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Regular Figures
- 1st Edition - July 10, 2014
- Author: L. Fejes Tóth
- Editors: I. N. Sneddon, S. Ulam, M. Stark
- Language: English
- Paperback ISBN:9 7 8 - 1 - 4 8 3 1 - 1 9 0 1 - 4
- eBook ISBN:9 7 8 - 1 - 4 8 3 1 - 5 1 4 3 - 4
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… Read more
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Request a sales quoteRegular 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.
Preface
Part One Systematology of the Regular Figures
I. Plane Ornaments
1. Isometrics
2. Symmetry Groups
3. Groups with Infinite Unit Cells
4. Groups with Finite Unit Cells
5. Remarks
II. Spherical Arrangements
6. Isometrics in Space
7. The Finite Rotation Groups
8. The Finite Symmetry Groups
9. Groups of Permutations
10. The Geometrical Crystal Classes
11. Remarks
III. Hyperbolic Tessellations
12. The Hyperbolic Plane
13. Hyperbolic Trigonometry
14. Hyperbolic Tessellations
15. Remarks
IV. Polyhedra
16. The Nine Regular Polyhedra
17. Semi-Regular Polyhedra
18. Parallelohedra
19. Remarks
V. Regular Polytopes
20. Geometry in More than Three Dimensions
21. The General Regular Polytope
22. The Convex Regular Polytopes
23. Remarks
Part Two Genetics of the Regular Figures
VI. Figures in the Euclidean Plane
24. Inequalities for Polygons
25. Packing and Covering Problems
26. Isoperimetric Problems in Cell-Aggregates
27. Packings and Coverings by Non-Congruent Circles
28. A Stability Problem for Circle-Packings
29. Remarks
VII. Spherical Figures
30. The Isoperimetric Property of the Regular Spherical Polygons
31. Shortest Spherical Net with Meshes of Equal Area
32. An Extremal Distribution of Great Circles
33. An Inequality for Star-Tessellations
34. A Covering Problem
35. Distribution of the Orifices on Pollen-Grains
36. Remarks
VIII. Problems in the Hyperbolic Plane
37. Circle-Packings and Circle-Coverings
38. Packing and Covering by Horocycles
39. An Extremum Property of the Tessellations {P, 3}
40. Remarks
IX. Problems in 3-Space
41. Volume Estimates for Polyhedra
42. Surface Area and Edge-Curvature
43. The Isoperimetric Problem for Polyhedra
44. Sphere-Clouds
45. Sphere-Packings and Sphere-Coverings
46. Honeycombs
47. Remarks
X. Problems in Higher Spaces
48. On the Volume of a Polyhedron in Non-euclidean 3-Space
49. Extremum Properties of the Regular Polytopes
50. Sphere-Packings and Sphere-Coverings in Spaces of Constant Curvature
51. Remarks
Postscript
Bibliography
Index
- No. of pages: 352
- Language: English
- Edition: 1
- Published: July 10, 2014
- Imprint: Pergamon
- Paperback ISBN: 9781483119014
- eBook ISBN: 9781483151434
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