Cannonballs and Honeycombs, Volume 47, Number 4
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Kepler and Hales: Conjectures and Proofs Dreams and Their
KEPLER AND HALES: CONJECTURES AND PROOFS DREAMS AND THEIR REALIZATION Josef Urban Czech Technical University in Prague Pittsburgh, June 21, 2018 1 / 31 This talk is an experiment Normally I give “serious” talks – I am considered crazy enough But this is Tom’s Birthday Party, so we are allowed to have some fun: This could be in principle a talk about our work on automation and AI for reasoning and formalization, and Tom’s great role in these areas. But the motivations and allusions go back to alchemistic Prague of 1600s and the (un)scientific pursuits of then versions of "singularity", provoking comparisons with our today’s funny attempts at building AI and reasoning systems for solving the great questions of Math, the Universe and Everything. I wonder where this will all take us. Part 1 (Proofs?): Learning automated theorem proving on top of Flyspeck and other corpora Learning formalization (autoformalization) on top of them Part 2 (Conjectures?): How did we get here? What were Kepler & Co trying to do in 1600s? What are we trying to do today? 2 / 31 The Flyspeck project – A Large Proof Corpus Kepler conjecture (1611): The most compact way of stacking balls of the same size in space is a pyramid. V = p 74% 18 Proved by Hales & Ferguson in 1998, 300-page proof + computations Big: Annals of Mathematics gave up reviewing after 4 years Formal proof finished in 2014 20000 lemmas in geometry, analysis, graph theory All of it at https://code.google.com/p/flyspeck/ All of it computer-understandable and verified in HOL Light: -
A Computer Verification of the Kepler Conjecture
ICM 2002 Vol. III 1–3 · · A Computer Verification of the Kepler Conjecture Thomas C. Hales∗ Abstract The Kepler conjecture asserts that the density of a packing of congruent balls in three dimensions is never greater than π/√18. A computer assisted verification confirmed this conjecture in 1998. This article gives a historical introduction to the problem. It describes the procedure that converts this problem into an optimization problem in a finite number of variables and the strategies used to solve this optimization problem. 2000 Mathematics Subject Classification: 52C17. Keywords and Phrases: Sphere packings, Kepler conjecture, Discrete ge- ometry. 1. Historical introduction The Kepler conjecture asserts that the density of a packing of congruent balls in three dimensions is never greater than π/√18 0.74048 .... This is the oldest problem in discrete geometry and is an important≈ part of Hilbert’s 18th problem. An example of a packing achieving this density is the face-centered cubic packing (Figure 1). A packing of balls is an arrangement of nonoverlapping balls of radius 1 in Euclidean space. Each ball is determined by its center, so equivalently it is a arXiv:math/0305012v1 [math.MG] 1 May 2003 collection of points in Euclidean space separated by distances of at least 2. The density of a packing is defined as the lim sup of the densities of the partial packings formed by the balls inside a ball with fixed center of radius R. (By taking the lim sup, rather than lim inf as the density, we prove the Kepler conjecture in the strongest possible sense.) Defined as a limit, the density is insensitive to changes in the packing in any bounded region. -
The Sphere Packing Problem in Dimension 8 Arxiv:1603.04246V2
The sphere packing problem in dimension 8 Maryna S. Viazovska April 5, 2017 8 In this paper we prove that no packing of unit balls in Euclidean space R has density greater than that of the E8-lattice packing. Keywords: Sphere packing, Modular forms, Fourier analysis AMS subject classification: 52C17, 11F03, 11F30 1 Introduction The sphere packing constant measures which portion of d-dimensional Euclidean space d can be covered by non-overlapping unit balls. More precisely, let R be the Euclidean d vector space equipped with distance k · k and Lebesgue measure Vol(·). For x 2 R and d r 2 R>0 we denote by Bd(x; r) the open ball in R with center x and radius r. Let d X ⊂ R be a discrete set of points such that kx − yk ≥ 2 for any distinct x; y 2 X. Then the union [ P = Bd(x; 1) x2X d is a sphere packing. If X is a lattice in R then we say that P is a lattice sphere packing. The finite density of a packing P is defined as Vol(P\ Bd(0; r)) ∆P (r) := ; r > 0: Vol(Bd(0; r)) We define the density of a packing P as the limit superior ∆P := lim sup ∆P (r): r!1 arXiv:1603.04246v2 [math.NT] 4 Apr 2017 The number be want to know is the supremum over all possible packing densities ∆d := sup ∆P ; d P⊂R sphere packing 1 called the sphere packing constant. For which dimensions do we know the exact value of ∆d? Trivially, in dimension 1 we have ∆1 = 1. -
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. -
REPLICATING TESSELLATIONS* ANDREW Vincet Abstract
SIAM J. DISC. MATH. () 1993 Society for Industrial and Applied Mathematics Vol. 6, No. 3, pp. 501-521, August 1993 014 REPLICATING TESSELLATIONS* ANDREW VINCEt Abstract. A theory of replicating tessellation of R is developed that simultaneously generalizes radix representation of integers and hexagonal addressing in computer science. The tiling aggregates tesselate Eu- clidean space so that the (m + 1)st aggregate is, in turn, tiled by translates of the ruth aggregate, for each m in exactly the same way. This induces a discrete hierarchical addressing systsem on R'. Necessary and sufficient conditions for the existence of replicating tessellations are given, and an efficient algorithm is provided to de- termine whether or not a replicating tessellation is induced. It is shown that the generalized balanced ternary is replicating in all dimensions. Each replicating tessellation yields an associated self-replicating tiling with the following properties: (1) a single tile T tesselates R periodically and (2) there is a linear map A, such that A(T) is tiled by translates of T. The boundary of T is often a fractal curve. Key words, tiling, self-replicating, radix representation AMS(MOS) subject classifications. 52C22, 52C07, 05B45, 11A63 1. Introduction. The standard set notation X + Y {z + y z E X, y E Y} will be used. For a set T c Rn denote by Tx z + T the translate of T to point z. Throughout this paper, A denotes an n-dimensional lattice in l'. A set T tiles a set R by translation by lattice A if R [-JxsA T and the intersection of the interiors of distinct tiles T and Tu is empty. -
An Overview of the Kepler Conjecture
AN OVERVIEW OF THE KEPLER CONJECTURE Thomas C. Hales 1. Introduction and Review The series of papers in this volume gives a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than π/√18 0.74048 . This is the oldest problem in discrete geometry and is an important≈ part of Hilbert’s 18th problem. An example of a packing achieving this density is the face-centered cubic packing. A packing of spheres is an arrangement of nonoverlapping spheres of radius 1 in Euclidean space. Each sphere is determined by its center, so equivalently it is a collection of points in Euclidean space separated by distances of at least 2. The density of a packing is defined as the lim sup of the densities of the partial packings formed by spheres inside a ball with fixed center of radius R. (By taking the lim sup, rather than lim inf as the density, we prove the Kepler conjecture in the strongest possible sense.) Defined as a limit, the density is insensitive to changes in the packing in any bounded region. For example, a finite number of spheres can be removed from the face-centered cubic packing without affecting its density. Consequently, it is not possible to hope for any strong uniqueness results for packings of optimal density. The uniqueness established by this work is as strong arXiv:math/9811071v2 [math.MG] 20 May 2002 as can be hoped for. It shows that certain local structures (decomposition stars) attached to the face-centered cubic (fcc) and hexagonal-close packings (hcp) are the only structures that maximize a local density function. -
Universität Regensburg Mathematik
UniversitÄatRegensburg Mathematik Parametric Approximation of Surface Clusters driven by Isotropic and Anisotropic Surface Energies J.W. Barrett, H. Garcke, R. NÄurnberg Preprint Nr. 04/2009 Parametric Approximation of Surface Clusters driven by Isotropic and Anisotropic Surface Energies John W. Barrett† Harald Garcke‡ Robert N¨urnberg† Abstract We present a variational formulation for the evolution of surface clusters in R3 by mean curvature flow, surface diffusion and their anisotropic variants. We intro- duce the triple junction line conditions that are induced by the considered gradient flows, and present weak formulations of these flows. In addition, we consider the case where a subset of the boundaries of these clusters are constrained to lie on an external boundary. These formulations lead to unconditionally stable, fully dis- crete, parametric finite element approximations. The resulting schemes have very good properties with respect to the distribution of mesh points and, if applicable, volume conservation. This is demonstrated by several numerical experiments, in- cluding isotropic double, triple and quadruple bubbles, as well as clusters evolving under anisotropic mean curvature flow and anisotropic surface diffusion, including for regularized crystalline surface energy densities. Key words. surface clusters, mean curvature flow, surface diffusion, soap bubbles, triple junction lines, parametric finite elements, anisotropy, tangential movement AMS subject classifications. 65M60, 65M12, 35K55, 53C44, 74E10, 74E15 1 Introduction Equilibrium soap bubble clusters are stationary solutions of the variational problem in which one seeks a least area way to enclose and separate a number of regions with pre- scribed volumes. The relevant energy in this case is given as the sum of the total surface area. -
On the Application of the Honeycomb Conjecture to the Bee's Honeycomb
On the Application of the Honeycomb Conjecture to the Bee’s Honeycomb∗y Tim Räzz July 29, 2014 Abstract In a recent paper, Aidan Lyon and Mark Colyvan have proposed an explanation of the structure of the bee’s honeycomb based on the mathematical Honeycomb Conjecture. This explanation has instantly become one of the standard examples in the philosophical debate on mathematical explanations of physical phenomena. In this critical note, I argue that the explanation is not scientifically adequate. The reason for this is that the explanation fails to do justice to the essen- tially three-dimensional structure of the bee’s honeycomb. Contents 1 Introduction 2 2 Lyon’s and Colyvan’s Explanation 2 3 Baker: A Philosophical Motivation 3 4 Why the Explanation Fails 4 4.1 The Explanation is Incomplete . .4 4.2 The Honeycomb Conjecture Is (Probably) Irrelevant . .7 ∗Acknowledgements: I am grateful to Mark Colyvan, Matthias Egg, Michael Esfeld, Martin Gasser, Marion Hämmerli, Michael Messerli, Antoine Muller, Christian Sachse, Tilman Sauer, Raphael Scholl, two anonymous referees, and the participants of the phi- losophy of science research seminar in the fall of 2012 in Lausanne for discussions and comments on previous drafts of this paper. The usual disclaimer applies. This work was supported by the Swiss National Science Foundation, grants (100011-124462/1), (100018- 140201/1) yThis is a pre-copyedited, author-produced PDF of an article accepted for publication in Philosophia Mathematica following peer review. The definitive publisher-authenticated version, Räz, T. (2013): On the Application of the Honeycomb Conjecture to the Bee’s Honeycomb. -
Nonisotropic Self-Assembly of Nanoparticles: from Compact Packing to Functional Aggregates Xavier Bouju, Etienne Duguet, Fabienne Gauffre, Claude Henry, Myrtil L
Nonisotropic Self-Assembly of Nanoparticles: From Compact Packing to Functional Aggregates Xavier Bouju, Etienne Duguet, Fabienne Gauffre, Claude Henry, Myrtil L. Kahn, Patrice Mélinon, Serge Ravaine To cite this version: Xavier Bouju, Etienne Duguet, Fabienne Gauffre, Claude Henry, Myrtil L. Kahn, et al.. Nonisotropic Self-Assembly of Nanoparticles: From Compact Packing to Functional Aggregates. Advanced Mate- rials, Wiley-VCH Verlag, 2018, 30 (27), pp.1706558. 10.1002/adma.201706558. hal-01795448 HAL Id: hal-01795448 https://hal.archives-ouvertes.fr/hal-01795448 Submitted on 18 Oct 2018 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. Non-isotropic self-assembly of nanoparticles: from compact packing to functional aggregates Xavier Bouju∗ CEMES-CNRS (UPR CNRS 8011), Bt F picoLab 29 Rue J. Marvig, BP 94347 31055 Toulouse Cedex 4, France Etienne´ Duguet∗ CNRS, Univ. Bordeaux, ICMCB, UPR 9048, 33600 Pessac, France Fabienne Gauffre∗ Institut des sciences chimiques de Rennes (ISCR), UMR CNRS 6226, 263 avenue du G´en´eral Leclerc, CS 74205, F-35000 Rennes, France Claude Henry∗ Centre interdisciplinaire de nanoscience de Marseille (CINAM), UMR CNRS 7325, Aix-Marseille Universit´e,Campus de Luminy, Case 913, F-13000 Marseille, France Myrtil Kahn∗ Laboratoire de chimie de coordination (LCC), UPR CNRS 8241, 205 route de Narbonne, F-31000 Toulouse, France Patrice M´elinon∗ Universit´ede Lyon, Universit´eClaude Bernard Lyon 1, UMR CNRS 5306, Institut Lumi`ere Mati`ere, F-69622, Villeurbanne, Francey Serge Ravaine CNRS, Univ. -
Contractions of Octagonal Tilings with Rhombic Tiles. Frédéric Chavanon, Eric Remila
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Archive Ouverte en Sciences de l'Information et de la Communication Contractions of octagonal tilings with rhombic tiles. Frédéric Chavanon, Eric Remila To cite this version: Frédéric Chavanon, Eric Remila. Contractions of octagonal tilings with rhombic tiles.. [Research Report] LIP RR-2003-44, Laboratoire de l’informatique du parallélisme. 2003, 2+12p. hal-02101891 HAL Id: hal-02101891 https://hal-lara.archives-ouvertes.fr/hal-02101891 Submitted on 17 Apr 2019 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. Laboratoire de l’Informatique du Parallelisme´ Ecole´ Normale Sup´erieure de Lyon Unit´e Mixte de Recherche CNRS-INRIA-ENS LYON no 5668 Contractions of octagonal tilings with rhombic tiles Fr´ed´eric Chavanon, Eric R´emila Septembre 2003 Research Report No 2003-44 Ecole´ Normale Superieure´ de Lyon, 46 All´ee d’Italie, 69364 Lyon Cedex 07, France T´el´ephone : +33(0)4.72.72.80.37 Fax : +33(0)4.72.72.80.80 Adresseelectronique ´ : [email protected] Contractions of octagonal tilings with rhombic tiles Fr´ed´eric Chavanon, Eric R´emila Septembre 2003 Abstract We prove that the space of rhombic tilings of a fixed octagon can be given a canonical order structure. -
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. -
Designing Modular Sculpture Systems
Bridges 2017 Conference Proceedings Designing Modular Sculpture Systems Christopher Carlson Wolfram Research, Inc 100 Trade Centre Drive Champaign, IL, 61820, USA E-mail: [email protected] Abstract We are interested in the sculptural possibilities of closed chains of modular units. One such system is embodied in MathMaker, a set of wooden pieces that can be connected end-to-end to create a fascinating variety of forms. MathMaker is based on a cubic honeycomb. We explore the possibilities of similar systems based on octahedral- tetrahedral, rhombic dodecahedral, and truncated octahedral honeycombs. Introduction The MathMaker construction kit consists of wooden parallelepipeds that can be connected end-to-end to make a great variety of sculptural forms [1]. Figure 1 shows on the left an untitled sculpture by Koos Verhoeff based on that system of modular units [2]. MathMaker derives from a cubic honeycomb. Each unit connects the center of one cubic face to the center of an adjacent face (Figure 1, center). Units connect via square planar faces which echo the square faces of cubes in the honeycomb. The square connecting faces can be matched in 4 ways via 90° rotation, so from any position each adjacent cubic face is accessible. A chain of MathMaker units jumps from cube to adjacent cube, adjacent units never traversing the same cube. A variant of MathMaker called “turned cross mitre” has units whose square cross-sections are rotated 45° about their central axes relative to the standard MathMaker units (Figure 1, right). Although the underlying cubic honeycomb structure is the same, the change in unit yields structures with strikingly different visual impressions.