In book IX of the Republic, Socrates offers a strange mathematical calculation, which claims to prove that the tyrant lives exactly 729 times less pleasantly than the king. For the first time, a complete and detailed reconstruction of this difficult text and its underlying structure is offered in the present article. It thereby proves that the distinction between ‘pleasure’ and the ‘image of pleasure’ is one among the keys to understanding the passage. It is furthermore shown how the whole calculation (...) is based on Plato’s ontology of dimensions and therefore proves that this theorem can be taken into account at least already for the time of the Republic. (shrink)