DOCSLIB.ORG

Bioengineered Metal Nanoparticles: Shape Control, Structure

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Bioengineered Metal Nanoparticles: Shape Control, Structure BIOENGINEERED METAL NANOPARTICLES: SHAPE CONTROL, STRUCTURE, AND CATALYTIC FUNCTIONALITY A Dissertation Presented to The Graduate Faculty of The University of Akron In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Hadi Ramezani-Dakhel May, 2015 BIOENGINEERED METAL NANOPARTICLES: SHAPE CONTROL, STRUCTURE, AND CATALYTIC FUNCTIONALITY Hadi Ramezani-Dakhel Dissertation Approved: Accepted: ______________________________ ______________________________ Advisor Department Chair Dr. Hendrik Heinz Dr. Robert Weiss ______________________________ ______________________________ Committee Member Dean of the College Dr. Alamgir Karim Dr. Eric J. Amis ______________________________ ______________________________ Committee Member Interim Dean of the Graduate School Dr. Kevin Cavicchi Dr. Rex D. Ramsier ______________________________ ______________________________ Committee Member Date Dr. Ali Dhinojwala ______________________________ Committee Member Dr. Jutta Luettmer-Strathman ii ABSTRACT Bioengineered colloidal noble metal nanoparticles have received much attention thanks to their superior functionality in variety of applications including catalysis, nanoelectronics, biosensors, and biomedicine. Size, shape, and surface features dictate the functionality while the underlying mechanisms of interactions at the interface of biomolecules and nanoscale metal substrates are not yet fully understood. Here, we carried out extensive parallel molecular dynamics simulations to explain how soft epitaxy determines facet specificity of several mutant peptides (S7: SSFPQPN as base sequence) on various facets of Pt crystals. Binding differentials between facets strongly depend on the presence of phenyl rings, including “lie-flat” attractive configurations on the {111} surface that match the hexagonal pattern of epitaxial sites and repulsive “stand-up” configurations on the {100} surface. We uncovered the molecular mechanism of specific recognition of Pt nanocubes and the evolution of cubic shapes from cuboctahedral seed crystals by combinatorially selected T7 peptide (TLTTLTN). Accordingly, T7 molecules are attracted to the edges of nanocubes due to multiple times higher mobility of water molecules compared to center portions of the cube, accompanied by a unique match of polarizable atoms in T7 to the square pattern of epitaxial sites. Synthesis, characterization, and atomistic simulations showed a preference of peptide T7 towards {100} facets over {111} facets at intermediate concentration, that explains a higher yield of cubes. Similar arguments explain control principles for the growth of twinned versus single crystals. The ratio of iii {111} and {100} facets differs 60/40 (for twinned crystals) versus 35/65 (for single crystals) and peptides with adequately balanced adsorption strength to these facets at different nucleation stages elucidates the mechanism of twin formation. We systematically analyzed the ratio of {h k l} facets on Pd nanoparticles of different size, identified different atom types, and calculated the relative reaction rate of nanoparticles in carbon-carbon Stille coupling reactions using the Boltzmann-averaged abstraction energies of individual atoms, in excellent agreement with measured turnover frequencies in experiment. Additionally, we developed a protocol using molecular dynamics simulations in conjunction with atomic pair distribution function (PDF) analysis of high-energy X-ray diffraction (HE-XRD) patterns and reverse Monte Carlo (RMC) simulations to obtain accurate 3D atomic-scale structure of nanocatalysts. The functionality of nanoparticles in the model systems of carbon-carbon coupling and allyl alcohol hydrogenation reactions were examined computationally and experimentally, leading to accurate predictions of relative reaction rates. The current research efforts provide specific guidance in the design of functional metal nanoparticles and introduces a new paradigm for the design of the next generation of catalytically active nanostructures with superior functionality. iv ACKNOWLEDGEMENTS The final stage of the formal educations for achieving a Ph.D. degree could be a long and complicated journey. This time period offers the opportunity to construct a key and open a door to a bright and progressive future. We need substantial assistance from the people around us to fulfil this goal. Here, I would like to thank all the people who encouraged me to work hard, be patient, distinct constructive comments from destructive inputs, and have a positive and professional attitude. I would like to thank my advisor Dr. Hendrik Heinz for his support and encouragement during the last four years. Dr. Heinz is a special advisor who never hesitates to elevate his students to the highest manageable levels. My second appreciation goes to our collaborators Dr. Yu Huang, and Dr. Lingyan Ruan from University of California in Los Angeles, and Dr. Marc R. Knecht and Dr. Nicholas Bedford from the University of Miami for a happy, efficient, and fruitful collaboration. Finally, I would like to express my sincere appreciation to my family for their unconditional love, support, patience, and kindness throughout my twenty three years of education. v TABLE OF CONTENTS Page LIST OF TABLES ............................................................................................................. ix LIST OF FIGURES ............................................................................................................ x CHAPTER I. INTRODUCTION ........................................................................................................... 1 1.1 Overview .............................................................................................................. 1 1.2 Shape-controlled synthesis of noble metal nanocrystals ...................................... 2 1.3 Stability, structure, and catalytic reactivity of palladium nanocrystals................ 4 II. LITERATURE REVIEW AND SIMULATION TECHNIQUES ................................. 7 2.1 Overview .............................................................................................................. 7 2.2 Metal nanoparticles: an overview on shape and size control and applications as catalyst ............................................................................................................................ 8 2.3 Overview of simulation techniques .................................................................... 34 2.4 CHARMM-INTERFACE force field ................................................................. 44 2.5 Summary of emerging simulation capabilities using CHARMM-INTERFACE force field ...................................................................................................................... 51 III. TOWARDS RATIONAL BIOMIMETIC DESIGN: PHENYL MOLECULAR SWITCH IDENTIFIED FOR PT {111} FACET RECOGNITION ................................. 53 3.1 Summary ............................................................................................................ 53 3.2 Overview ............................................................................................................ 53 3.3 Results and discussions ...................................................................................... 54 3.4 Conclusions ........................................................................................................ 68 vi IV. MOLECULAR MECHANISM OF SPECIFIC RECOGNITION OF CUBIC PLATINUM NANOCRYSTALS BY PEPTIDES AND OF THE CONCENTRATION- DEPENDENT FORMATION FROM SEED CRYSTALS ............................................. 70 4.1 Summary ............................................................................................................ 70 4.2 Overview ............................................................................................................ 71 4.3 Results and discussions ...................................................................................... 75 4.4 Conclusions ...................................................................................................... 101 V. STABILIZATION MECHANISM OF PEPTIDE MEDIATED PLATINUM SINGLE- TWINNED NANOCRYSTALS ..................................................................................... 104 5.1 Summary .......................................................................................................... 104 5.2 Overview .......................................................................................................... 105 5.3 Results and discussions .................................................................................... 107 5.4 Conclusions ...................................................................................................... 116 VI. STABILITY, SURFACE STRUCTURE, AND ATOMIC ABSTRACTION ENERGIES OF PALLADIUM NANOPARTICLES: TOWARD PREDICTION OF CATALYTIC FUNCTIONALITY ................................................................................. 117 6.1 Summary .......................................................................................................... 117 6.2 Results and Discussions ................................................................................... 118 6.3 Conclusions ...................................................................................................... 132 VII. ELUCIDATION OF PEPTIDE-DIRECTED PALLADIUM STRUCTURE FOR BIOLOGICALLY-TUNABLE CATALYSTS ............................................................... 133 7.1 Summary .........................................................................................................
Load more

Recommended publications
  • 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.
    [Show full text]
  • Arxiv:2105.14305V1 [Cs.CG] 29 May 2021
    Efficient Folding Algorithms for Regular Polyhedra ∗ Tonan Kamata1 Akira Kadoguchi2 Takashi Horiyama3 Ryuhei Uehara1 1 School of Information Science, Japan Advanced Institute of Science and Technology (JAIST), Ishikawa, Japan fkamata,[email protected] 2 Intelligent Vision & Image Systems (IVIS), Tokyo, Japan [email protected] 3 Faculty of Information Science and Technology, Hokkaido University, Hokkaido, Japan [email protected] Abstract We investigate the folding problem that asks if a polygon P can be folded to a polyhedron Q for given P and Q. Recently, an efficient algorithm for this problem has been developed when Q is a box. We extend this idea to regular polyhedra, also known as Platonic solids. The basic idea of our algorithms is common, which is called stamping. However, the computational complexities of them are different depending on their geometric properties. We developed four algorithms for the problem as follows. (1) An algorithm for a regular tetrahedron, which can be extended to a tetramonohedron. (2) An algorithm for a regular hexahedron (or a cube), which is much efficient than the previously known one. (3) An algorithm for a general deltahedron, which contains the cases that Q is a regular octahedron or a regular icosahedron. (4) An algorithm for a regular dodecahedron. Combining these algorithms, we can conclude that the folding problem can be solved pseudo-polynomial time when Q is a regular polyhedron and other related solid. Keywords: Computational origami folding problem pseudo-polynomial time algorithm regular poly- hedron (Platonic solids) stamping 1 Introduction In 1525, the German painter Albrecht D¨urerpublished his masterwork on geometry [5], whose title translates as \On Teaching Measurement with a Compass and Straightedge for lines, planes, and whole bodies." In the book, he presented each polyhedron by drawing a net, which is an unfolding of the surface of the polyhedron to a planar layout without overlapping by cutting along its edges.
    [Show full text]
  • 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
    [Show full text]
  • 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.
    [Show full text]
  • Solid Geometry Object Instruction Manual
    Solid Geometry Object Instruction Manual Swivel-Snaps® Solid Geometry Object Instruction Manual Fundamental Shapes Type Definition/Figure 2D Layout Edges all same length; same number edges at every vertex; all faces same Platonic Solids shape and size. Cube 6 12 Tetrahedron 4 6 Octahedron 8 12 Cannot be made with one 74 piece Dodecahedron Swivel-Snaps kit. Icosahedron 20 30 2 | P a g e Copyright Creative Toys LLC, all rights reserved Swivel-Snaps® Solid Geometry Object Instruction Manual Pyramids Triangular sides. Polygon base. Triangle Base For equilateral triangles, See Platonic solids Pyramid this is a tetrahedron. above Equilateral Square Base 3 1 8 Pyramid Pentagonal or 5 5 Base Pyramid (base not included) Archimedean Same as Platonic solids except two different face types. Solids Cuboctahedron 8 6 24 Rhombicub- 8 18 48 octahedron Two kits needed Snub 32 6 60 hexahedron Two kits needed Other Archimedean solids with pentagon faces cannot be made with Others the 74 piece Swivel-Snaps® kit. 3 | P a g e Copyright Creative Toys LLC, all rights reserved Swivel-Snaps® Solid Geometry Object Instruction Manual Convex All faces equilateral triangles; no adjacent faces in same plane. Deltahedra Regular tetrahedron, octahedron, These are also Platonic solids. See above. Deltahedra icosahedron Johnson Five shown below These are not Platonic solids. Deltahedra Triangular 6 9 Bipyramid Pentagonal 10 15 Bipyramid Snub 12 18 Disphenoid Triaugmented Triangular 14 21 Prism Gyroelongated Square 16 24 Bipyramid 4 | P a g e Copyright Creative Toys LLC, all rights reserved Swivel-Snaps® Solid Geometry Object Instruction Manual Composite Shapes Type Definition/Figure 3D Layout All faces equilateral triangles; two or more adjacent faces in the same plane.
    [Show full text]
  • Wythoffian Skeletal Polyhedra
    Wythoffian Skeletal Polyhedra by Abigail Williams B.S. in Mathematics, Bates College M.S. in Mathematics, Northeastern 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 Dedication I would like to dedicate this dissertation to my Meme. She has always been my loudest cheerleader and has supported me in all that I have done. Thank you, Meme. ii Abstract of Dissertation Wythoff's construction can be used to generate new polyhedra from the symmetry groups of the regular polyhedra. In this dissertation we examine all polyhedra that can be generated through this construction from the 48 regular polyhedra. We also examine when the construction produces uniform polyhedra and then discuss other methods for finding uniform polyhedra. iii Acknowledgements I would like to start by thanking Professor Schulte for all of the guidance he has provided me over the last few years. He has given me interesting articles to read, provided invaluable commentary on this thesis, had many helpful and insightful discussions with me about my work, and invited me to wonderful conferences. I truly cannot thank him enough for all of his help. I am also very thankful to my committee members for their time and attention. Additionally, I want to thank my family and friends who, for years, have supported me and pretended to care everytime I start talking about math. Finally, I want to thank my husband, Keith.
    [Show full text]
  • Construction and Analysis of Double Helix for Triangular Bipyramid and Pentangular Bipyramid
    Hindawi Computational and Mathematical Methods in Medicine Volume 2020, Article ID 5609593, 9 pages https://doi.org/10.1155/2020/5609593 Research Article Construction and Analysis of Double Helix for Triangular Bipyramid and Pentangular Bipyramid Tao Deng 1,2,3 1Key Laboratory of China’s Ethnic Languages and Information Technology of Ministry of Education, Northwest Minzu University, Lanzhou 730030, China 2Key Laboratory of Streaming Data Computing Technologies and Application, Northwest Minzu University, Lanzhou 730030, China 3School of Mathematics and Computer Science, Northwest Minzu University, Lanzhou 730030, China Correspondence should be addressed to Tao Deng; [email protected] Received 20 December 2019; Revised 8 February 2020; Accepted 14 February 2020; Published 14 May 2020 Academic Editor: Nadia A. Chuzhanova Copyright © 2020 Tao Deng. (is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. DNA cages can be joined together to make larger 3D nanostructures on which molecular electronic circuits and tiny containers are built for drug delivery. (e mathematical models for these promising nanomaterials play important roles in clarifying their assembly mechanism and understanding their structures. In this study, we propose a mathematical and computer method to construct permissible topological structures with double-helical edges for a triangular bipyramid and pentangular bipyramid. Furthermore, we remove the same topological links, without eliminating the nonrepeated ones for a triangular bipyramid and pentangular bipyramid. By analyzing characteristics of these unique links, some self-assembly and statistic rules are discussed. (is study may obtain some new insights into the DNA assembly from the viewpoint of mathematics, promoting the com- prehending and design efficiency of DNA polyhedra with required topological structures.
    [Show full text]
  • Supporting Information
    Supporting Information Triple-ringed Luminescent Heptanuclear Zn(II) Cluster for Efficient Ag(I) Ion Sensing Materials Qian-Jun Deng,*,1 Min Chen,1 Dong-Chu Chen,1 Zhong-Hong Zhu,2,* Hua-Hong Zou,2,* 1 School of Material Science and Energy Engineering, Foshan University, Foshan 528000, Guangdong, People’s Republic of China 2 State Key Laboratory for Chemistry and Molecular Engineering of Medicinal Resources, School of Chemistry & Pharmacy of Guangxi Normal University, Guilin 541004, P. R. China Supporting Tables Table S1 Selected bond lengths (Å) and angles (°) of 1. Table S2 SHAPE analysis in complex 1. Supporting Figures 1 2 Figure S1 The scheme with the structures of H2L and H2L . Figure S2 The coordination pattern diagram of Zn(II) ions in compound 1. Thermogravimetry of the compounds at a heating rate of 5 °C/min Figure S3 under N2 atmosphere for 1. The UV-visible absorption spectrum of the ligand HL1 and compound Figure S4 1 dissolved in DMF, respectively. Fluorescence spectra of ligand HL1 (a) and compound 1 (b) dissolved Figure S5 in DMF, respectively. The complex 1 and different metal ions were dissolved in an Figure S6 ultraviolet-visible absorption test in DMF. Crystals 2019, 9, 374; doi: 10.3390/cryst9070374 www.mdpi.com/journal/crystals Crystals 2019, 9, 374 2 of 7 Table S1. Selected bond lengths (Å) and angles (°) of 1. 1 Zn5—O15 1.914 (5) Zn2—N4 1.961 (6) Zn6—N12 2.001 (6) Zn5—O7 1.946 (6) Zn2—O4 2.047 (5) Zn6—N11 2.316 (7) Zn5—O10 2.190 (6) Zn2—N3 2.420 (6) Zn1—O3 1.924 (6) Zn5—N9 2.400 (7) Zn4—O8 2.196 (6) Zn1—O8 1.938
    [Show full text]
  • STATISTICAL TMEPTIODYLT^Ilcs and HOMOGENEOUS NUCLEATION of ATOMIC MICROCLUSTERS JOHN A. Mcinnes DEPARTMENT of PHYSICS BEDFORD CO
    STATISTICAL TMEPTIODYLT^ilCS AND HOMOGENEOUS NUCLEATION OF ATOMIC MICROCLUSTERS JOHN A. McINNES DEPARTMENT OF PHYSICS BEDFORD COLLEGE UNIVERSITY OF LONDON A THESIS PRESENTED IN FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY OF THE UNIVERSITY OF LONDON ProQuest Number: 10098962 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest. ProQuest 10098962 Published by ProQuest LLC(2016). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code. Microform Edition © ProQuest LLC. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 Acknowledgements This research has been .carried out under the supervision of Dr. M.R. Hoare, Reader in Theoretical Physics, at Bedford College, University of London, I owe Dr. Hoare a great debt of gratitude for his guidance, friendship and encouragement over the last few years• I am grateful to the advisory and operations staff both past and present at Bedford College Computer Unit, and especially to Dr. P. Pal for many hours of useful discussion and an extremely generous allocation of computing time. 1 would also like to thank Dr. J.A. Barker for advice on the computer generation of stereo images. Acknowledgement of the facilities provided by the University of London Computing Centre, the Atomic Energy Research Establishment, Harwell, and the Quantum Chemistry Program Exchange, University of Indiana, U*S.A., is made, and thanks given to the Science Research Council, London, who financed this research.
    [Show full text]
  • Generalized Bipyramids and Hyperbolic Volumes of Alternating $ K $-Uniform Tiling Links
    GENERALIZED BIPYRAMIDS AND HYPERBOLIC VOLUMES OF ALTERNATING k-UNIFORM TILING LINKS COLIN ADAMS, AARON CALDERON, AND NATHANIEL MAYER Abstract. We present explicit geometric decompositions of the hyperbolic complements of alternating k- 2 2 uniform tiling links, which are alternating links whose projection graphs are k-uniform tilings of S , E , 2 or H . A consequence of this decomposition is that the volumes of spherical alternating k-uniform tiling links are precisely twice the maximal volumes of the ideal Archimedean solids of the same combinatorial description, and the hyperbolic structures for the hyperbolic alternating tiling links come from the equilateral 2 realization of the k-uniform tiling on H . In the case of hyperbolic tiling links, we are led to consider links embedded in thickened surfaces Sg × I with genus g ≥ 2 and totally geodesic boundaries. We generalize the bipyramid construction of Adams to truncated bipyramids and use them to prove that the set of possible volume densities for all hyperbolic links in Sg × I, ranging over all g ≥ 2, is a dense subset of the interval [0; 2voct], where voct ≈ 3:66386 is the volume of the ideal regular octahedron. 1. Introduction For many links L embedded in a 3-manifold M, the complement M n L admits a unique hyperbolic struc- ture. In such cases we say that the link is hyperbolic, and denote the hyperbolic volume of the complement by vol(L), leaving the identity of M implicit. Computer programs like SnapPy allow for easy numerical computation of these volumes, but exact theoretical computations are rare. In [CKP15], Champanerkar, Kofman, and Purcell present an explicit geometric decomposition of the complement of the infinite square weave, the infinite alternating link whose projection graph is the square lattice W (see Figure 1) into regular ideal octahedra, one for each square face of the projection.
    [Show full text]
  • From Klein's Platonic Solids to Kepler's
    Dessin d'Enfants Examples due to Magot and Zvonkin Moduli Spaces From Klein's Platonic Solids to Kepler's Archimedean Solids: Elliptic Curves and Dessins d'Enfants Part II Edray Herber Goins Department of Mathematics Purdue University September 7, 2012 Number Theory Seminar From Klein's Platonic Solids to Kepler's Archimedean Solids Dessin d'Enfants Examples due to Magot and Zvonkin Moduli Spaces Abstract In 1884, Felix Klein wrote his influential book, \Lectures on the Icosahedron," where he explained how to express the roots of any quintic polynomial in terms of elliptic modular functions. His idea was to relate rotations of the icosahedron with the automorphism group of 5-torsion points on a suitable elliptic curve. In fact, he created a theory which related rotations of each of the five regular solids (the tetrahedron, cube, octahedron, icosahedron, and dodecahedron) with the automorphism groups of 3-, 4-, and 5-torsion points. Using modern language, the functions which relate the rotations with elliptic curves are Bely˘ımaps. In 1984, Alexander Grothendieck introduced the concept of a Dessin d'Enfant in order to understand Galois groups via such maps. We will complete a circle of ideas by reviewing Klein's theory with an emphasis on the octahedron; explaining how to realize the five regular solids (the Platonic solids) as well as the thirteen semi-regular solids (the Archimedean solids) as Dessins d'Enfant; and discussing how the corresponding Bely˘ımaps relate to moduli spaces of elliptic curves. Number Theory Seminar From Klein's
    [Show full text]
  • Clusters of Polyhedra in Spherical Confinement PNAS PLUS
    Clusters of polyhedra in spherical confinement PNAS PLUS Erin G. Teicha, Greg van Andersb, Daphne Klotsab,1, Julia Dshemuchadseb, and Sharon C. Glotzera,b,c,d,2 aApplied Physics Program, University of Michigan, Ann Arbor, MI 48109; bDepartment of Chemical Engineering, University of Michigan, Ann Arbor, MI 48109; cDepartment of Materials Science and Engineering, University of Michigan, Ann Arbor, MI 48109; and dBiointerfaces Institute, University of Michigan, Ann Arbor, MI 48109 Contributed by Sharon C. Glotzer, December 21, 2015 (sent for review December 1, 2015; reviewed by Randall D. Kamien and Jean Taylor) Dense particle packing in a confining volume remains a rich, largely have addressed 3D dense packings of anisotropic particles inside unexplored problem, despite applications in blood clotting, plas- a container. Of these, almost all pertain to packings of ellipsoids monics, industrial packaging and transport, colloidal molecule design, inside rectangular, spherical, or ellipsoidal containers (56–58), and information storage. Here, we report densest found clusters of and only one investigates packings of polyhedral particles inside a the Platonic solids in spherical confinement, for up to N = 60 constit- container (59). In that case, the authors used a numerical algo- uent polyhedral particles. We examine the interplay between aniso- rithm (generalizable to any number of dimensions) to generate tropic particle shape and isotropic 3D confinement. Densest clusters densest packings of N = ð1 − 20Þ cubes inside a sphere. exhibit a wide variety of symmetry point groups and form in up to In contrast, the bulk densest packing of anisotropic bodies has three layers at higher N. For many N values, icosahedra and do- been thoroughly investigated in 3D Euclidean space (60–65).
    [Show full text]