2) For $$i \neq j, \phi_i(A_i) \cap \phi_j(A_j) = \varnothing \ $$