Polyhedra in Physics, Chemistry and Geometry
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Symmetry of Fulleroids Stanislav Jendrol’, František Kardoš
Symmetry of Fulleroids Stanislav Jendrol’, František Kardoš To cite this version: Stanislav Jendrol’, František Kardoš. Symmetry of Fulleroids. Klaus D. Sattler. Handbook of Nanophysics: Clusters and Fullerenes, CRC Press, pp.28-1 - 28-13, 2010, 9781420075557. hal- 00966781 HAL Id: hal-00966781 https://hal.archives-ouvertes.fr/hal-00966781 Submitted on 27 Mar 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Symmetry of Fulleroids Stanislav Jendrol’ and Frantiˇsek Kardoˇs Institute of Mathematics, Faculty of Science, P. J. Saf´arikˇ University, Koˇsice Jesenn´a5, 041 54 Koˇsice, Slovakia +421 908 175 621; +421 904 321 185 [email protected]; [email protected] Contents 1 Symmetry of Fulleroids 2 1.1 Convexpolyhedraandplanargraphs . .... 3 1.2 Polyhedral symmetries and graph automorphisms . ........ 4 1.3 Pointsymmetrygroups. 5 1.4 Localrestrictions ............................... .. 10 1.5 Symmetryoffullerenes . .. 11 1.6 Icosahedralfulleroids . .... 12 1.7 Subgroups of Ih ................................. 15 1.8 Fulleroids with multi-pentagonal faces . ......... 16 1.9 Fulleroids with octahedral, prismatic, or pyramidal symmetry ........ 19 1.10 (5, 7)-fulleroids .................................. 20 1 Chapter 1 Symmetry of Fulleroids One of the important features of the structure of molecules is the presence of symmetry elements. -
Prvních Deset Abelových Cen Za Matematiku
Prvních deset Abelových cen za matematiku The first ten Abel Prizes for mathematics [English summary] In: Michal Křížek (author); Lawrence Somer (author); Martin Markl (author); Oldřich Kowalski (author); Pavel Pudlák (author); Ivo Vrkoč (author); Hana Bílková (other): Prvních deset Abelových cen za matematiku. (English). Praha: Jednota českých matematiků a fyziků, 2013. pp. 87–88. Persistent URL: http://dml.cz/dmlcz/402234 Terms of use: © M. Křížek © L. Somer © M. Markl © O. Kowalski © P. Pudlák © I. Vrkoč Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://dml.cz Summary The First Ten Abel Prizes for Mathematics Michal Křížek, Lawrence Somer, Martin Markl, Oldřich Kowalski, Pavel Pudlák, Ivo Vrkoč The Abel Prize for mathematics is an international prize presented by the King of Norway for outstanding results in mathematics. It is named after the Norwegian mathematician Niels Henrik Abel (1802–1829) who found that there is no explicit formula for the roots of a general polynomial of degree five. The financial support of the Abel Prize is comparable with the Nobel Prize, i.e., about one million American dollars. Niels Henrik Abel (1802–1829) M. Křížek a kol.: Prvních deset Abelových cen za matematiku, JČMF, Praha, 2013 87 Already in 1899, another famous Norwegian mathematician Sophus Lie proposed to establish an Abel Prize, when he learned that Alfred Nobel would not include a prize in mathematics among his five proposed Nobel Prizes. -
1 Getting a Knighthood
Getting a knighthood (Bristol, June 1996) It's unreal, isn't it? I've always had a strong sense of the ridiculous, of the absurd, and this is working overtime now. Many of you must have been thinking, ‘Why Berry?’ Well, since I am the least knightly person I know, I have been asking the same question, and although I haven't come up with an answer I can offer a few thoughts. How does a scientist get this sort of national recognition? One way is to do something useful to the nation, that everyone agrees is important to the national well- being or even survival. Our home-grown example is of course Sir Charles Frank's war work. Or, one can sacrifice several years of one's life doing high-level scientific administration, not always received with gratitude by the scientific hoi polloi but important to the smooth running of the enterprise. Sir John Kingman is in this category. Another way – at least according to vulgar mythology – is to have the right family background. Well, my father drove a cab in London and my mother ruined her eyes as a dressmaker. Another way is to win the Nobel Prize or one of the other huge awards like the Wolf or King Faisal prizes, like Sir George Porter as he then was, or Sir Michael Atiyah. The principle here I suppose is, to those that have, more shall be given. I don't qualify on any of these grounds. The nearest I come is to have have won an fairly large number of smaller awards. -
Cadenza Document
2017/18 Whole Group Timetable Semester 1 Mathematics Module CMIS Room Class Event ID Code Module Title Event Type Day Start Finish Weeks Lecturer Room Name Code Capacity Size 3184417 ADM003 Yr 1 Basic Skills Test Test Thu 09:00 10:00 18-20 Tonks, A Dr Fielding Johnson South Wing FJ SW 485 185 Basement Peter Williams PW Lecture Theatre 3198490 ADM003 Yr 1 Basic Skills Test Test Thu 09:00 10:00 18-20 Leschke K Dr KE 323, 185 KE 101, ADR 227, MSB G62, KE 103 3203305 ADM003 Yr 1 Basic Skills Test Test Thu 09:00 10:00 18-20 Hall E J C Dr Attenborough Basement ATT LT1 204 185 Lecture Theatre 1 3216616 ADM005 Yr 2 Basic Skills Test Test Fri 10:00 11:00 18 Brilliantov N BEN 149 Professor F75b, MSB G62, KE 103, KE 323 2893091 MA1012 Calculus & Analysis 1 Lecture Mon 11:00 12:00 11-16, Tonks, A Dr Ken Edwards First Floor KE LT1 249 185 18-21 Lecture Theatre 1 2893092 MA1012 Calculus & Analysis 1 Lecture Tue 11:00 12:00 11-16, Tonks, A Dr Attenborough Basement ATT LT1 204 185 18-21 Lecture Theatre 1 2893129 MA1061 Probability Lecture Thu 10:00 11:00 11-16, Hall E J C Dr Ken Edwards First Floor KE LT1 249 185 18-21 Lecture Theatre 1 2893133 MA1061 Probability Lecture Fri 11:00 12:00 11-16, Hall E J C Dr Bennett Upper Ground Floor BEN LT1 244 185 18-21 Lecture Theatre 1 2902659 MA1104 ELEMENTS OF NUMBER Lecture Tue 13:00 14:00 11-16, Goedecke J Ken Edwards Ground Floor KE LT3 150 100 THEORY 18-21 Lecture Theatre 3 2902658 MA1104 ELEMENTS OF NUMBER Lecture Wed 10:00 11:00 11-16, Goedecke J Astley Clarke Ground Floor AC LT 110 100 THEORY 18-21 Lecture -
Systematics of Atomic Orbital Hybridization of Coordination Polyhedra: Role of F Orbitals
molecules Article Systematics of Atomic Orbital Hybridization of Coordination Polyhedra: Role of f Orbitals R. Bruce King Department of Chemistry, University of Georgia, Athens, GA 30602, USA; [email protected] Academic Editor: Vito Lippolis Received: 4 June 2020; Accepted: 29 June 2020; Published: 8 July 2020 Abstract: The combination of atomic orbitals to form hybrid orbitals of special symmetries can be related to the individual orbital polynomials. Using this approach, 8-orbital cubic hybridization can be shown to be sp3d3f requiring an f orbital, and 12-orbital hexagonal prismatic hybridization can be shown to be sp3d5f2g requiring a g orbital. The twists to convert a cube to a square antiprism and a hexagonal prism to a hexagonal antiprism eliminate the need for the highest nodality orbitals in the resulting hybrids. A trigonal twist of an Oh octahedron into a D3h trigonal prism can involve a gradual change of the pair of d orbitals in the corresponding sp3d2 hybrids. A similar trigonal twist of an Oh cuboctahedron into a D3h anticuboctahedron can likewise involve a gradual change in the three f orbitals in the corresponding sp3d5f3 hybrids. Keywords: coordination polyhedra; hybridization; atomic orbitals; f-block elements 1. Introduction In a series of papers in the 1990s, the author focused on the most favorable coordination polyhedra for sp3dn hybrids, such as those found in transition metal complexes. Such studies included an investigation of distortions from ideal symmetries in relatively symmetrical systems with molecular orbital degeneracies [1] In the ensuing quarter century, interest in actinide chemistry has generated an increasing interest in the involvement of f orbitals in coordination chemistry [2–7]. -
Chapter 1 – Symmetry of Molecules – P. 1
Chapter 1 – Symmetry of Molecules – p. 1 - 1. Symmetry of Molecules 1.1 Symmetry Elements · Symmetry operation: Operation that transforms a molecule to an equivalent position and orientation, i.e. after the operation every point of the molecule is coincident with an equivalent point. · Symmetry element: Geometrical entity (line, plane or point) which respect to which one or more symmetry operations can be carried out. In molecules there are only four types of symmetry elements or operations: · Mirror planes: reflection with respect to plane; notation: s · Center of inversion: inversion of all atom positions with respect to inversion center, notation i · Proper axis: Rotation by 2p/n with respect to the axis, notation Cn · Improper axis: Rotation by 2p/n with respect to the axis, followed by reflection with respect to plane, perpendicular to axis, notation Sn Formally, this classification can be further simplified by expressing the inversion i as an improper rotation S2 and the reflection s as an improper rotation S1. Thus, the only symmetry elements in molecules are Cn and Sn. Important: Successive execution of two symmetry operation corresponds to another symmetry operation of the molecule. In order to make this statement a general rule, we require one more symmetry operation, the identity E. (1.1: Symmetry elements in CH4, successive execution of symmetry operations) 1.2. Systematic classification by symmetry groups According to their inherent symmetry elements, molecules can be classified systematically in so called symmetry groups. We use the so-called Schönfliess notation to name the groups, Chapter 1 – Symmetry of Molecules – p. 2 - which is the usual notation for molecules. -
Can Every Face of a Polyhedron Have Many Sides ?
Can Every Face of a Polyhedron Have Many Sides ? Branko Grünbaum Dedicated to Joe Malkevitch, an old friend and colleague, who was always partial to polyhedra Abstract. The simple question of the title has many different answers, depending on the kinds of faces we are willing to consider, on the types of polyhedra we admit, and on the symmetries we require. Known results and open problems about this topic are presented. The main classes of objects considered here are the following, listed in increasing generality: Faces: convex n-gons, starshaped n-gons, simple n-gons –– for n ≥ 3. Polyhedra (in Euclidean 3-dimensional space): convex polyhedra, starshaped polyhedra, acoptic polyhedra, polyhedra with selfintersections. Symmetry properties of polyhedra P: Isohedron –– all faces of P in one orbit under the group of symmetries of P; monohedron –– all faces of P are mutually congru- ent; ekahedron –– all faces have of P the same number of sides (eka –– Sanskrit for "one"). If the number of sides is k, we shall use (k)-isohedron, (k)-monohedron, and (k)- ekahedron, as appropriate. We shall first describe the results that either can be found in the literature, or ob- tained by slight modifications of these. Then we shall show how two systematic ap- proaches can be used to obtain results that are better –– although in some cases less visu- ally attractive than the old ones. There are many possible combinations of these classes of faces, polyhedra and symmetries, but considerable reductions in their number are possible; we start with one of these, which is well known even if it is hard to give specific references for precisely the assertion of Theorem 1. -
Correspondences Between the Classical Electrostatic Thomson
Correspondences Between the Classical Electrostatic Thomson Problem and Atomic Electronic Structure Tim LaFave Jr.1,∗ Abstract Correspondences between the Thomson Problem and atomic electron shell-filling patterns are observed as systematic non-uniformities in the distribution of potential energy necessary to change configurations of N ≤ 100 electrons into discrete geometries of neighboring N − 1 systems. These non-uniformities yield electron energy pairs, intra-subshell pattern similarities with empirical ionization energy, and a salient pattern that coincides with size-normalized empirical ionization energies. Spatial symmetry limitations on discrete charges constrained to a spherical volume are conjectured as underlying physical mechanisms responsible for shell-filling patterns in atomic electronic structure and the Periodic Law. 1. Introduction present paper builds on this previous work by iden- tifying numerous correspondences between the elec- Quantum mechanical treatments of electrons in trostatic Thomson Problem of distributing equal spherical quantum dots, or “artificial atoms”, 1 rou- point charges on a unit sphere and atomic electronic tinely exhibit correspondences to atomic-like shell- structure. filling patterns by the appearance of abrupt jumps Despite the diminished stature of J.J. Thomson’s or dips in calculated energy or capacitance distribu- classical “plum-pudding” model 19 among more ac- tions as electrons are added to or removed from the curate atomic models, the Thomson Problem has 2–10 system. Additionally, shell-filling -
In Boron Chemistry
Bull. Mater. Sci., Vol. 22, No. 5, August 1999, pp. 863-867. © Indian Academy of Sciences. The ubiquitous icosahedral B12 in boron chemistry E D JEMMIS* and M M BALAKRISHNARAJAN School of Chemistry, University of Hyderabad, Hyderabad500 046. India Abstract. Though boranes exhibit a wide variety of polyhedral structures, all the three polymorpbs of elemental boron essentially contain icosahedral Bn units as the predominant building block in their unit cell. Theoretical and experimental studies on boranes show that the icosuhedral arrangement leads to most stable boranes and borane anions. This paper attempts to explain the phenomenal stability associated with the icosahedral Bn structure. Using fragment molecular orbital theory, the remarkable stability of BI2Ht2~ among closo boranes are explained. The preferential selection icosahedral B12 unit by elemental boron is explained by improvising a contrived B84 sub-unit of the ~-rhombohedron, the most stable polymorph. This also leads to a novel covalent way of stuffing fnilerenes with icosahedral symmetry. Keywords. Boron; polyhedral closo borane dianions; icosahedral B12; ~-rhombohedral polymorph; orbital compatibility. 1. Introduction Elemental boron exists in three different polymorphic forms (Wells 1979). The thermodynamically most stable The chemistry of boron formed a vast discipline by itself [t-rhombohedral form has 105 boron atoms in its unit cell. and helped to alter the concept of structure and bonding The next stable ~-rhombohedral form has 12 boron atoms. radically (Lipscomb 1963). Starting from the famous The meta stable ~-tetragonal form has fifty boron atoms three-centre two-electron bond, many of the structural in its unit cell. The icosahedral BI2 unit is the primary features exhibited by boron compounds (most signifi- building block in all of these polymorphic forms. -
Banknote Character Advisory Committee Minutes
BANKNOTE CHARACTER ADVISORY COMMITTEE (1) MINUTES Bank of England, Threadneedle Street 13.45 ‐ 15.00: 14 March 2018 Advisory Committee Members: Ben Broadbent (Chair), Sir David Cannadine, Victoria Cleland, Sandy Nairne, Baroness Lola Young Also present: Head of Banknote Resilience Programme Manager, Banknote Transformation Programme Senior Manager, Banknote Development Secretary to the Committee Minutes 1 Introduction and background 1.1 The Chair welcomed the Committee. He reminded members that their discussions should remain confidential. 1.2 Following the successful launch of the £5 and £10, the Bank was considering plans to launch a polymer £50. Governors and Court had approved a polymer £50 and the Banknote Character Advisory Committee had been re‐convened. 1.3 The Chair advised that the Chancellor’s Spring Statement the previous day had announced a call for evidence relating to the future of cash. The results of this exercise could impact the future of the £50 and it would be inappropriate for the Bank to press forward with a polymer £50 while the Government was considering the future denominational mix. 1.4 The closing date for receipt of evidence is 5 June and a decision might be expected around the Autumn budget. If HM Treasury decided to continue with the denomination and the Bank chose to develop a polymer £50, the Bank could presumptively make an announcement in early 2019 to avoid the public nomination process spanning Christmas 2018. 1 2 Equality considerations for selecting the field 2.1 The Bank reminded the Committee that equality considerations need to be consciously considered during the character selection process. -
Uniform Panoploid Tetracombs
Uniform Panoploid Tetracombs George Olshevsky TETRACOMB is a four-dimensional tessellation. In any tessellation, the honeycells, which are the n-dimensional polytopes that tessellate the space, Amust by definition adjoin precisely along their facets, that is, their ( n!1)- dimensional elements, so that each facet belongs to exactly two honeycells. In the case of tetracombs, the honeycells are four-dimensional polytopes, or polychora, and their facets are polyhedra. For a tessellation to be uniform, the honeycells must all be uniform polytopes, and the vertices must be transitive on the symmetry group of the tessellation. Loosely speaking, therefore, the vertices must be “surrounded all alike” by the honeycells that meet there. If a tessellation is such that every point of its space not on a boundary between honeycells lies in the interior of exactly one honeycell, then it is panoploid. If one or more points of the space not on a boundary between honeycells lie inside more than one honeycell, the tessellation is polyploid. Tessellations may also be constructed that have “holes,” that is, regions that lie inside none of the honeycells; such tessellations are called holeycombs. It is possible for a polyploid tessellation to also be a holeycomb, but not for a panoploid tessellation, which must fill the entire space exactly once. Polyploid tessellations are also called starcombs or star-tessellations. Holeycombs usually arise when (n!1)-dimensional tessellations are themselves permitted to be honeycells; these take up the otherwise free facets that bound the “holes,” so that all the facets continue to belong to two honeycells. In this essay, as per its title, we are concerned with just the uniform panoploid tetracombs. -
Assembly of One-Patch Colloids Into Clusters Via Emulsion Droplet Evaporation
materials Article Assembly of One-Patch Colloids into Clusters via Emulsion Droplet Evaporation Hai Pham Van 1,2, Andrea Fortini 3 and Matthias Schmidt 1,* 1 Theoretische Physik II, Physikalisches Institut, Universität Bayreuth, Universitätsstraße 30, D-95440 Bayreuth, Germany 2 Department of Physics, Hanoi National University of Education, 136 Xuanthuy, Hanoi 100000, Vietnam; [email protected] 3 Department of Physics, University of Surrey, Guildford GU2 7XH, UK; [email protected] * Correspondence: [email protected]; Tel.: +49-921-55-3313 Academic Editors: Andrei V. Petukhov and Gert Jan Vroege Received: 23 February 2017; Accepted: 27 March 2017; Published: 29 March 2017 Abstract: We study the cluster structures of one-patch colloidal particles generated by droplet evaporation using Monte Carlo simulations. The addition of anisotropic patch–patch interaction between the colloids produces different cluster configurations. We find a well-defined category of sphere packing structures that minimize the second moment of mass distribution when the attractive surface coverage of the colloids c is larger than 0.3. For c < 0.3, the uniqueness of the packing structures is lost, and several different isomers are found. A further decrease of c below 0.2 leads to formation of many isomeric structures with less dense packings. Our results could provide an explanation of the occurrence of uncommon cluster configurations in the literature observed experimentally through evaporation-driven assembly. Keywords: one-patch colloid; Janus colloid; cluster; emulsion-assisted assembly; Pickering effect 1. Introduction Packings of colloidal particles in regular structures are of great interest in colloid science. One particular class of such packings is formed by colloidal clusters which can be regarded as colloidal analogues to small molecules, i.e., “colloidal molecules” [1,2].