In this paper we present a new way for proving the existence of non-measurable sets using a convenient operation of a discrete group on the Euclidian sphere. The only choice assumption used in this construction is the Hahn-Banach theorem, a weaker hypothesis than the Boolean Prime Ideal Theorem. Our construction proves that the Hahn-Banach theorem implies the existence of a non-Lebesgue-measurable set of reals. In fact we prove (under Hahn-Banach theorem) that there is no finitely additive, rotation invariant extension of Lebesgue measure to all subsets of the three-dimensional Euclidean space.