Least action principle of crystal formation of dense packing type and Kepler's conjecture:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Singapore [u.a.]
World Scientific
2001
|
| Series: | Nankai tracts in mathematics
3 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXI, 402 S. graph. Darst. |
| ISBN: | 9810246706 |
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MARC
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| 245 | 1 | 0 | |a Least action principle of crystal formation of dense packing type and Kepler's conjecture |c Wu-Yi Hsiang |
| 264 | 1 | |a Singapore [u.a.] |b World Scientific |c 2001 | |
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| 650 | 4 | |a Kepler's conjecture | |
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Record in the Search Index
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| adam_text | Contents
Foreword vii
Acknowledgment ix
List of Symbols xix
Chapter 1 Introduction 1
1.1 Sphere Packings and the Sphere Packing Problem ... 1
1.1.1 Density of an infinite packing 2
1.2 Kepler s Conjecture on Sphere Packings 4
1.2.1 Another mathematical formulation of the
sphere packing problem and
Kepler s Conjecture 5
1.3 Density of Finite Packings without a Container and the
Least Action Principle of Crystal Formation 6
1.3.1 Hexagonal close packings 7
1.3.2 Least action principle and crystal formation .... 8
1.3.3 Local cell decomposition, local density and the
Dodecahedron Conjecture 10
1.4 Locally Averaged Density and the Intrinsic Density of
the Second Kind 13
1.5 Main Theorems on Sphere Packings 14
xii Contents
1.5.1 Relationship between the two kinds of relative
densities and the proof of Kepler s Conjecture . . 16
Chapter 2 The Basics of Euclidean and Spherical
Geometries and a New Proof of the Problem
of Thirteen Spheres 19
2.1 Vector Algebra and Basic Spherical Geometry 20
2.1.1 Basic properties of the unit sphere 21
2.1.2 Vector algebra and spherical trigonometry 29
2.1.3 Some further results on areas of spherical
triangles and quadrilaterals 37
2.1.3.1 Spherical quadrilaterals 43
2.1.3.2 Shearing deformations 46
2.2 Spherical Configurations, Area Estimates and
a New Proof of the Impossibility of Thirteen
Touching Neighbors 61
2.2.1 Examples of problems on the distribution of point
on 52(1) 61
2.2.2 Spherical configuration and some basic
techniques of area estimation 65
2.2.3 Techniques of area estimates 68
2.2.4 Star configurations 73
2.2.5 Another proof of the impossibility of thirteen
touching neighbors 81
Chapter 3 Circle Packings and Sphere Packings 83
3.1 The Problem of Circle Packings 83
3.2 Sphere Packings and Crystal Formations, Kepler s
Conjecture and a Least Action Principle of Crystal
Formation 89
3.2.1 Three kinds of sphere packings and the concepts of
their densities 89
Contents xiii
3.2.1.1 Some natural problems on the density of
the first kind and a generalized Dodeca¬
hedron Conjecture 94
3.2.2 The sphere packing problem and
Kepler s Conjecture 99
3.2.2.1 A brief comparison between the sphere
packing and the circle packing problem 100
3.2.3 Sphere packings and crystal formation 101
3.2.3.1 The least action principle and crystal
formation 103
3.2.4 The least action principle of crystal formation
and the localization of the proof of
Kepler s Conjecture 105
3.3 Some Basic Ideas and Crucial Understanding which the
Proofs of the Major Theorems are Based 106
3.3.1 Volume estimates of local cells 106
3.3.1.1 Basic geometry of local cells 106
3.3.1.2 Basic strategy of volume estimations of
local cells 108
3.3.1.3 Volume estimation techniques specifi¬
cally developed for local cells 112
3.3.1.4 Some examples of volume estimation . 113
3.3.2 Geometry of single layer local packings 115
3.3.2.1 Quantitative refinements of the problem
of thirteen spheres 115
3.3.2.2 The non tightness of local packings
with twelve touching or almost touching
neighbors 116
3.3.3 Geometry of double layer local packing 119
Chapter 4 Geometry of Local Cells and Specific Vol¬
ume Estimation Techniques for Local Cells 123
4.1 Basic Geometry of Local Packings and Local Cells ... 123
xiv Contents
4.1.1 Associated spherical configurations and
polyhedrons of (£, {hj}) 125
4.1.2 Geometric correlations between spherical
configurations and associated polyhedrons 126
4.1.2.1 The correlation between 5(S) and
T(£) 126
4.1.2.2 The basic geometry of T(S, {hj}) ... 128
4.1.2.3 The relationships between T(£) and
T(E,{/ij}) (resp. 5*(£) and
S*(^,{hj})) 132
4.1.2.4 The peripheral part and its rectilinear
slabs 134
4.1.3 Basic strategies of volume estimation of
local cells 135
4.1.3.1 The separation of a local cell into its core
part and its peripheral part 135
4.1.3.2 The subdivision of the peripheral part
into rectilinear slabs 136
4.1.3.3 Spherical configuration and a basic vol¬
ume formula of T(S), the basic strategy
for the estimation of the core part . . 137
4.2 Technique of Volume Estimation of the Core Part ... 138
4.2.1 The volume function of tangent subpolyhedron . . 139
4.2.1.1 A basic volume formula 140
4.2.2 A volume formula of T(£) 142
4.2.2.1 A gradient formula of volT(S) 147
4.2.3 Two basic lemmas of volume estimation 151
4.2.4 Some direct applications 158
4.3 Volume Estimation of a Rectilinear Slab 164
4.3.1 The local geometry surrounding a given
neighbor of C {S0) 164
4.3.2 Lower bound volume estimates of
rectilinear slabs 172
Contents xv
4.4 Volume Estimation of Local Cells 187
Chapter 5 Estimates of Total Buckling Height 201
5.1 The Correlation Between Buckling Heights and Area
Estimates 204
5.2 Area Estimates of Star Configurations 215
5.2.1 Area decreasing deformations of
star configurations 217
5.2.2 Area estimates of star configurations 219
5.2.2.1 Examples of non deformable 6A stars
with a given set of edge length bounds 220
5.2.2.2 7A star with individual lower bounds on
edge lengths close to | 223
5.3 Estimation of Total Buckling Height 225
5.3.1 The proof of Lemma 5.3.3 226
5.3.2 The proof of Lemma 5.3.2 228
5.3.3 The proof of Lemma 5.3.1 229
Chapter 6 The Proof of the Dodecahedron
Conjecture 235
6.1 The Proof of Case 1: m = 12 236
6.2 The Proof of Case 2: m 11 237
6.3 The Proof of Case 3: 13 m 15 238
Chapter 7 Geometry of Type I Configurations and
Local Extensions 239
7.1 Geometry of Type I Local Packings and Type I
Configurations 241
7.1.1 A strong local characterization of f.c.c. and h.c.p.
configurations 242
7.1.2 Type I configuration with at least one
4 fork vertex 253
7.1.3 The moduli space of type I configuration 278
xvi Contents
7.1.3.1 Deformationsofthef.ee. and the h.c.p.
configurations 278
7.1.3.2 Type I configurations with quadrilater¬
als or 4 fork vertices 280
7.1.4 Geometry of type I icosahedrons 285
7.1.4.1 Total area excess and its distribution 285
7.1.4.2 Geometry of 5A star configurations . 287
7.1.4.3 5A star configurations with a lopsided
distribution of edge excess 292
7.1.4.4 A characterization of type I icosahe¬
drons which are small deformations of
the 5D ones 296
7.2 Geometry of Local Extension 300
7.2.1 Basic geometry of local extensions 300
7.2.1.1 Complementary regions and type I local
extensions 301
7.2.1.2 Buckling effects 306
7.2.2 Local extensions and volume estimations 309
7.2.2.1 Collective volume estimate and localized
estimates 309
7.2.2.2 Local extensions with local cells of vol¬
umes close to 4 /2 311
7.2.2.3 Local extensions with at most a small
amount of total buckling effect and
volume estimation of their local cells . 314
Chapter 8 The Proof of Main Theorem I 327
8.1 The Proof of Main Theorem I for Core Packings with at
Least Thirteen Close Neighbors 329
8.2 The Proof of Main Theorem I for Core Packings with at
Most Twelve Close Neighbors 332
8.2.1 Case I: Type I core packing (i.e. with twelve
touching neighbors) 333
Contents xvii
8.2.2 Case II: Non type I core packings with twelve
close neighbors 355
8.2.3 Case III: Core packings with at most eleven
close neighbors 363
Chapter 9 Retrospects and Prospects 383
9.1 Retrospects 383
9.1.1 Main results and their significances 383
9.1.2 Localizations and reductions 384
9.1.3 Subdivision of a local cell and basic techniques
of volume estimation of local cells 386
9.1.4 Vector algebra and spherical geometry 387
9.2 Prospects 387
9.2.1 Problems in solid geometry 388
9.2.1.1 Isoperimetric problem for polyhedra
with a given number of faces
(cf. Sec. 2.2) 388
9.2.1.2 Problems on the optimal distribution of
points on 52(1) 389
9.2.1.3 Generalized Dodecahedron Conjecture 390
9.2.1.4 Another characterization of hexagonal
double layer local packings and a con¬
jecture of Fejes Toth 391
9.2.2 Problems in higher dimensional Euclidean and
spherical geometry 392
9.2.2.1 Sphere packings in Sn(l) and the prob¬
lem of kissing numbers 393
9.2.2.2 Sphere packings in En, n 4 394
References 397
Index 401
|
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| id | DE-604.BV014461275 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:02:55Z |
| institution | BVB |
| isbn | 9810246706 |
| language | English |
| lccn | 2001045504 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009875500 |
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| owner_facet | DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-11 |
| physical | XXI, 402 S. graph. Darst. |
| publishDate | 2001 |
| publishDateSearch | 2001 |
| publishDateSort | 2001 |
| publisher | World Scientific |
| record_format | marc |
| series | Nankai tracts in mathematics |
| series2 | Nankai tracts in mathematics |
| spelling | Hsiang, Wu Yi 1937- Verfasser (DE-588)172146119 aut Least action principle of crystal formation of dense packing type and Kepler's conjecture Wu-Yi Hsiang Singapore [u.a.] World Scientific 2001 XXI, 402 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Nankai tracts in mathematics 3 Crystallography, Mathematical Kepler's conjecture Sphere packings Kristallmathematik (DE-588)4125615-3 gnd rswk-swf Kugelpackung (DE-588)4165929-6 gnd rswk-swf Kugelpackung (DE-588)4165929-6 s Kristallmathematik (DE-588)4125615-3 s DE-604 Nankai tracts in mathematics 3 (DE-604)BV014017593 3 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009875500&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hsiang, Wu Yi 1937- Least action principle of crystal formation of dense packing type and Kepler's conjecture Nankai tracts in mathematics Crystallography, Mathematical Kepler's conjecture Sphere packings Kristallmathematik (DE-588)4125615-3 gnd Kugelpackung (DE-588)4165929-6 gnd |
| subject_GND | (DE-588)4125615-3 (DE-588)4165929-6 |
| title | Least action principle of crystal formation of dense packing type and Kepler's conjecture |
| title_auth | Least action principle of crystal formation of dense packing type and Kepler's conjecture |
| title_exact_search | Least action principle of crystal formation of dense packing type and Kepler's conjecture |
| title_full | Least action principle of crystal formation of dense packing type and Kepler's conjecture Wu-Yi Hsiang |
| title_fullStr | Least action principle of crystal formation of dense packing type and Kepler's conjecture Wu-Yi Hsiang |
| title_full_unstemmed | Least action principle of crystal formation of dense packing type and Kepler's conjecture Wu-Yi Hsiang |
| title_short | Least action principle of crystal formation of dense packing type and Kepler's conjecture |
| title_sort | least action principle of crystal formation of dense packing type and kepler s conjecture |
| topic | Crystallography, Mathematical Kepler's conjecture Sphere packings Kristallmathematik (DE-588)4125615-3 gnd Kugelpackung (DE-588)4165929-6 gnd |
| topic_facet | Crystallography, Mathematical Kepler's conjecture Sphere packings Kristallmathematik Kugelpackung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009875500&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV014017593 |
| work_keys_str_mv | AT hsiangwuyi leastactionprincipleofcrystalformationofdensepackingtypeandkeplersconjecture |