Regular Figures

1st Edition - January 1, 1964
Author: L. Fejes Tóth
Editors: I. N. Sneddon, S. Ulam, M. Stark
Language: English
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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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.

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