A fullerene F n is a 3-regular (or cubic) polyhedral carbon molecule for which the n vertices - the carbons atoms - are arranged in 12 pentagons and ( n 2 \Gamma 10) hexagons. Only a finite number of fullerenes are expected to be, up to scale, isometrically embeddable into a hypercube. Looking for the list of such fullerenes, we first check the embeddability of all fullerenes F n for n ! 60 and of all preferable fullerenes C n for n ! 86 and their duals. Then, we consider some infinite families, including fullerenes with icosahedral symmetry, which describe virus capsids, onion-like metallic clusters and geodesic domes. Quasi-embeddings and fullerene analogues are considered. We also present some results on chemically relevant polyhedra such as coordination polyhedra and cluster polyhedra. Finally we conjecture that the list of known embeddable fullerenes is complete and present its relevance to the Katsura model for vesicles cells. Contents 1 Introduction and Basic Properties 2 1...

. In this note, a short report is presented concerning a symbolic approach to tiling theory which has been developed in Bielefeld over the last 15 years. 0. Introduction Spatial structures displaying --- one way or the other --- some kind of repetitive regularity have attracted the attention of the human mind from prehistoric times on. Yet, it took more than two thousand years after the first complete classification and enumeration results had been obtained by the Pythagorean school before mathematicians conceived of and explicitly used the group concept as an elegant conceptual framework for formalizing the arguments on which such classification results had been based. Since then, phenomenal progress has been achieved. Still, for a long time, grouptheoretical concepts appeared to detect phenomena related to the algebraic properties of symmetry operations (that is, the various ways they combine to make up a group) only, and to be unable to represent the individual geometric-topologic...