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Interactive Topological Drawing
, 1998
"... The research presented here examines topological drawing, a new mode of constructing and interacting with mathematical objects in threedimensional space. In topological drawing, issues such as adjacency and connectedness, which are topological in nature, take precedence over purely geometric issues ..."
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Cited by 18 (1 self)
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The research presented here examines topological drawing, a new mode of constructing and interacting with mathematical objects in threedimensional space. In topological drawing, issues such as adjacency and connectedness, which are topological in nature, take precedence over purely geometric issues. Because the domain of application is mathematics, topological drawing is also concerned with the correct representation and display of these objects on a computer. By correctness we mean that the essential topological features of objects are maintained during interaction. We have chosen to limit the scope of topological drawing to knot theory, a domain that consists essentially of one class of object (embedded circles in threedimensional space) yet is rich enough to contain a wide variety of difficult problems of research interest. In knot theory, two embedded circles (knots) are considered equivalent if one may be smoothly deformed into the other without any cuts or selfintersections. This notion of equivalence may be thought of as the heart of knot theory. We present methods for the computer construction and interactive manipulation of a
Belyi Functions for Archimedean Solids
 Discrete Math
, 1996
"... The notion of a Belyi function is a main technical tool which relates the combinatorics of maps (i.e., graphs embedded into surfaces) with Galois theory, algebraic number theory, and the theory of Riemann surfaces. In this paper we compute Belyi functions for a class of semiregular maps which co ..."
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Cited by 8 (0 self)
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The notion of a Belyi function is a main technical tool which relates the combinatorics of maps (i.e., graphs embedded into surfaces) with Galois theory, algebraic number theory, and the theory of Riemann surfaces. In this paper we compute Belyi functions for a class of semiregular maps which correspond to the socalled Archimedean solids. R#sum# La notion de fonction de Belyi est un outil technique qui relie la combinatoire des cartes (c'est#dire, des graphes plong#s sur des surfaces) avec la th#orie de Galois, la th#orie des nombres alg#briques et la th#orie des surfaces de Riemann. Dans cet article nous calculons les fonctions de Belyi pour une classe des cartes semireguli#res, correspondant # ce qu'on appelle les solides d'Archim#de. 1 Introduction The title of the present paper attempts to link together traditional and contemporary mathematics. The name of Archimedes represents tradition; that of Belyi, one of the most recent advances in Galois theory, known (even i...
On Equicut Graphs
 QUARTERLY JOURNAL OF MATHEMATICS OXFORD
, 2000
"... The size sz() of an `1graph = (V; E) is the minimum of n f =tf over all the possible `1embeddings f into n f dimensional hypercube with scale t f . The sum of distances between all the pairs of vertices of is at most sz()dv=2ebv=2c (v = jV j). The latter is an equality if and only if is equic ..."
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Cited by 7 (3 self)
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The size sz() of an `1graph = (V; E) is the minimum of n f =tf over all the possible `1embeddings f into n f dimensional hypercube with scale t f . The sum of distances between all the pairs of vertices of is at most sz()dv=2ebv=2c (v = jV j). The latter is an equality if and only if is equicut graph, that is, admits an `1 embedding f that for any 1 i n f satis es x2V f(x) i 2 fdv=2e; bv=2cg. Basic properties of equicut graphs are investigated. A construction of equicut graphs from `1graphs via a natural doubling construction is given. It generalizes several wellknown constructions of polytopes and distanceregular graphs. Finally, large families of examples, mostly related to polytopes and distanceregular graphs, are presented.
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...
THE SPECIAL CUTS OF 600CELL
, 708
"... Abstract. A polytope is called regular faced if every one of its facets is a regular polytope. The 4dimensional regularfaced polytopes were determined by [2, 3, 1]. The last class of such poytopes are the one obtained by removing a set of nonadjacent vertices of 600cell. Here we determine all su ..."
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Abstract. A polytope is called regular faced if every one of its facets is a regular polytope. The 4dimensional regularfaced polytopes were determined by [2, 3, 1]. The last class of such poytopes are the one obtained by removing a set of nonadjacent vertices of 600cell. Here we determine all such independent sets up to isomorphism and find 314248344 polytopes. 1.
JUNCTION OF NONCOMPOSITE POLYHEDRA
"... To my son’s coming of age Abstract. All 3dimensional convex regularhedra are found, i.e., the convex polyhedra having positive curvature of each vertex and such that every face is either a regular polygon or is composed of two regular polygons. The algorithm for constructing such solids is based o ..."
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To my son’s coming of age Abstract. All 3dimensional convex regularhedra are found, i.e., the convex polyhedra having positive curvature of each vertex and such that every face is either a regular polygon or is composed of two regular polygons. The algorithm for constructing such solids is based on calculation of the corresponding symmetry groups and gives a listing of all possible adjoins along entire faces of convex regularhedra that cannot be cut by any plane into smaller regularhedra.