Calabi-Yau Three-folds: Poincare Polynomials and Fractals

We study the Poincar´e polynomials of all known Calabi-Yau three-folds as constrained polynomials of Littlewood type, thus generalising the wellknown investigation into the distribution of the Euler characteristic and Hodge numbers. We find interesting fractal behaviour in the roots of these polynomials in relation to the existence of isometries, distribution versus typicality, and mirror symmetry