We review 28 uniform partitions of 3-space in order to find out which of them have graphs (skeletons) embeddable isometrically (or with scale 2) into some cubic lattice Zn. We also consider some relatives of those 28 partitions, including Achimedean 4-polytopes of Conway-Guy, non-compact uniform partitions, Kelvin partitions and those with unique vertex figure (i.e. Delaunay star). Among last ones we indicate two continuums of aperiodic tilings by semi-regular 3-prisms with cubes or with regular tetrahedra and regular octahedra. On the way many new partitions are added to incomplete cases considered here. 1

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...

We review the regular tilings of d-sphere, Euclidean d-space, hyperbolic d-space and Coxeter's regular hyperbolic honeycombs (with infinite or star-shaped cells or vertex figures) with respect of possible embedding, isometric up to a scale, of their skeletons into a m-cube or m-dimensional cubic lattice. In section 2 the last remaining 2-dimensional case is decided: for any odd m 7, star-honeycombs m m 2 are embeddable while m 2 m are not (unique case of non-embedding for dimension 2). As a spherical analogue of those honeycombs, we enumerate, in section 3, 36 Riemann surfaces representing all nine regular polyhedra on the sphere. In section 4, non-embeddability of all remaining star-honeycombs (on 3-sphere and hyperbolic 4-space) is proved. In the last section 5, all cases of embedding for dimension d ? 2 are identified. Besides hyper-simplices and hyper-octahedra, they are exactly those with bipartite skeleton: hyper-cubes, cubic lattices and 8, 2, 1 tilings of hyperbolic 3-, ...

We complete here the study of l 1 -polyhedra started in our previous paper on this subject, [DeGr97a]. New classes are considered, especially small polyhedra, some operations on Platonic solids and k-valent polyhedra with only two types of faces. 1

We review mosaics T (tilings of Euclidean plane by regular polygons) with respect to possible embedding, isometric up to a scale, of their skeletons or the skeletons of their duals T , into some cubic lattice Z n . The main result of this paper is the classication, given in Table 1, of all 58 such mosaics among all 165 mosaics of the catalog, given in [Cha89] and including all main classications of mosaics. We consider mosaics T (i.e. edge-to-edge planar tilings by regular polygons), such that the skeleton graph of T or of T embeds isometrically, up to a scale , into the skeleton of n-dimensional cubic lattice Z n for some n 1. Such embedding will be denoted by T ! 1 Z n and T ! 1 Z n . For planar embeddable graph, we have = 1; 2 and such embedding, if the graph does not contain K 4 , is essentially unique; see [CDG97]. The following examples illustrate those notions. Example 1. A path P n (with n vertices) embeds into an hypercube H n 1 , as well as into...