scholarly journals Arithmetic proof of the multiplicity-weighted Euler characteristic for symmetrically arranged space-filling polyhedra

2021 ◽  
Vol 77 (2) ◽  
pp. 126-129
Author(s):  
Bartosz Naskręcki ◽  
Zbigniew Dauter ◽  
Mariusz Jaskolski
Keyword(s):  
Euler Characteristic ◽  
Three Dimensional ◽  
Unit Cell ◽  
Asymmetric Unit ◽  
Space Filling ◽  
Simple Argument ◽  
General Validity ◽  
A Value ◽  
Euler’S Formula

The puzzling observation that the famous Euler's formula for three-dimensional polyhedra V − E + F = 2 or Euler characteristic χ = V − E + F − I = 1 (where V, E, F are the numbers of the bounding vertices, edges and faces, respectively, and I = 1 counts the single solid itself) when applied to space-filling solids, such as crystallographic asymmetric units or Dirichlet domains, are modified in such a way that they sum up to a value one unit smaller (i.e. to 1 or 0, respectively) is herewith given general validity. The proof provided in this paper for the modified Euler characteristic, χm = V m − E m + F m − I m = 0, is divided into two parts. First, it is demonstrated for translational lattices by using a simple argument based on parity groups of integer-indexed elements of the lattice. Next, Whitehead's theorem, about the invariance of the Euler characteristic, is used to extend the argument from the unit cell to its asymmetric unit components.

scholarly journals The Euler characteristic as a basis for teaching topology concepts to crystallographers

2022 ◽  
Vol 55 (1) ◽  
Author(s):  
Bartosz Naskręcki ◽  
Mariusz Jaskolski ◽  
Zbigniew Dauter
Keyword(s):  
Euler Characteristic ◽  
Unit Cell ◽  
Fundamental Concept ◽  
Closed Space ◽  
Asymmetric Unit ◽  
Crystal Lattices ◽  
Space Groups ◽  
Topological Interpretation ◽  
Infinite Crystal ◽  
Finite Space

The simple Euler polyhedral formula, expressed as an alternating count of the bounding faces, edges and vertices of any polyhedron, V − E + F = 2, is a fundamental concept in several branches of mathematics. Obviously, it is important in geometry, but it is also well known in topology, where a similar telescoping sum is known as the Euler characteristic χ of any finite space. The value of χ can also be computed for the unit polyhedra (such as the unit cell, the asymmetric unit or Dirichlet domain) which build, in a symmetric fashion, the infinite crystal lattices in all space groups. In this application χ has a modified form (χm) and value because the addends have to be weighted according to their symmetry. Although derived in geometry (in fact in crystallography), χm has an elegant topological interpretation through the concept of orbifolds. Alternatively, χm can be illustrated using the theorems of Harriot and Descartes, which predate the discovery made by Euler. Those historical theorems, which focus on angular defects of polyhedra, are beautifully expressed in the formula of de Gua de Malves. In a still more general interpretation, the theorem of Gauss–Bonnet links the Euler characteristic with the general curvature of any closed space. This article presents an overview of these interesting aspects of mathematics with Euler's formula as the leitmotif. Finally, a game is designed, allowing readers to absorb the concept of the Euler characteristic in an entertaining way.

scholarly journals Crystal structure of 4,4′-diethynylbiphenyl

2015 ◽  
Vol 71 (7) ◽  
pp. 816-820
Author(s):  
Tei Tagg ◽  
C. John McAdam ◽  
Brian H. Robinson ◽  
Jim Simpson
Keyword(s):  
Crystal Structure ◽  
Network Structure ◽  
Title Compound ◽  
Triple Bond ◽  
Three Dimensional ◽  
Unit Cell ◽  
Asymmetric Unit ◽  
Compound C ◽  
Dimensional Network ◽  
Benzene Rings

The title compound, C16H10, crystallizes with four unique molecules, designated 1–4, in the asymmetric unit of the monoclinic unit cell. None of the molecules is planar, with the benzene rings of molecules 1–4 inclined to one another at angles of 42.41 (4), 24.07 (6), 42.59 (4) and 46.88 (4)°, respectively. In the crystal, weak C—H...π(ring) interactions, augmented by even weaker C[triple-bond]C—H...π(alkyne) contacts, generate a three-dimensional network structure with interlinked columns of molecules formed along thec-axis direction.

scholarly journals Multiplicity-weighted Euler's formula for symmetrically arranged space-filling polyhedra

2020 ◽  
Vol 76 (5) ◽  
pp. 580-583
Author(s):  
Zbigniew Dauter ◽  
Mariusz Jaskolski
Keyword(s):  
Three Dimensional ◽  
Group Symmetry ◽  
General Character ◽  
Two Dimensional ◽  
Asymmetric Unit ◽  
Space Filling ◽  
Space Group Symmetry ◽  
Space Group ◽  
Space Groups ◽  
Modified Formula

The famous Euler's rule for three-dimensional polyhedra, F − E + V = 2 (F, E and V are the numbers of faces, edges and vertices, respectively), when extended to many tested cases of space-filling polyhedra such as the asymmetric unit (ASU), takes the form Fn − En + Vn = 1, where Fn, En and Vn enumerate the corresponding elements, normalized by their multiplicity, i.e. by the number of times they are repeated by the space-group symmetry. This modified formula holds for the ASUs of all 230 space groups and 17 two-dimensional planar groups as specified in the International Tables for Crystallography, and for a number of tested Dirichlet domains, suggesting that it may have a general character. The modification of the formula stems from the fact that in a symmetrical space-filling arrangement the polyhedra (such as the ASU) have incomplete bounding elements (faces, edges, vertices), since they are shared (in various degrees) with the space-filling neighbors.

scholarly journals Crystal structure of a new monoclinic polymorph ofN-(4-methylphenyl)-3-nitropyridin-2-amine

2014 ◽  
Vol 70 (8) ◽  
pp. 58-61
Author(s):  
Aina Mardia Akhmad Aznan ◽  
Zanariah Abdullah ◽  
Vannajan Sanghiran Lee ◽  
Edward R. T. Tiekink
Keyword(s):  
Three Dimensional ◽  
Basis Set ◽  
Dihedral Angles ◽  
Asymmetric Unit ◽  
Acta Cryst ◽  
Common Features ◽  
Compound C ◽  
B3lyp Level ◽  
The Common ◽  
Independent Molecules

The title compound, C12H11N3O2, is a second monoclinic polymorph (P21, withZ′ = 4) of the previously reported monoclinic (P21/c, withZ′ = 2) form [Akhmad Aznanet al.(2010).Acta Cryst.E66, o2400]. Four independent molecules comprise the asymmetric unit, which have the common features of asyndisposition of the pyridine N atom and the toluene ring, and an intramolecular amine–nitro N—H...O hydrogen bond. The differences between molecules relate to the dihedral angles between the rings which range from 2.92 (19) to 26.24 (19)°. The geometry-optimized structure [B3LYP level of theory and 6–311 g+(d,p) basis set] has the same features except that the entire molecule is planar. In the crystal, the three-dimensional architecture is consolidated by a combination of C—H...O, C—H...π, nitro-N—O...π and π–π interactions [inter-centroid distances = 3.649 (2)–3.916 (2) Å].

scholarly journals Crystal structures of {[Cu(Lpn)2][Fe(CN)5(NO)]·H2O}nand {[Cu(Lpn)2]3[Cr(CN)6]2·5H2O}n[where Lpn = (R)-propane-1,2-diamine]: two heterometallic chiral cyanide-bridged coordination polymers

2015 ◽  
Vol 71 (4) ◽  
pp. 392-397
Author(s):  
Olha Sereda ◽  
Helen Stoeckli-Evans
Keyword(s):  
Hydrogen Bonds ◽  
Coordination Polymers ◽  
Three Dimensional ◽  
Water Molecules ◽  
Two Dimensional ◽  
Asymmetric Unit ◽  
Compound I ◽  
Distorted Octahedral ◽  
Coordination Spheres ◽  
Compound Ii

The title compounds,catena-poly[[[bis[(R)-propane-1,2-diamine-κ2N,N′]copper(II)]-μ-cyanido-κ2N:C-[tris(cyanido-κC)(nitroso-κN)iron(III)]-μ-cyanido-κ2C:N] monohydrate], {[Cu(Lpn)2][Fe(CN)5(NO)]·H2O}n, (I), and poly[[hexa-μ-cyanido-κ12C:N-hexacyanido-κ6C-hexakis[(R)-propane-1,2-diamine-κ2N,N′]dichromium(III)tricopper(II)] pentahydrate], {[Cu(Lpn)2]3[Cr(CN)6]2·5H2O}n, (II) [where Lpn = (R)-propane-1,2-diamine, C3H10N2], are new chiral cyanide-bridged bimetallic coordination polymers. The asymmetric unit of compound (I) is composed of two independent cation–anion units of {[Cu(Lpn)2][Fe(CN)5)(NO)]} and two water molecules. The FeIIIatoms have distorted octahedral geometries, while the CuIIatoms can be considered to be pentacoordinate. In the crystal, however, the units align to form zigzag cyanide-bridged chains propagating along [101]. Hence, the CuIIatoms have distorted octahedral coordination spheres with extremely long semicoordination Cu—N(cyanido) bridging bonds. The chains are linked by O—H...N and N—H...N hydrogen bonds, forming two-dimensional networks parallel to (010), and the networks are linkedviaN—H...O and N—H...N hydrogen bonds, forming a three-dimensional framework. Compound (II) is a two-dimensional cyanide-bridged coordination polymer. The asymmetric unit is composed of two chiral {[Cu(Lpn)2][Cr(CN)6]}−anions bridged by a chiral [Cu(Lpn)2]2+cation and five water molecules of crystallization. Both the CrIIIatoms and the central CuIIatom have distorted octahedral geometries. The coordination spheres of the outer CuIIatoms of the asymmetric unit can be considered to be pentacoordinate. In the crystal, these units are bridged by long semicoordination Cu—N(cyanide) bridging bonds forming a two-dimensional network, hence these CuIIatoms now have distorted octahedral geometries. The networks, which lie parallel to (10-1), are linkedviaO—H...O, O—H...N, N—H...O and N—H...N hydrogen bonds involving all five non-coordinating water molecules, the cyanide N atoms and the NH2groups of the Lpn ligands, forming a three-dimensional framework.

A two-dimensional CdII coordination polymer with 2,2′-(disulfanediyl)dibenzoate and 1,10-phenanthroline ligands

2014 ◽  
Vol 70 (5) ◽  
pp. 517-521
Author(s):  
Yu-Xiu Jin ◽  
Fang Yang ◽  
Li-Min Yuan ◽  
Chao-Guo Yan ◽  
Wen-Long Liu
Keyword(s):  
Coordination Polymer ◽  
Three Dimensional ◽  
Phen Ligand ◽  
Supramolecular Architecture ◽  
Two Dimensional ◽  
Asymmetric Unit ◽  
Π Interactions ◽  
Carboxylate Groups ◽  
Phenanthroline Ligand ◽  
Phenanthroline Ligands

In poly[[μ3-2,2′-(disulfanediyl)dibenzoato-κ5 O:O,O′:O′′,O′′′](1,10-phenanthroline-κ2 N,N′)cadmium(II)], [Cd(C14H8O4S2)(C12H8N2)] n , the asymmetric unit contains one CdII cation, one 2,2′-(disulfanediyl)dibenzoate anion (denoted dtdb2−) and one 1,10-phenanthroline ligand (denoted phen). Each CdII centre is seven-coordinated by five O atoms of bridging/chelating carboxylate groups from three dtdb2− ligands and by two N atoms from one phen ligand, forming a distorted pentagonal–bipyramidal geometry. The CdII cations are bridged by dtdb2− anions to give a two-dimensional (4,4) layer. The layers are stacked to generate a three-dimensional supramolecular architecture via a combination of aromatic C—H...π and π–π interactions. The thermogravimetric and luminescence properties of this compound were also investigated.

scholarly journals 2-Amino-3-carboxypyridinium perchlorate

2012 ◽  
Vol 68 (6) ◽  
pp. o1601-o1602 ◽  
Author(s):  
Fadila Berrah ◽  
Sofiane Bouacida ◽  
Hayet Anana ◽  
Thierry Roisnel
Keyword(s):  
Hydrogen Bonds ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Asymmetric Unit ◽  
Dimensional Network

The asymmetric unit includes two crystallographically independent equivalents of the title salt, C6H7N2O2 +·ClO4 −. The cations and anions form separate layers alternating along the c axis, which are linked by N—H...O, O—H...O and C—H...O hydrogen bonds into a two-dimensional network parallel to (100). Further C—H...O contacts connect these layers, forming a three-dimensional network, in which R 4 4(20) rings and C 2 2(11) infinite chains can be identified.

scholarly journals 1-(4-Hydroxyphenyl)piperazine-1,4-diium tetrachloridocobalt(II) monohydrate

2014 ◽  
Vol 70 (2) ◽  
pp. m75-m75 ◽  
Author(s):  
Marwa Mghandef ◽  
Habib Boughzala
Keyword(s):  
Hydrogen Bonds ◽  
Water Molecule ◽  
Three Dimensional ◽  
Organic Cations ◽  
Water Molecules ◽  
Asymmetric Unit ◽  
Hybrid Compound ◽  
Organic Hybrid ◽  
Compound C ◽  
Dimensional Network

The asymmetric unit of the title inorganic–organic hybrid compound, (C10H16N2O)[CoCl4]·H2O, consists of a tetrahedral [CoCl4]2−anion, together with a [C10H18N2O]2+cation and a water molecule. Crystal cohesion is achieved through N—H...Cl, O—H...Cl and N—H...O hydrogen bonds between organic cations, inorganic anions and the water molecules, building up a three-dimensional network.

Iodo-nitroarenesulfonamides: interplay of hard and soft hydrogen bonds, I...O interactions and aromatic π...π stacking interactions

2001 ◽  
Vol 58 (1) ◽  
pp. 94-108 ◽  
Author(s):  
Craig J. Kelly ◽  
Janet M. S. Skakle ◽  
James L. Wardell ◽  
Solange M. S. V. Wardell ◽  
John N. Low ◽  
...  
Keyword(s):  
Hydrogen Bond ◽  
Hydrogen Bonds ◽  
Three Dimensional ◽  
Stacking Interactions ◽  
Asymmetric Unit ◽  
Space Group ◽  
Rotation Axes ◽  
Π Stacking ◽  
Structural Database ◽  
A Chain

Molecules of N-(4′-iodophenylsulfonyl)-4-nitroaniline, 4-O2NC6H4NHSO2C6H4I-4′ (1), are linked by three-centre I...O2N interactions into chains and these chains are linked into a three-dimensional framework by C—H...O hydrogen bonds. In the isomeric N-(4′-nitrophenylsulfonyl)-4-iodoaniline, 4-IC6H4NHSO2C6H4NO2-4′ (2), the chains generated by the I...O2N interactions are again linked into a three-dimensional framework by C—H...O hydrogen bonds. Molecules of N,N-bis(3′-nitrophenylsulfonyl)-4-iodoaniline, 4-IC6H4N(SO2C6H4NO2-3′)2 (3), lie across twofold rotation axes in space group C2/c and they are linked into chains by paired I...O=S interactions: these chains are linked into sheets by a C—H...O hydrogen bond, and the sheets are linked into a three-dimensional framework by aromatic π...π stacking interactions. In N-(4′-iodophenylsulfonyl)-3-nitroaniline, 3-O2NC6H4NHSO2C6H4I-4′ (4), there are R^2_2(8) rings formed by hard N—H...O=S hydrogen bonds and R^2_2(24) rings formed by two-centre I...nitro interactions, which together generate a chain of fused rings: the combination of a C—H...O hydrogen bond and aromatic π...π stacking interactions links the chains into sheets. Molecules of N-(4′-iodophenylsulfonyl)-4-methyl-2-nitroaniline, 4-CH3-2-O2NC6H3NHSO2C6H4I-4′ (5), are linked by N—H...O=S and C—H...O(nitro) hydrogen bonds into a chain containing alternating R^2_2(8) and R^2_2(10) rings, but there are no I...O interactions of either type. There are two molecules in the asymmetric unit of N-(4′-iodophenylsulfonyl)-2-nitroaniline, 2-O2NC6H4NHSO2C6H4I-4′ (6), and the combination of an I...O=S interaction and a hard N—H...O(nitro) hydrogen bond links the two types of molecule to form a cyclic, centrosymmetric four-component aggregate. C—H...O hydrogen bonds link these four-molecule aggregates to form a molecular ladder. Comparisons are made with structures retrieved from the Cambridge Structural Database.

Bending of an Infinite Beam on an Elastic Foundation

1937 ◽  
Vol 4 (1) ◽  
pp. A1-A7 ◽  
Author(s):  
M. A. Biot
Keyword(s):  
Elastic Foundation ◽  
Poisson Ratio ◽  
Elementary Theory ◽  
Three Dimensional ◽  
Bending Moment ◽  
Two Dimensional ◽  
Concentrated Load ◽  
Elastic Continuum ◽  
Infinite Beam ◽  
A Value

Abstract The elementary theory of the bending of a beam on an elastic foundation is based on the assumption that the beam is resting on a continuously distributed set of springs the stiffness of which is defined by a “modulus of the foundation” k. Very seldom, however, does it happen that the foundation is actually constituted this way. Generally, the foundation is an elastic continuum characterized by two elastic constants, a modulus of elasticity E, and a Poisson ratio ν. The problem of the bending of a beam resting on such a foundation has been approached already by various authors. The author attempts to give in this paper a more exact solution of one aspect of this problem, i.e., the case of an infinite beam under a concentrated load. A notable difference exists between the results obtained from the assumptions of a two-dimensional foundation and of a three-dimensional foundation. Bending-moment and deflection curves for the two-dimensional case are shown in Figs. 4 and 5. A value of the modulus k is given for both cases by which the elementary theory can be used and leads to results which are fairly acceptable. These values depend on the stiffness of the beam and on the elasticity of the foundation.

Sign in / Sign up

Export Citation Format

Share Document