Results 1 
5 of
5
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 ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
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. Contents 1 Introduction and Basic Properties 2 1...
The minimal perimeter for N confined deformable bubbles of equal area
"... Candidates to the least perimeter partition of various polygonal shapes into N planar connected equalarea regions are calculated for N � 42, compared to partitions of the disc, and discussed in the context of the energetic groundstate of a twodimensional monodisperse foam. The total perimeter and ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Candidates to the least perimeter partition of various polygonal shapes into N planar connected equalarea regions are calculated for N � 42, compared to partitions of the disc, and discussed in the context of the energetic groundstate of a twodimensional monodisperse foam. The total perimeter and the number of peripheral regions are presented, and the patterns classified according to the number and position of the topological defects, that is nonhexagonal regions (bubbles). The optimal partitions of an equilateral triangle are found to follow a pattern based on the position of no more than one defect pair, and this pattern is repeated for many of the candidate partitions of a hexagon. Partitions of a square and a pentagon show greater disorder. Candidates to the least perimeter partition of the surface of the sphere into N connected equalarea regions are also calculated. For small N these can be related to simple polyhedra and for N � 14 they consist of 12 pentagons and N −12 hexagons. 1
FaceRegular Bifaced Polyhedra
, 1997
"... Call bifaced any kvalent polyhedron, whose faces are p a agons and p b bgons only, where 3 a ! b, 0 ! p a , 0 p b . We consider the case b 2k k\Gamma2 covering applications; so either k = 3 a ! b 6, or (k; a; b; p a ) = (4; 3; 4; 8). For all these cases p a 12. Call such a polyhedron aR ..."
Abstract
 Add to MetaCart
Call bifaced any kvalent polyhedron, whose faces are p a agons and p b bgons only, where 3 a ! b, 0 ! p a , 0 p b . We consider the case b 2k k\Gamma2 covering applications; so either k = 3 a ! b 6, or (k; a; b; p a ) = (4; 3; 4; 8). For all these cases p a 12. Call such a polyhedron aR i (resp. bR j ) if each of its agonal (resp. bgonal) faces is adjacent to exactly i agonal (resp. j bgonal) faces. The preferable (i.e. with isolated pentagons) fullerenes are the case aR 0 for (k; a; b) = (3; 5; 6). We classify bifaced polyhedra which are both aR i and bR j , and also all a or bface regular bifaced polyhedra (except aR 0 , aR 1 for (k; a)= (3,4), (3,5), (4,3), and, for fullerenes, 6R 4 with 52 n 78, or 6R 3 with 52 n 58). 1 Introduction Denote by (k; a; b; p a ; p b ) and call bifaced any kvalent polyhedron whose faces are only p a agons and p b bgons with 3 a ! b and 0 ! p a , 0 p b . Any polyhedron (k; a; b; p a ; p b ) with n vertices has 1 2 kn = ...
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 ..."
Abstract
 Add to MetaCart
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
THE INFIMUM OF THE VOLUMES OF CONVEX POLYTOPES OF ANY GIVEN FACET AREAS IS 0
, 1304
"... Abstract. We prove the theorem mentioned in the title, for R n, where n ≥ 3. The case of the simplex was known previously. Also, the case n = 2 was settled, but there the infimum was some welldefined function of the side lengths. We also consider the cases of spherical and hyperbolic nspaces. Ther ..."
Abstract
 Add to MetaCart
Abstract. We prove the theorem mentioned in the title, for R n, where n ≥ 3. The case of the simplex was known previously. Also, the case n = 2 was settled, but there the infimum was some welldefined function of the side lengths. We also consider the cases of spherical and hyperbolic nspaces. There we give some necessary conditions for the existence of a convex polytope with given facet areas, and some partial results about sufficient conditions for the existence of (convex) tetrahedra.