Dense Sphere Packings : A Blueprint for Formal Proofs
Thomas. Hales
Bok Engelsk 2012 · Electronic books.
| Annen tittel | |
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| Utgitt | Cambridge : Cambridge University Press , 2012
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| Omfang | 1 online resource (287 p.)
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| Opplysninger | Description based upon print version of record.. - Cover; Title; Copyright; Contents; Preface; The Kepler Conjecture; Formal Proofs; Conventions; A Blueprint; Simplifications; PART ONE: OVERVIEW; 1 Close Packing; 1.1 History; 1.1.1 Sanskrit sources; 1.1.2 Harriot and Kepler; 1.1.3 Newton and Gregory; 1.2 Face-Centered Cubic; 1.3 Hexagonal-Close Packing; 1.4 Gauss; 1.5 Thue; 1.6 Dense Packings in a Nutshell; 1.6.1 geometric partition; 1.6.2 contravening packing; 1.6.3 tame hypermap; 1.6.4 linear programming; PART TWO: FOUNDATIONS; 2 Trigonometry; 2.1 Background Knowledge; 2.1.1 formal proof; 2.1.2 real analysis; 2.1.3 Tarski arithmetic. - 2.2 Trig Identities2.2.1 sine and cosine; 2.2.2 periodicity; 2.2.3 tangent; 2.2.4 arctangent; 2.2.5 inverse trig; 2.3 Vector Geometry; 2.3.1 Euclidean space; 2.3.2 affine geometry; 2.3.3 parallelepiped; 2.4 Angle; 2.4.1 cross product; 2.4.2 dihedral angle; 2.4.3 Euler triangle; 2.5 Coordinates; 2.5.1 azimuth angle; 2.5.2 zenith angle; 2.6 Cycle; 2.6.1 polar cycle; 2.6.2 azimuth cycle; 2.6.3 spherical triangle inequality; 2.7 Chapter Summary; 3 Volume; 3.1 Background in Measure; 3.2 Primitive Volume; 3.2.1 radial set; 3.2.2 wedge; 3.2.3 primitive types; 3.2.4 volume calculations. - 3.3 Finiteness and Volume4 Hypermap; 4.1 Background on Permutations; 4.2 Definitions; 4.3 Walkup; 4.3.1 single; 4.3.2 double; 4.4 Planarity; 4.5 Path; 4.5.1 contour; 4.5.2 Möbius; 4.6 Subquotient; 4.6.1 definition; 4.6.2 properties; 4.6.3 example; 4.7 Generation; 4.7.1 flag; 4.7.2 markup; 4.7.3 transform; 4.7.4 algorithm; 5 Fan; 5.1 Definitions; 5.1.1 basic properties; 5.1.2 hypermap; 5.2 Topology; 5.2.1 background; 5.2.2 topological component and dart; 5.3 Planarity; 5.3.1 face attributes; 5.3.2 conformance; 5.3.3 existence; 5.4 Polyhedron; 5.4.1 background on convex sets. - 5.4.2 background on polyhedra5.4.3 fan and polyhedron; PART THREE: THE KEPLER CONJECTURE; 6 Packing; 6.1 The Primitive State of Our Subject Revealed; 6.1.1 definition; 6.1.2 Voronoi cell; 6.1.3 reduction to a finite packing; 6.2 Rogers Simplex; 6.2.1 faces; 6.2.2 partitioning space; 6.2.3 circumcenter; 6.2.4 Delaunay simplex; 6.3 Cells; 6.3.1 definition; 6.3.2 informal discussion; 6.3.3 cell partition; 6.3.4 edges of cells; 6.3.5 A conjecture; 6.4 Clusters; 6.5 Counting Spheres; 6.5.1 solid angle; 6.5.2 a polyhedral bound; 7 Local Fan; 7.1 Localization; 7.1.1 basics; 7.1.2 geometric type. - 7.2 Modification7.2.1 deformation; 7.2.2 slicing; 7.3 Polarity; 7.3.1 construction; 7.3.2 perimeter; 7.4 Main Estimate; 7.4.1 statement of results; 7.4.2 constraints; 7.4.3 minimality; 7.4.4 reducing dimension; 7.4.5 computer proof of main estimate; construction of S; triangles and quadrilaterals; pentagons; hexagons; instabilities; 8 Tame Hypermap; 8.1 Definition; 8.1.1 weight assignment; 8.1.2 hypermap property; 8.2 Contravening Hypermap; 8.2.1 standard fan; 8.2.2 surrounded and isolated nodes; 8.3 Contravention is Tame; 8.3.1 general properties; 8.3.2 properties of nodes; 8.3.3 faces. - 8.4 Admissibility. - The definitive account of the recent computer solution of the oldest problem in discrete geometry.
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| Emner | Kepler's conjecture
Sphere packings. Algebra Mathematics Vis mer... Physical Sciences & Mathematics
keplers formodning diskret geometri kombinatorikk datorbevis formelle beviser |
| Sjanger | |
| Dewey | |
| ISBN | 9780521617703
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