Least Action Principle of Crystal Formation of Dense Packing Type and Kepler's ConjectureWorld Scientific, 2001 - 424 pages The dense packing of microscopic spheres (i.e. atoms) is the basic geometric arrangement in crystals of mono-atomic elements with weak covalent bonds, which achieves the optimal known density of p/OeU18. In 1611, Johannes Kepler had already conjectured that p/OeU18 should be the optimal density of sphere packings. Thus, the central problems in the study of sphere packings are the proof of Kepler''s conjecture that p/OeU18 is the optimal density, and the establishing of the least action principle that the hexagonal dense packings in crystals are the geometric consequence of optimization of density. This important book provides a self-contained proof of both, using vector algebra and spherical geometry as the main techniques and in the tradition of classical geometry." |
Table des matières
The Basics of Euclidean and Spherical | 19 |
Circle Packings and Sphere Packings | 83 |
Geometry of Local Cells and Specific Vol | 123 |
Estimates of Total Buckling Height | 201 |
The Proof of the Dodecahedron | 235 |
configurations | 242 |
The Proof of Main Theorem I | 327 |
Retrospects and Prospects | 383 |
| 397 | |
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Expressions et termes fréquents
6-stars area estimates area-excess big-hole buckled quadrilaterals Ĉ(So cells central angles circle packings circumcenter circumcircle circumradius circumscribing close neighbors cluster cocircular quadrilateral complementary region contains convex core packing cos² crystal formation diagonal Dodecahedron Dodecahedron Conjecture double-layer local packings edge-lengths Euclidean example extension faces of S(E finite formula geometric given global density h.c.p. configuration h₁ h₂ Hence hexagonal close packing icosahedron indicated in Figure infinite packings Kepler's Conjecture kind least action principle least equal length local cells locally averaged density lower bound estimate Main Theorem minimal moduli space Moreover namely pair pentagon polyhedron radial edge-excess radial edges rectilinear slabs resp septuple side-lengths sin² single-layer small deformation sphere packing problem spherical configuration spherical geometry spherical triangle St(A star configuration subcase Sublemma subset tan² tangent total buckling height touching neighbors triangular type I configuration unit sphere vector vertex vertices