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Uniform Partitions of 3space, their Relatives and Embedding
 European J. of Combinatorics
, 2000
"... We review 28 uniform partitions of 3space 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 4polytopes of ConwayGuy, noncompact uniform par ..."
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We review 28 uniform partitions of 3space 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 4polytopes of ConwayGuy, noncompact 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 semiregular 3prisms with cubes or with regular tetrahedra and regular octahedra. On the way many new partitions are added to incomplete cases considered here. 1
Fullerenes and Coordination Polyhedra versus HalfCubes Embeddings
, 1997
"... A fullerene F n is a 3regular (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 fo ..."
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A fullerene F n is a 3regular (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, onionlike metallic clusters and geodesic domes. Quasiembeddings 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.
A Zoo of l_1embeddable Polyhedra II
"... 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 kvalent polyhedra with only two types of faces. 1 ..."
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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 kvalent polyhedra with only two types of faces. 1
Embedding of All Regular Tilings and StarHoneycombs
, 1998
"... We review the regular tilings of dsphere, Euclidean dspace, hyperbolic dspace and Coxeter's regular hyperbolic honeycombs (with infinite or starshaped cells or vertex figures) with respect of possible embedding, isometric up to a scale, of their skeletons into a mcube or mdimensional cub ..."
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We review the regular tilings of dsphere, Euclidean dspace, hyperbolic dspace and Coxeter's regular hyperbolic honeycombs (with infinite or starshaped cells or vertex figures) with respect of possible embedding, isometric up to a scale, of their skeletons into a mcube or mdimensional cubic lattice. In section 2 the last remaining 2dimensional case is decided: for any odd m 7, starhoneycombs m m 2 are embeddable while m 2 m are not (unique case of nonembedding 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, nonembeddability of all remaining starhoneycombs (on 3sphere and hyperbolic 4space) is proved. In the last section 5, all cases of embedding for dimension d ? 2 are identified. Besides hypersimplices and hyperoctahedra, they are exactly those with bipartite skeleton: hypercubes, cubic lattices and 8, 2, 1 tilings of hyperbolic 3, ...
Isometric Embedding of Mosaics Into Cubic Lattices
, 2001
"... 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 suc ..."
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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.