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Encoding fullerenes and geodesic domes
 SIAM. J. Discrete Math
, 2004
"... Abstract. Coxeter’s classification of the highly symmetric geodesic domes (and, by duality, the highly symmetric fullerenes) is extended to a classification scheme for all geodesic domes and fullerenes. Each geodesic dome is characterized by its signature: a plane graph on twelve vertices with label ..."
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Abstract. Coxeter’s classification of the highly symmetric geodesic domes (and, by duality, the highly symmetric fullerenes) is extended to a classification scheme for all geodesic domes and fullerenes. Each geodesic dome is characterized by its signature: a plane graph on twelve vertices with labeled angles and edges. In the case of the Coxeter geodesic domes, the plane graph is the icosahedron, all angles are labeled one, and all edges are labeled by the same pair of integers (p, q). Edges with these “Coxeter coordinates ” correspond to straight line segments joining two vertices of Λ, the regular triangular tessellation of the plane, and the faces of the icosahedron are filled in with equilateral triangles from Λ whose sides have coordinates (p, q). We describe the construction of the signature for any geodesic dome. In turn, we describe how each geodesic dome may be reconstructed from its signature: the angle and edge labels around each face of the signature identify that face with a polygonal region of Λ and, when the faces are filled by the corresponding regions, the geodesic dome is reconstituted. The signature of a fullerene is the signature of its dual. For each fullerene, the separation of its pentagons, the numbers of its vertices, faces, and edges, and its symmetry structure are easily computed directly from its signature. Also, it is easy to identify nanotubes by their signatures.
Fullerenes as tilings of surfaces
 J. Chem. Inf. Comput. Sci
"... If a fullerene is defined as a finite trivalent graph made up solely of pentagons and hexagons, embedding in only four surfaces is possible: the sphere, torus, Klein bottle, and projective (elliptic) plane. The usual spherical fullerenes have 12 pentagons; elliptic fullerenes, 6; and toroidal and Kl ..."
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Cited by 5 (1 self)
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If a fullerene is defined as a finite trivalent graph made up solely of pentagons and hexagons, embedding in only four surfaces is possible: the sphere, torus, Klein bottle, and projective (elliptic) plane. The usual spherical fullerenes have 12 pentagons; elliptic fullerenes, 6; and toroidal and Kleinbottle fullerenes, none. Kleinbottle and elliptic fullerenes are the antipodal quotients of centrosymmetric toroidal and spherical fullerenes, respectively. Extensions to infinite systems (plane fullerenes, cylindrical fullerenes, and space fullerenes) are indicated. Eigenvalue spectra of all four classes of finite fullerenes, are reviewed. Leapfrog fullerenes have equal numbers of positive and negative eigenvalues, with 0, 0, 2, or 4 eigenvalues zero for spherical, elliptic, Kleinbottle, and toroidal cases, respectively.
Zigzag structure of simple twofaced polyhedra
 Combinatorics, Probability & Computing 14
, 2005
"... A zigzag in a plane graph is a circuit of edges, such that any two, but no three, consecutive edges belong to the same face. A railroad in a plane graph is a circuit of hexagonal faces, such that any hexagon is adjacent to its neighbors on opposite edges. A graph without a railroad is called tight. ..."
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Cited by 4 (2 self)
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A zigzag in a plane graph is a circuit of edges, such that any two, but no three, consecutive edges belong to the same face. A railroad in a plane graph is a circuit of hexagonal faces, such that any hexagon is adjacent to its neighbors on opposite edges. A graph without a railroad is called tight. We consider the zigzag and railroad structures of general 3valent plane graph and, especially, of simple twofaced polyhedra, i.e., 3valent 3polytopes with only agonal and bgonal faces, where 3 ≤ a < b ≤ 6; the main cases are (a,b) = (3,6), (4,6) and (5,6) (the fullerenes). We completely describe the zigzag structure for the case (a,b)=(3,6). For the case (a,b)=(4,6) we describe symmetry groups, classify all tight graphs with simple zigzags and give the upper bound 9 for the number of zigzags in general tight graphs. For the remaining case (a,b)=(5,6) we give a construction realizing a prescribed zigzag structure.
4valent plane graphs with 2, 3 and 4gonal faces, satellite conference (of congress
 of ISM in Beijing) on algebra and combinatorics
"... Call ihedrite any 4valent nvertex plane graph, whose faces are 2, 3 and 4gons only and p2+p3 = i. The edges of an ihedrite, as of any Eulerian plane graph, are partitioned by its central circuits, i.e. those, which are obtained by starting with an edge and continuing at each vertex by the edg ..."
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Cited by 3 (3 self)
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Call ihedrite any 4valent nvertex plane graph, whose faces are 2, 3 and 4gons only and p2+p3 = i. The edges of an ihedrite, as of any Eulerian plane graph, are partitioned by its central circuits, i.e. those, which are obtained by starting with an edge and continuing at each vertex by the edge opposite the entering one. So, any ihedrite is a projection of an alternating link, whose components correspond to its central circuits. Call an ihedrite irreducible, if it has no railroad, i.e. a circuit of 4gonal faces, in which every 4gon is adjacent to two of its neighbors on opposite edges. We present the list of all ihedrites with at most 15 vertices. Examples of other results: (i) All ihedrites, which are not 3connected, are identified. (ii) Any irreducible ihedrite has at most i − 2 central circuits. (iii) All ihedrites without selfintersecting central circuits are listed. (iv) All symmetry group of ihedrites are listed.
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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Cited by 2 (0 self)
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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. Contents 1 Introduction and Basic Properties 2 1...
Zigzag structure of complexes
, 2008
"... Inspired by Coxeter’s notion of Petrie polygon for dpolytopes (see [Cox73]), we consider a generalization of the notion of zigzag circuits on complexes and compute the zigzag structure for several interesting families of dpolytopes, including semiregular, regularfaced, Wythoff Archimedean ones, C ..."
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Inspired by Coxeter’s notion of Petrie polygon for dpolytopes (see [Cox73]), we consider a generalization of the notion of zigzag circuits on complexes and compute the zigzag structure for several interesting families of dpolytopes, including semiregular, regularfaced, Wythoff Archimedean ones, Conway’s 4polytopes, halfcubes, folded cubes. Also considered are regular maps and Lins triality relations on maps.
Twisted Domes
"... The most usual polyhedra with large numbers of triangular faces are geodesic domes, having nonregular triangles chosen so that the polyhedron approximates to a sphere. If the faces are equilateral triangles more interesting forms result, particularly if there are no planes of mirror symmetry, and t ..."
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The most usual polyhedra with large numbers of triangular faces are geodesic domes, having nonregular triangles chosen so that the polyhedron approximates to a sphere. If the faces are equilateral triangles more interesting forms result, particularly if there are no planes of mirror symmetry, and the polyhedron has a twisted appearance. Some techniques for producing such polyhedra are described, and illustrated with examples. Twisted Tilings Of the various kinds of tiling that Grünbaum and Shephard[1] call not edgetoedge, some (the right hand tilings figure 1 for example) retain rotational symmetry but lose mirror symmetry, and occur in right and left handed forms. Although they can be seen as the result of sliding faces, they can also be seen as the result of twisting the faces of an edgetoedge tiling. This has the effect of producing additional faces at the vertices of the edgetoedge tiling. A dual to a tiling has a vertex corresponding to each face of the original, and a face coresponding to each vertex of the original. Vertices in the dual are joined by an edge if and only if faces in the original have an edge in common (and conversely). The new faces in the twisted tiling must be faces of the dual tiling, because of their relation to the original vertices: the edges meeting at the vertex are sides of the new
Zigzag Structure of Simple Bifaced Polyhedra
, 2008
"... A zigzag in a plane graph is a circuit of edges such that any two, but no three, consecutive edges belong to the same face. A railroad in a plane graph is a circuit of hexagonal faces, such that any hexagon is adjacent to its neighbors on opposite edges. A graph without railroad is called tight. We ..."
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A zigzag in a plane graph is a circuit of edges such that any two, but no three, consecutive edges belong to the same face. A railroad in a plane graph is a circuit of hexagonal faces, such that any hexagon is adjacent to its neighbors on opposite edges. A graph without railroad is called tight. We consider zigzag and railroad structure of general 3valent plane graph and, especially, of simple bifaced polyhedra, i.e. 3valent 3polytopes with only agonal and bgonal faces, where 3 ≤ a < b ≤ 6; the main cases are (a,b) = (3,6), (4,6) and (5,6) (the fullerenes). We describe completely zigzag structure and symmetry groups for the case (a,b)=(3,6). For the case (a,b)=(4,6) we describe symmetry groups, classify all tight graphs with simple zigzags and give upper bound 9 for the number of zigzags in general tight graph. For the remaining case (a,b)=(5,6) we give a construction realizing prescribed zigzag structure.
ZIGZAG AND CENTRAL CIRCUIT STRUCTURE OF ({1, 2, 3}, 6)SPHERES
"... Abstract. We consider 6regular plane graphs whose faces have size 1, 2 or 3. In Section 2 a practical enumeration method is given that allowed us to enumerate them up to 53 vertices. Subsequently, in Section 3 we enumerate all possible symmetry groups of the spheres that showed up. In Section 4 we ..."
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Abstract. We consider 6regular plane graphs whose faces have size 1, 2 or 3. In Section 2 a practical enumeration method is given that allowed us to enumerate them up to 53 vertices. Subsequently, in Section 3 we enumerate all possible symmetry groups of the spheres that showed up. In Section 4 we introduce a new GoldbergCoxeter construction that takes a 6regular plane graph G0, two integers k and l and returns two 6regular plane graphs. Then in the final section, we consider the notions of zigzags and central circuits for the considered graphs. We introduced the notions of tightness and weak tightness for them and prove an upper bound on the number of zigzags and central circuits of such tight graphs. We also classify the tight and weakly tight graphs with simple zigzags or central circuits.