In this paper we consider the well-known Thurstone box problem in exploratory factor analysis. Initial loadings and components are extracted using principal component analysis. Rotating the components towards independence rather than rotating the loadings towards simplicity allows one to accurately recover the dimensions of each box and also produce simple loadings. It is shown how this may be done using an appropriate rotation criterion and a general rotation algorithm. Methods from independent component analysis are used, and this paper may be viewed as an introduction to independent component analysis from the perspective of factor analysis.