Assembly of One-Patch Colloids Into Clusters Via Emulsion Droplet Evaporation
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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 -
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. -
Order in Space
Order in space A.K. van der Vegt VSSD Other textbooks on mathematics, published by VSSD Mathematical Systems Theory, G. Olsder and J.W. van der Woude, x + 210 pp. See http://www.vssd.nl/hlf/a003.htm Numerical Methods in Scientific Computing, J. van Kan, A. Segal and F. Vermolen, xii + 279 pp. See http://www.vssd.nl/hlf/a002.htm By the author of Order in space: From Plastics to Polymers, A.K. van der Vegt, 268 pp. See http://www.vssd.nl/hlf/m028.htm © VSSD Edition on the internet, 2001 Printed edition 2006 Published by: VSSD Leeghwaterstraat 42, 2628 CA Delft, The Netherlands tel. +31 15 27 82124, telefax +31 15 27 87585, e-mail: [email protected] internet: http://www.vssd.nl/hlf URL about this book: http://www.vssd.nl/hlf/a017.htm A collection of digital pictures and/or an electronic version can be made available for lecturers who adopt this book. Please send a request by e-mail to [email protected] All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photo-copying, recording, or otherwise, without the prior written permission of the publisher. NUR 918 ISBN-10: 90-71301-61-3 ISBN-13: 978-90-71301-61-2 Keywords: geometry 3 PREFACE This book deals with a very old subject. Many centuries ago some people were already fascinated by polyhedra, and they spent much time in investigating regular spatial structures. In the terminology of polyhedra we, therefore, meet the names of Archimedes, Pythagoras, Plato, Kepler etc. -
Convex Polytopes and Tilings with Few Flag Orbits
Convex Polytopes and Tilings with Few Flag Orbits by Nicholas Matteo B.A. in Mathematics, Miami University M.A. in Mathematics, Miami University A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the degree of Doctor of Philosophy April 14, 2015 Dissertation directed by Egon Schulte Professor of Mathematics Abstract of Dissertation The amount of symmetry possessed by a convex polytope, or a tiling by convex polytopes, is reflected by the number of orbits of its flags under the action of the Euclidean isometries preserving the polytope. The convex polytopes with only one flag orbit have been classified since the work of Schläfli in the 19th century. In this dissertation, convex polytopes with up to three flag orbits are classified. Two-orbit convex polytopes exist only in two or three dimensions, and the only ones whose combinatorial automorphism group is also two-orbit are the cuboctahedron, the icosidodecahedron, the rhombic dodecahedron, and the rhombic triacontahedron. Two-orbit face-to-face tilings by convex polytopes exist on E1, E2, and E3; the only ones which are also combinatorially two-orbit are the trihexagonal plane tiling, the rhombille plane tiling, the tetrahedral-octahedral honeycomb, and the rhombic dodecahedral honeycomb. Moreover, any combinatorially two-orbit convex polytope or tiling is isomorphic to one on the above list. Three-orbit convex polytopes exist in two through eight dimensions. There are infinitely many in three dimensions, including prisms over regular polygons, truncated Platonic solids, and their dual bipyramids and Kleetopes. There are infinitely many in four dimensions, comprising the rectified regular 4-polytopes, the p; p-duoprisms, the bitruncated 4-simplex, the bitruncated 24-cell, and their duals. -
[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. -
Regulating the Distortion Degree of the Square Antiprism Coordination
Electronic Supplementary Material (ESI) for CrystEngComm. This journal is © The Royal Society of Chemistry 2021 Electronic Supplementary Information (ESI) for CrystEngComm Regulating the Distortion Degree of the Square Antiprism Coordination Geometry in the Dy-Na Single Ion Magnets Hui-Sheng Wang,*a Ke Zhang,a Jia Wang,b Zhaobo Hu,b You Song,*b Zaichao Zhang,c and Zhi-Quan Pana aSchool of Chemistry and Environmental Engineering, Key Laboratory of Green Chemical Process of Ministry of Education, Key Laboratory of Novel Reactor and Green Chemical Technology of Hubei Province, Wuhan Institute of Technology, Wuhan 430205, P. R. China. bState Key Laboratory of Coordinate Chemistry, Nanjing National Laboratory of Microstructures, School of Chemistry and Chemical Engineering, Nanjing University, Nanjing 210023, P. R. China. cJiangsu Key Laboratory for the Chemistry of Low-dimensional Materials, School of Chemistry and Chemical Engineering, Huaiyin Normal University, P. R. China. *Email: [email protected] (H.-S. W.). *Email: [email protected] (Y. S.) -S1- Contents Table S1. Crystal data and structure refinement parameters for complexes 1-2. Table S2. Selected bond lengths (Å) and angles (°) for 1. Table S3. Selected bond lengths (Å) and angles (°) for 2. Table S4. The possible geometries of octa-coordination metal centers and Deviation parameters from each ideal polyhedron for Dy in complex 1. Table S5. The possible geometries of octa-coordination metal centers and Deviation parameters from each ideal polyhedron for Dy in complex 2. Table S6. The details for parameters involved in SAP geometry for 1 and 2. Table S7. Best fitted parameters obtained for the extended Debye model with ac susceptibility data from SQUID magnetometer of compound 1 under a 1000 Oe dc applied field. -
This Is a Set of Activities Using Both Isosceles and Equilateral Triangles
This is a set of activities using both Isosceles and Equilateral Triangles. 3D tasks Technical information This pack includes 25 Equilateral Triangles and 25 Isosceles Triangles. You will also need a small tube of Copydex. If you are new to making models with Copydex we suggest you start by watching www.atm.org.uk/Using-ATM-MATs Different polyhedra to make Kit A: 4 Equilateral Triangle MATs. Place one triangle on your desk and join the other three triangles to its three sides, symmetrically. Now add glue to one edge of each of the three added triangles so that all can be joined into a polyhedron This is your first polyhedron. Kit B: 3 Isosceles triangles and 1 equilateral triangle Kit C: 4 Isosceles triangles Kit D: 6 Equilateral Triangles Kit E: 6 Isosceles Triangles Kit F: 3 Equilateral and 3 Isosceles Triangles. (are there different possible polyhedra this time?) Kit G: 8 Equilateral Triangles. One possibility is a Regular Octahedron, but is there more than one Octahedron? Kit H: 4 Equilateral and 4 Isosceles Triangles (are there different possible polyhedra this time?) Kit I: 2 Equilateral Triangles and 6 Isosceles Triangles Kit J: Go back to the Regular Octahedron (from Kit G) and add a Regular Tetrahedron to one of its faces, then add another … until four have been added. Keep track (in a table) of how many faces and vertices the Regular Octahedron and the next four polyhedra have. Hint: if you spaced the added Tetrahedra equally around the Octahedron you will find a bigger version of a previously made solid. -
[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. -
Unit Origami: Star-Building on Deltahedra Heidi Burgiel Bridgewater State University, [email protected]
Bridgewater State University Virtual Commons - Bridgewater State University Mathematics Faculty Publications Mathematics Department 2015 Unit Origami: Star-Building on Deltahedra Heidi Burgiel Bridgewater State University, [email protected] Virtual Commons Citation Burgiel, Heidi (2015). Unit Origami: Star-Building on Deltahedra. In Mathematics Faculty Publications. Paper 47. Available at: http://vc.bridgew.edu/math_fac/47 This item is available as part of Virtual Commons, the open-access institutional repository of Bridgewater State University, Bridgewater, Massachusetts. Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture Unit Origami: Star-Building on Deltahedra Heidi Burgiel Dept. of Mathematics, Bridgewater State University 24 Park Avenue, Bridgewater, MA 02325, USA [email protected] Abstract This workshop provides instructions for folding the star-building unit – a modification of the Sonobe module for unit origami. Geometric questions naturally arise during this process, ranging in difficulty from middle school to graduate levels. Participants will learn to fold and assemble star-building units, then explore the structure of the eight strictly convex deltahedra. Introduction Many authors and educators have used origami, the art of paper folding, to provide concrete examples moti- vating mathematical problem solving. [3, 2] In unit origami, multiple sheets are folded and combined to form a whole; the Sonobe unit is a classic module in this art form. This workshop describes the construction and assembly of the star-building unit1, highlights a small selection of the many geometric questions motivated by this process, and introduces participants to the strictly convex deltahedra. Proficiency in geometry and origami is not required to enjoy this event. About the Unit Three Sonobe units interlock to form a pyramid with an equilateral triangular base as shown in Figure 1.