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A '''rigid motion''' of a set is a translation, rotation, or a composition of those two kinds of motion.  A rigid motion of a set <math>A \ </math> is often written <math>g(A) \ </math>.  

A '''rigid motion''' of a set is a translation, rotation, or a composition of those two kinds of motion.  A rigid motion of a set <math>A \ </math> is often written <math>g(A) \ </math>.  

Let <math>A, B \ </math> be sets.  We say there is a '''rigid body equivalence''' <math>A~B \ </math> if <math>A \ </math> can be partitioned into finitely many sets <math>A_1, ..., A_n \ </math> and there are <math>n \ </math> rigid motions <math>g_1, ..., g_n \ </math> such that <math>g_1(A_1) \cup ... \cup g_n(A_n) = B \ </math>.

Let <math>A, B \ </math> be sets.  We say there is a '''rigid body equivalence''' <math>A \cong B \ </math> if <math>A \ </math> can be partitioned into finitely many sets <math>A_1, ..., A_n \ </math> and there are <math>n \ </math> rigid motions <math>g_1, ..., g_n \ </math> such that <math>g_1(A_1) \cup ... \cup g_n(A_n) = B \ </math>.

We say there exists a '''paradoxical decomposition''' of a set <math>A \ </math> if <math>A = B \cup C \ </math>, and <math>A~B \ </math> and <math>A~C \ </math>.

We say there exists a '''paradoxical decomposition''' of a set <math>A \ </math> if <math>A = B \cup C \ </math>, and <math>A\cong B \ </math> and <math>A\cong C \ </math>.

The Banach-Tarski theorem (aka the Banach-Tarski paradox) states that the solid ball admits a paradoxical decomposition.

The Banach-Tarski theorem (aka the Banach-Tarski paradox) states that the solid ball admits a paradoxical decomposition.