On topological form in structures

Bucknum, Michael J.; Castro, Eduardo A.

doi:10.1007/s10910-005-9002-8

Cited by 12 publications

(37 citation statements)

References 5 publications

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“…In this paper, the authors describe the enumeration of a network in the hexagonal symmetry class, lying in space group P-6m2 (#187), that possesses a rare quaternary stoichiometry of AB 3 C 3 D 3 , as is readily revealed by consideration of its Wells point symbol, cited below. Table 1 lists the fractional hexagonal crystallographic coordinates of the 10 atoms within The overall trigohexagonite network, shown in Figures 2 and 3, can thus be seen to have the relatively complex quaternary stoichiometry [2,5] reflected in the Wells point symbol given by (7 3 )(6 3 8 3 ) 3 (6.7 2 ) 3 (3.7 4 .8) 3 . This Wellsean point symbol [2] can readily be translated into its corresponding Schläfli symbol of (n, p) = (6.75, 3.6) by the method already outlined by Bucknum et al previously [5].…”

Section: Trigohexagonite Structural-typementioning

confidence: 99%

“…Table 1 lists the fractional hexagonal crystallographic coordinates of the 10 atoms within The overall trigohexagonite network, shown in Figures 2 and 3, can thus be seen to have the relatively complex quaternary stoichiometry [2,5] reflected in the Wells point symbol given by (7 3 )(6 3 8 3 ) 3 (6.7 2 ) 3 (3.7 4 .8) 3 . This Wellsean point symbol [2] can readily be translated into its corresponding Schläfli symbol of (n, p) = (6.75, 3.6) by the method already outlined by Bucknum et al previously [5]. Finally, there is an alternative scheme for identifying the topology of arbitrary networks with a so-called vertexsymbol notation.…”

Section: Trigohexagonite Structural-typementioning

confidence: 99%

“…monograph on novel networks of 1984 [2] as, (8 3 ) 2 (6 3 ) 6 (6 3 8 3 ) 6 . And this ternary Wells point symbol can therefore be translated into its corresponding Schläfli symbol [5] as (6 4/5 , 3.4285…. ).…”

Section: Figurementioning

confidence: 99%

“…It can be seen from Figure 5 that all circuits in the Waserite structure-type are 8-gons, giving the lattice a classification as Catalan [5]. The Wells point symbol for this lattice is easy to write down as 8 4…”

Section: Figurementioning

confidence: 99%

“…Building on the work of Wells in outlining the scope of the problem of the systematic enumeration of networks, Bucknum took the concept of the topology map in a Schläfli space (a space of n and p, the Schläfli symbols for networks) and extended it to include the 3D crystalline networks explicitly [4]. Bucknum showed in 2005 [5] how one could use the topological device that Wells had earlier invented to classify polyhedra, tessellations and networks [1,2], called the Wells point symbol, and translate this notation into an unambiguous and unique Schläfli symbol (n, p) for a structure, and thereby map that object in the Schläfli space described above. Such a topology map, in the approximation of the mapping out of the regular or Platonic structures, is indicated in Figure 1 below.…”

Section: Introductionmentioning

confidence: 99%

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Trigohexagonite: A Rare Quaternary Stoichiometry Network

et al. 2008

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A novel 3-,4-connected network is reported in this communication that is a rare example of a three-dimensional (3D) network with a quaternary stoichiometry, in which 4 distinct sites of bonding occur in the unit of pattern of the material. The net lies in the hexagonal space group P-6m2 (#187) and there are 10 vertices (or atoms) in the unit cell, including 4 sites of approximately trigonal planar coordination and 6 sites of approximately tetrahedral coordination. All vertices (atoms) in the network sit on special positions in the P-6m2 space group, giving the network high 6-fold symmetry axes parallel to the crystallographic c-axis. The structure has thus been named trigohexagonite in a loose analogy with its hexagonite crystalline homolog, sitting in the hexagonal space group P6/mmm (#191). The Wells point symbol for the trigohexagonite network is given by (7 3)(6 3 8 3) 3 (6.7 2) 3 (3.7 4 .8) 3 where this symbol indicates the quaternary stoichiometry of the network. The Wells point symbol also reveals that the only structural strain present in the network comes from the presence of the 3-gon, cyclopropane-like moieties in it, built on tetrahedral vertices. This structural motif of cyclopropane-like rings has precedent in organic chemistry and it adds character to the overall 6-fold symmetry of the trigohexagonite pattern. Also, the overall network contains rare trimethylenemethane-like clusters of 4 trigonal planar vertices (atoms), bonded together, that constitute the 3connected component of the network. Both C 10 and B 10 realizations are briefly described.

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Section: Trigohexagonite Structural-typementioning

confidence: 99%

Section: Trigohexagonite Structural-typementioning

confidence: 99%

Section: Figurementioning

confidence: 99%

Section: Figurementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Trigohexagonite: A Rare Quaternary Stoichiometry Network

et al. 2008

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Some Comments on the Topological Features of Some 3-,4-connected Networks and their Relationships with the Numbers e and π

Bucknum

Castro

2006

J Math Chem

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Chemical topology of crystalline matter and the transcendental numbers ϕ, e and π

Bucknum

Castro

2008

J Math Chem

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In this paper, we describe certain rational approximations to the transcendental mathematical constants φ, e and π, that arise out of considerations of both: (1) the Euler relation for the division of the sphere into vertices, V, faces, F, and edges, E, and: (2) its simple algebraic transformation into the so-called Schläfli relation, which is an equivalent mathematical statement for the polyhedra, in terms of parameters known as the polygonality, defined as n = 2E/F, and the connectivivty, defined as p = 2E/V. It is thus the transformation to the Schläfli relation from the Euler relation, in particular, that enables one to move from a simple heuristic mapping of the polyhedra in the space of V, F and E, into a corresponding heuristic mapping into Schläfli-space, the space circumscribed by the parameters of n and p. It is also true, that this latter transformation equation, the Schläfli relation, applies only directly to the polyhedra, again, with their corresponding Schläfli symbols (n, p), but as a bonus, there is a direct 1-to-1 mapping result for the polyhedra, that can be seen to also be extendable to the tessellations in 2dimensions, and the networks in 3-dimensions, in terms of coordinates in a 2-dimensional Cartesian grid, represented as the Schläfli symbols (n, p), as discussed above, which do not involve rigorous solutions to the Schläfli relation. For while one could never identify the triplet set of integers (V, F, E) for the tessellations and networks, that would fit as a rational solution within the Euler relation, it is in fact possible for one to identify the corresponding values of the ordered pair (n, p) for any tessellation or network. The identification of the Schläfli symbol (n, p) for the tessellations and networks emerges from the formulation of its so-called Well's point symbol, through the proper translation of that Well's point symbol into an equivalent and unambiguous Schläfli symbol (n, p) for a given tessellation or network, as has been shown by Bucknum et al. previously. What we report in this communication, are the computations of some, certain Schläfli symbols (n, p) for the so-called Waserite (also called platinate, Pt 3 O 4 , a 3-,4-connected cubic pattern), Moravia (A 3 B 8 , a 3-,8-connected cubic pattern) and Kentuckia (ABC 2 , a 4-,6-,8-connected tetragonal pattern) networks, and some topological descriptors of other relevant structures. It is thus seen, that the computations of the polygonality and connectivity indexes, n and p, that are found as a consequence of identifying the Schläfli symbols for these relatively simple networks, lead to simple and direct connections to certain rational approximations to the transcendental mathematical constants φ, e and π, that, to the author's knowledge, have not been identified previously. Such rational approximations lead to elementary and straightforward methods to estimate these mathematical constants to an accuracy of better than 99 parts in 100.

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On topological form in structures

Cited by 12 publications

References 5 publications

Trigohexagonite: A Rare Quaternary Stoichiometry Network

Trigohexagonite: A Rare Quaternary Stoichiometry Network

Some Comments on the Topological Features of Some 3-,4-connected Networks and their Relationships with the Numbers e and π

Chemical topology of crystalline matter and the transcendental numbers ϕ, e and π

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