Problems at the Nexus of Geometry and Soft Matter: Rings, Ribbons and Shells
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The Volumes of a Compact Hyperbolic Antiprism
The volumes of a compact hyperbolic antiprism Vuong Huu Bao joint work with Nikolay Abrosimov Novosibirsk State University G2R2 August 6-18, Novosibirsk, 2018 Vuong Huu Bao (NSU) The volumes of a compact hyperbolic antiprism Aug. 6-18,2018 1/14 Introduction Calculating volumes of polyhedra is a classical problem, that has been well known since Euclid and remains relevant nowadays. This is partly due to the fact that the volume of a fundamental polyhedron is one of the main geometrical invariants for a 3-dimensional manifold. Every 3-manifold can be presented by a fundamental polyhedron. That means we can pair-wise identify the faces of some polyhedron to obtain a 3-manifold. Thus the volume of 3-manifold is the volume of prescribed fundamental polyhedron. Theorem (Thurston, Jørgensen) The volumes of hyperbolic 3-dimensional hyperbolic manifolds form a closed non-discrete set on the real line. This set is well ordered. There are only finitely many manifolds with a given volume. Vuong Huu Bao (NSU) The volumes of a compact hyperbolic antiprism Aug. 6-18,2018 2/14 Introduction 1835, Lobachevsky and 1982, Milnor computed the volume of an ideal hyperbolic tetrahedron in terms of Lobachevsky function. 1993, Vinberg computed the volume of hyperbolic tetrahedron with at least one vertex at infinity. 1907, Gaetano Sforza; 1999, Yano, Cho ; 2005 Murakami; 2005 Derevnin, Mednykh gave different formulae for general hyperbolic tetrahedron. 2009, N. Abrosimov, M. Godoy and A. Mednykh found the volumes of spherical octahedron with mmm or 2 m-symmetry. | 2013, N. Abrosimov and G. Baigonakova, found the volume of hyperbolic octahedron with mmm-symmetry. -
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]. -
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 -
Chains of Antiprisms
Chains of antiprisms Citation for published version (APA): Verhoeff, T., & Stoel, M. (2015). Chains of antiprisms. In Bridges Baltimore 2015 : Mathematics, Music, Art, Architecture, Culture, Baltimore, MD, USA, July 29 - August 1, 2015 (pp. 347-350). The Bridges Organization. Document status and date: Published: 01/01/2015 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. -
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]. -
Hwsolns5scan.Pdf
Math 462: Homework 5 Paul Hacking 3/27/10 (1) One way to describe a polyhedron is by cutting along some of the edges and folding it flat in the plane. The diagram obtained in this way is called a net. For example here is a net for the cube: Draw nets for the tetrahedron, octahedron, and dodecahedron. +~tfl\ h~~1l1/\ \(j [ NJte: T~ i) Mq~ u,,,,, cMi (O/r.(J 1 (2) Another way to describe a polyhedron is as follows: imagine that the faces of the polyhedron arc made of glass, look through one of the faces and draw the image you see of the remaining faces of the polyhedron. This image is called a Schlegel diagram. For example here is a Schlegel diagram for the octahedron. Note that the face you are looking through is not. drawn, hut the boundary of that face corresponds to the boundary of the diagram. Now draw a Schlegel diagram for the dodecahedron. (:~) (a) A pyramid is a polyhedron obtained from a. polygon with Rome number n of sides by joining every vertex of the polygon to a point lying above the plane of the polygon. The polygon is called the base of the pyramid and the additional vertex is called the apex, For example the Egyptian pyramids have base a square (so n = 4) and the tetrahedron is a pyramid with base an equilateral 2 I triangle (so n = 3). Compute the number of vertices, edges; and faces of a pyramid with base a polygon with n Rides. (b) A prism is a polyhedron obtained from a polygon (called the base of the prism) as follows: translate the polygon some distance in the direction normal to the plane of the polygon, and join the vertices of the original polygon to the corresponding vertices of the translated polygon. -
Hexagonal Antiprism Tetragonal Bipyramid Dodecahedron
Call List hexagonal antiprism tetragonal bipyramid dodecahedron hemisphere icosahedron cube triangular bipyramid sphere octahedron cone triangular prism pentagonal bipyramid torus cylinder squarebased pyramid octagonal prism cuboid hexagonal prism pentagonal prism tetrahedron cube octahedron square antiprism ellipsoid pentagonal antiprism spheroid Created using www.BingoCardPrinter.com B I N G O hexagonal triangular squarebased tetrahedron antiprism cube prism pyramid tetragonal triangular pentagonal octagonal cube bipyramid bipyramid bipyramid prism octahedron Free square dodecahedron sphere Space cuboid antiprism hexagonal hemisphere octahedron torus prism ellipsoid pentagonal pentagonal icosahedron cone cylinder prism antiprism Created using www.BingoCardPrinter.com B I N G O triangular pentagonal triangular hemisphere cube prism antiprism bipyramid pentagonal hexagonal tetragonal torus bipyramid prism bipyramid cone square Free hexagonal octagonal tetrahedron antiprism Space antiprism prism squarebased dodecahedron ellipsoid cylinder octahedron pyramid pentagonal icosahedron sphere prism cuboid spheroid Created using www.BingoCardPrinter.com B I N G O cube hexagonal triangular icosahedron octahedron prism torus prism octagonal square dodecahedron hemisphere spheroid prism antiprism Free pentagonal octahedron squarebased pyramid Space cube antiprism hexagonal pentagonal triangular cone antiprism cuboid bipyramid bipyramid tetragonal cylinder tetrahedron ellipsoid bipyramid sphere Created using www.BingoCardPrinter.com B I N G O -
[ENTRY POLYHEDRA] Authors: Oliver Knill: December 2000 Source: Translated Into This Format from Data Given In
ENTRY POLYHEDRA [ENTRY POLYHEDRA] Authors: Oliver Knill: December 2000 Source: Translated into this format from data given in http://netlib.bell-labs.com/netlib tetrahedron The [tetrahedron] is a polyhedron with 4 vertices and 4 faces. The dual polyhedron is called tetrahedron. cube The [cube] is a polyhedron with 8 vertices and 6 faces. The dual polyhedron is called octahedron. hexahedron The [hexahedron] is a polyhedron with 8 vertices and 6 faces. The dual polyhedron is called octahedron. octahedron The [octahedron] is a polyhedron with 6 vertices and 8 faces. The dual polyhedron is called cube. dodecahedron The [dodecahedron] is a polyhedron with 20 vertices and 12 faces. The dual polyhedron is called icosahedron. icosahedron The [icosahedron] is a polyhedron with 12 vertices and 20 faces. The dual polyhedron is called dodecahedron. small stellated dodecahedron The [small stellated dodecahedron] is a polyhedron with 12 vertices and 12 faces. The dual polyhedron is called great dodecahedron. great dodecahedron The [great dodecahedron] is a polyhedron with 12 vertices and 12 faces. The dual polyhedron is called small stellated dodecahedron. great stellated dodecahedron The [great stellated dodecahedron] is a polyhedron with 20 vertices and 12 faces. The dual polyhedron is called great icosahedron. great icosahedron The [great icosahedron] is a polyhedron with 12 vertices and 20 faces. The dual polyhedron is called great stellated dodecahedron. truncated tetrahedron The [truncated tetrahedron] is a polyhedron with 12 vertices and 8 faces. The dual polyhedron is called triakis tetrahedron. cuboctahedron The [cuboctahedron] is a polyhedron with 12 vertices and 14 faces. The dual polyhedron is called rhombic dodecahedron. -
The Scottish Bubble and the Irish Bubble a Maths Club Project the Elementary Mathematics in an Advanced Problem
The Scottish Bubble and the Irish Bubble A Maths Club Project The Elementary Mathematics in an Advanced Problem Overview The contents of this CD are intended for you, the leader of the maths club. The material is of 3 different kinds: history: the problems in their historical setting: how they arose, how people first tackled them. Many of you will be specialists in a science other than maths. I hope you find something to interest you personally among the references I have cited. I also hope you take this project as an opportunity to work with colleagues from across the curriculum. calculation: what students of school age can work out for themselves; something not for club-time but as a private challenge for the keen. This work is pitched at the 16-18 age range. However, elementary algebra, Pythagoras’ Theorem, trigonometric identities and their use in solving triangles is almost all the maths the students will need. experiment: what the students can do and make. This is the project core and can involve students from right across the secondary age range. With a weekly session of an hour and a half, the work will take a full term. As you see, these headings appear as possible bookmarks down the left side of the screen. Introduction When scientists turn to mathematicians for answers, they expect them to have ready-made solutions, something they can pull down off the shelf. In one area, packing, this has not been the case. The reason lies in the problems’ untidiness. To identify a packing as the best, one must exclude all others possible. -
The Maximally Entangled Symmetric State in Terms of the Geometric Measure
Home Search Collections Journals About Contact us My IOPscience The maximally entangled symmetric state in terms of the geometric measure This content has been downloaded from IOPscience. Please scroll down to see the full text. 2010 New J. Phys. 12 073025 (http://iopscience.iop.org/1367-2630/12/7/073025) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 130.63.180.147 This content was downloaded on 13/08/2014 at 08:38 Please note that terms and conditions apply. New Journal of Physics The open–access journal for physics The maximally entangled symmetric state in terms of the geometric measure Martin Aulbach1,2,3,6, Damian Markham1,4 and Mio Murao1,5 1 Department of Physics, Graduate School of Science, The University of Tokyo, Tokyo 113-0033, Japan 2 The School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UK 3 Department of Physics, University of Oxford, Clarendon Laboratory, Oxford OX1 3PU, UK 4 CNRS, LTCI, Telecom ParisTech, 37/39 rue Dareau, 75014 Paris, France 5 Institute for Nano Quantum Information Electronics, The University of Tokyo, Tokyo 113-0033, Japan E-mail: [email protected], [email protected] and [email protected] New Journal of Physics 12 (2010) 073025 (34pp) Received 22 April 2010 Published 22 July 2010 Online at http://www.njp.org/ doi:10.1088/1367-2630/12/7/073025 Abstract. The geometric measure of entanglement is investigated for permutation symmetric pure states of multipartite qubit systems, in particular the question of maximum entanglement. -
[Math.MG] 7 Oct 2008 Cat(F)Udrgatsh 1503/4-1
EXPERIMENTAL STUDY OF ENERGY-MINIMIZING POINT CONFIGURATIONS ON SPHERES BRANDON BALLINGER, GRIGORIY BLEKHERMAN, HENRY COHN, NOAH GIANSIRACUSA, ELIZABETH KELLY, AND ACHILL SCHURMANN¨ Abstract. In this paper we report on massive computer experiments aimed at finding spherical point configurations that minimize potential energy. We present experimental evidence for two new universal optima (consisting of 40 points in 10 dimensions and 64 points in 14 dimensions), as well as evidence that there are no others with at most 64 points. We also describe several other new polytopes, and we present new geometrical descriptions of some of the known universal optima. [T]he problem of finding the configurations of stable equilibrium for a number of equal particles acting on each other according to some law of force...is of great interest in connexion with the rela- tion between the properties of an element and its atomic weight. Unfortunately the equations which determine the stability of such a collection of particles increase so rapidly in complexity with the number of particles that a general mathematical investigation is scarcely possible. J. J. Thomson, 1897 Contents 1. Introduction 2 1.1. Experimental results 4 1.2. New universal optima 8 2. Methodology 9 2.1. Techniques 9 2.2. Example 14 3. Experimental phenomena 15 arXiv:math/0611451v3 [math.MG] 7 Oct 2008 3.1. Analysis of Gram matrices 15 3.2. Other small examples 18 3.3. 2n + 1 points in Rn 20 3.4. 2n + 2 points in Rn 21 3.5. 48 points in R4 22 3.6. Hopf structure 23 3.7. -
OPTIMA Mathematical Optimization Society Newsletter 100 MOS Chair’S Column
OPTIMA Mathematical Optimization Society Newsletter 100 MOS Chair’s Column May 15, 2016. In June 1980, the chair of the MPS Publica- As of March 31, 2016, the society has $794,610 in total assets, tions Committee, Michael Held, wrote “Thus the decision was with $25,711 restricted for the Fulkerson Prize and $19,257 re- made to establish a new Newsletter – OPTIMA.” You can find stricted for the Lagrange Prize. This is a substantial increase over the text of his article in Optima 1, available on the Web page our balance of $550,261 at the end of 2012. The MOS is a learned www.mathopt.org/?nav=optima_details together with full copies of society, with no professional staff. Our main expense is an adminis- all 100 issues of Optima. It is great fun to click through the old copies trative fee paid to SIAM, to maintain our membership list, provide – give it a try. And please join me in thanking the current and past email service, and handle our financial matters. Over the past three Editors of Optima, Donald Hearn, issues 1–55 (!), Karen Aardal, is- years, the SIAM fee amounted to a total of $125,728. This was more sues 56–65, Jens Clausen, issues 66–72 , Andrea Lodi, issues 73–84, than offset by our $155,966 royalty income over the three-year pe- Katya Scheinberg, issues 85–93, and Volker Kaibel, issues 94–100. riod, derived mainly from the publication of MPA, MPB, and MPC. Their work has, of course, been strongly supported by a long line of Thanks again to everyone for your support over the past three Co-Editors, starting with Achim Bachem and continuing through the years.