The dimension for the Europen Union Council under the Nice rules

In this note it is proved that the European Union enlargement voting system under two different decision rules (simple majority of countries or qualified majority of 2/3) adopted in the Nice European summit, held in December 2000, has exactly dimension 3. This result clarifies that these voting systems cannot realize either as a weighted voting system or as the interaction of two weighted voting systems and, therefore, it is useful to prove the existence of well-known real voting systems of dimension 3.