Tóth / Sneddon / Ulam Regular Figures

1. Auflage 2014
ISBN: 978-1-4831-5143-4
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 352 Seiten, Web PDF

Reihe: International Series in Pure and Applied Mathematics

Regular Figures concerns the systematology and genetics of regular figures. The first part of the book deals with the classical theory of the regular figures. This topic includes description of plane ornaments, spherical arrangements, hyperbolic tessellations, polyhedral, and regular polytopes. The problem of geometry of the sphere and the two-dimensional hyperbolic space are considered. Classical theory is explained as describing all possible symmetrical groupings in different spaces of constant curvature. The second part deals with the genetics of the regular figures and the inequalities found in polygons; also presented as examples are the packing and covering problems of a given circle using the most or least number of discs. The problem of distributing n points on the sphere for these points to be placed as far as possible from each other is also discussed. The theories and problems discussed are then applied to pollen-grains, which are transported by animals or the wind. A closer look into the exterior composition of the grain shows many characteristics of uniform distribution of orifices, as well as irregular distribution. A formula that calculates such packing density is then explained. More advanced problems such as the genetics of the protean regular figures of higher spaces are also discussed. The book is ideal for physicists, mathematicians, architects, and students and professors in geometry.

Autoren/Hrsg.

Tóth, L. Fejes

Sneddon, I. N.

Ulam, S.

Stark, M.

Inhaltsverzeichnis

1;Front Cover;1
2;Regular Figures;6
3;Copyright Page;7
4;Table of Contents;8
5;Preface;10
6;PART ONE: SYSTEMATOLOGY OF THE REGULAR FIGURES;14
6.1;CHAPTER I. PLANE ORNAMENTS;16
6.1.1;1. Isometries;16
6.1.2;2. Symmetry groups;21
6.1.3;3. Groups with infinite unit cells;24
6.1.4;4. Groups with finite unit cells;34
6.1.5;5. Remarks;51
6.2;CHAPTER
II. SPHERICAL ARRANGEMENTS;75
6.2.1;6. Isometries in space;75
6.2.2;7. The finite rotation groups;77
6.2.3;8. The finite symmetry groups;82
6.2.4;9. Groups of permutations;89
6.2.5;10. The geometrical crystal classes;97
6.2.6;11. Remarks;100
6.3;CHAPTER
III. HYPERBOLIC TESSELLATIONS;103
6.3.1;12. The hyperbolic plane;103
6.3.2;13. Hyperbolic trigonometry;110
6.3.3;14. Hyperbolic tessellations;114
6.3.4;15. Remarks;116
6.4;CHAPTER
IV. POLYHEDRA;121
6.4.1;16. The nine regular polyhedra;121
6.4.2;17. Semi-regular polyhedra;129
6.4.3;18. Parallelohedra;135
6.4.4;19. Remarks;140
6.5;CHAPTER
V. REGULAR POLYTOPES;145
6.5.1;20. Geometry in more than three dimensions;145
6.5.2;21. The general regular polytope;153
6.5.3;22. The convex regular polytopes;158
6.5.4;23. Remarks;165
7;PART TWO: GENETICS OF THE REGULAR FIGURES;170
7.1;CHAPTER
VI. FIGURES IN THE EUCLIDEAN PLANE;172
7.1.1;24. Inequalities for polygons;172
7.1.2;25. Packing and covering problems;181
7.1.3;26. Isoperimetric problems in cell-aggregates;193
7.1.4;27. Packings and coverings by non-congruent circles;204
7.1.5;28. A stability problem for circle-packings;219
7.1.6;29. Remarks;221
7.2;CHAPTER
VII. SPHERICAL FIGURES;234
7.2.1;30. The isoperimetric property of the regular spherical polygons;234
7.2.2;31. Shortest spherical net with meshes of equal
area;236
7.2.3;32. An extremal distribution of great circles;237
7.2.4;33. An inequality for star-tessellations;240
7.2.5;34. A covering problem;245
7.2.6;35. Distribution of the orifices on pollen-grains;247
7.2.7;36. Remarks;256
7.3;CHAPTER
VIII. PROBLEMS IN THE HYPERBOLIC PLANE;259
7.3.1;37. Circle-packings and circle-coverings;259
7.3.2;38. Packing and covering by horocycles;268
7.3.3;39. An extremum property of the tessellations {p, 3};275
7.3.4;40. Remarks;282
7.4;CHAPTER
IX. PROBLEMS IN 3-SPACE;284
7.4.1;41. Volume estimates for polyhedra;284
7.4.2;42. Surface area and edge-curvature;299
7.4.3;43. The isoperimetric problem for polyhedra;303
7.4.4;44. Sphere-clouds;309
7.4.5;45. Sphere-packings and sphere-coverings;313
7.4.6;46. Honeycombs;321
7.4.7;47. Remarks;325
7.5;CHAPTER
X. PROBLEMS IN HIGHER SPACES;329
7.5.1;48. On the volume of a polyhedron in non-Euclidean 3-space;329
7.5.2;49. Extremum properties of the regular polytopes;333
7.5.3;50. Sphere-packings and sphere-coverings in spaces of constant curvature;338
7.5.4;51. Remarks;344
8;Postscript;347
9;Bibliography;348
10;Index;354