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El. knyga: Geometry Revealed: A Jacob's Ladder to Modern Higher Geometry

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  • Formatas: PDF+DRM
  • Išleidimo metai: 23-Jul-2010
  • Leidėjas: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Kalba: eng
  • ISBN-13: 9783540709978
Kitos knygos pagal šią temą:
Geometry Revealed: A Jacob's Ladder to Modern Higher GeometryDidesnis paveikslėlis
  • Formatas: PDF+DRM
  • Išleidimo metai: 23-Jul-2010
  • Leidėjas: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Kalba: eng
  • ISBN-13: 9783540709978
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Both classical geometry and modern differential geometry have been active subjects of research throughout the 20th century and lie at the heart of many recent advances in mathematics and physics. The underlying motivating concept for the present book is that it offers readers the elements of a modern geometric culture by means of a whole series of visually appealing unsolved (or recently solved) problems that require the creation of concepts and tools of varying abstraction. Starting with such natural, classical objects as lines, planes, circles, spheres, polygons, polyhedra, curves, surfaces, convex sets, etc., crucial ideas and above all abstract concepts needed for attaining the results are elucidated. These are conceptual notions, each built "above" the preceding and permitting an increase in abstraction, represented metaphorically by Jacob's ladder with its rungs: the 'ladder' in the Old Testament, that angels ascended and descended...

In all this, the aim of the book isto demonstrate to readers the unceasingly renewed spirit of geometry and that even so-called "elementary" geometry is very much alive and at the very heart of the work of numerous contemporary mathematicians. It is also shown that there are innumerable paths yet to be explored and concepts to be created. The book is visually rich and inviting, so that readers may open it at random places and find much pleasure throughout according their own intuitions and inclinations.

Marcel Berger is t









he author of numerous successful books on geometry, this book once again is addressed to all students and teachers of mathematics with an affinity for geometry.

Recenzijos

It is a must own book for anyone serious about developing a conceptual understanding of the interconnected web of modern geometry and the ever-growing intertwining of geometry with practically all other branches of mathematics. It is remarkable for a book to provide such a detailed glimpse of contemporary geometry via well developed discussions of so many questions of current interest. It provides the most extensive exposition of geometric thinking Ive ever seen in a book at this level. (William H. Barker, MAA Reviews, August, 2017)

Geometry Revealed is to give the reader a feel for the conceptual frameworks of modern geometry, attempting to reach as far as possible with a minimum of assumed knowledge and formal scaffolding. Geometry Revealed being useful for research mathematicians as a still reasonably up-to-date survey. Geometry Revealed offered anascent into the wonders of a new world. (Danny Yee, Danny Yees Book Reviews, dannyreviews.com, July, 2015)

By considering a hierarchy of natural geometrical objects it sets out to investigate significant geometrical problems which are either unsolved or were solved only recently. it is undoubtedly a major tour de force, and if you really want to gain an idea of where geometry is going in the 21st century, you will find plenty of exquisite material here. (Gerry Leversha, The Mathematical Gazette, Vol. 96 (356), July, 2012)

The book contains twelve chapters, each of them is a collection of such problems about geometric objects with more and more complexity . The chapters are independent from each other, any of them can serve as a course. Researchers in geometry can use it as a source for further research. the book is accessible to a wide audience of people who are interested in geometry. (Jįnos Kincses, Acta Scientiarum Mathematicarum (Szeged), Vol. 78 (1-2),2012)

Geometry Revealed is a massive text of 831 pages which is organized in twelve chapters and which additionally provides indices for names, subjects and symbols throughout the author quite carefully lays out the historical perspective. a typical chapter starts with an observation or a problem in elementary geometry. Large parts of the text are very accessible, and a reader who likes (mathematical) physics will often get something extra. (Michael Joswig, Zentralblatt MATH, Vol. 1232, 2012)













The author provides the reader with an enormous amount of detailed information and thus yields deep insight into the various topics. All in all an overwhelming book which is a must for everyone having sufficient mathematical knowledge. (G. Kowol, Monatshefte für Mathematik, Vol. 164 (2), October, 2011)

The book is a very readable account of several branches of geometry, classical and modern, elementary and advanced. Every chapter is extremelyinteresting and alive. The book is rich in ideas, written in an informal style, with no formulae and no unnecessary technical details. Every part of this book is interesting and should be accessible to a wide audience of mathematicians. Every mathematician will experience great pleasure in reading this book. (Athanase Papadopoulos, Mathematical Reviews, Issue 2011 m)

About the Author v
Introduction vii
Chapter I Points and lines in the plane
1(60)
I.1 In which setting and in which plane are we working? And right away an utterly simple problem of Sylvester about the collinerarity of points
1(5)
I.2 Another naive problem of sylvester, this time on the geometric probabilities of four points
6(6)
I.3 The essence of affine geometry and the fundamental theorem
12(5)
I.4 Three configurations of the affine plane and what has happened to them: Pappus, Desargues and Perles
17(6)
I.5 The irresistible necessity of projective geometry and the construction of the projective plane
23(5)
I.6 Intermezzo: the projective line and the cross ratio
28(3)
I.7 Return to the projective plane: continuation and conclusion
31(9)
I.8 The complex case and, better still, Sylvester in the complex case: Serre's conjecture
40(3)
I.9 Three configurations of space (of three dimensions): Reye, Mobius and Schlafli
43(4)
I.10 Arrangements of hyperplanes
47(1)
I XYZ
48(9)
Bibliography
57(4)
Chapter II Circles and spheres
61(80)
II.1 Introduction and Borsuk's conjecture
61(5)
II.2 A choice of circle configurations and a critical view of them
66(12)
II.3 A solitary inversion and what can be done with it
78(4)
II.4 How do we compose inversions? First solution: the conformal group on the disk and the geometry of the hyperbolic plane
82(5)
II.5 Second solution: the conformal group of the sphere, first seen algebraically, then geometrically, with inversions in dimension 3 (and three-dimensional hyperboic geometry). Historical appearance of the first fractals
87(4)
II.6 Inversion in space: the sextuple and its generalization thanks to the sphere of dimension 3
91(5)
II.7 Higher up the ladder: the global geometry of circles and spheres
96(7)
II.8 Hexagonal packings of circles and conformal representation
103(10)
II.9 Circles of Apollonius
113(3)
II XYZ
116(21)
Bibliography
137(4)
Chapter III The sphere by itself: can we distribute points on it evenly?
141(40)
III.1 The metric of the sphere and spherical trigonometry
141(6)
III.2 The Mobius group: applications
147(2)
III.3 Mission impossible: to uniformly distribute points on the sphere S2: ozone, electrons, enemy dictators, golf balls, virology, physics of condensed matter
149(21)
III.4 The Kissing number of S2 alias the hard problem of the thirteenth shphere
170(2)
III.5 Four open problems for the sphere S3
172(2)
III.6 A Problem of Banach-Ruziewicz: the uniquencess of canonical measure
174(1)
III.7 A conceptual approach for the kissing number in arbitrary dimension
175(2)
III XYZ
177(1)
Bibliography
178(3)
Chapter IV Conics and quadrics
181(68)
IV.1 Motivations, a Definition parachuted from the ladder, and why
181(2)
IV.2 Before Descartes: the real Euclidean conics. Definition and some classical properties
183(15)
IV.3 The coming of Descrates and the birth of algebriac geometry
198(2)
IV.4 Real projective theory of conics; duality
200(5)
IV.5 Klein's philosophy comes quite naturally
205(3)
IV.6 Playing with two conics, necessitating once again complexification
208(4)
IV.7 Complex projective conics and the space of all conics
212(4)
IV.8 The most beautiful theorem on conics: the Poncelet polygons
216(10)
IV.9 The most difficult theorem on the conics: the 3264 conics of Chasles
226(6)
IV.10 The quadrics
232(10)
IV XYZ
242(3)
Bibliography
245(4)
Chapter V Plane curves
249(92)
V.1 Plain curves and the person in the street: the jordan curve theorem, the turning tangent theorem and the isoperimetric inequality
249(5)
V.2 What is a curve? Geometric curves and kinematic curves
254(3)
V.3 The classification of geometric curves and the degree of mappings of the circle onto itself
257(2)
V.4 The Jordan theorem
259(1)
V.5 The turning tangent theorem and global convexity
260(3)
V.6 Euclidean invaariants: length (theorem of the peripheral boulevard) and curvature (scalar and algebraic): Winding number
263(6)
V.7 The algebraic curvature is a characteristic invariant: manufacture of rulers, control by the curvature
269(2)
V.8 The four vertex theorem and its converse; an application to physics
271(7)
V.9 Generalizations of the four vertex theorem: Arnold I
278(3)
V.10 Toward a classification of closed curves: Whitney and Arnold II
281(14)
V.11 Isoperimetric inequality: Steiner's attempts
295(3)
V.12 The isoperimetric inequality: proofs on all rungs
298(7)
V.13 Plane algebraic curves: generalities
305(3)
V.14 The cubics, their addition law and abstract elliptic curves
308(12)
V.15 Real and Euclidean algebraic curves
320(8)
V.16 Finite order geometry
328(3)
V XYZ
331(5)
Bibliography
336(5)
Chapter VI Smooth surfaces
341(68)
VI.1 Which objects are involved and why? Classification of compact surfaces
341(4)
VI.2 The intrinsic metric and the problem of the shortest path
345(2)
VI.3 The geodesics, the cut locus and the recalcitrant ellipsoids
347(10)
VI.4 An indispensable abstract concept: Riemannian surfaces
357(4)
VI.5 Problems of isometries: abstract surfaces versus surfaces of E3
361(3)
VI.6 Local shape of surfaces: the second fundamental form, total curvature and mean curvature, their geometric interpretation, the theorema egregium, the manufacture of precise balls
364(9)
VI.7 What is known about the total curvature (of Gauss)
373(7)
VI.8 What we know how to do with the mean curvature, all about soap bubbles and lead balls
380(6)
VI.9 What we don't entirely know how to do for surfaces
386(5)
VI.10 Surfaces and genericity
391(6)
VI.11 The isoperimetric inequality for surfaces
397(2)
VI XYZ
399(4)
Bibliography
403(6)
Chapter VII Convexity and convex sets
409(96)
VII.1 History and introduction
409(3)
VII.2 Convex functions, examples and first applications
412(3)
VII.3 Convex functions of several variables, an important example
415(2)
VII.4 Examples of convex sets
417(3)
VII.5 Three essential operations on convex sets
420(8)
VII.6 Volume and area of (compacts) convex sets, classical volumes: Can the volume be calculated in polynomial time?
428(9)
VII.7 Volume, area, diameter and symmetrizations: first proof of the isoperimetric inequality and other applications
437(2)
VII.8 Volume and Minkowski addition: the Brunn-Minkowski theorem and a second proof of the isoperimetric inequality
439(5)
VII.9 Volume and polarity
444(2)
VII.10 The appearance of convex sets, their degree of badness
446(13)
VII.11 Volumes of slices of convex sets
459(11)
VII.12 Sections of low dimension: the concentration phenomenon and the Dvoretsky theorem on the existence of almost spherical sections
470(7)
VII.13 Miscellany
477(16)
VII.14 Intermezzo: can we dispose of the isoperimetric inequality?
493(6)
Bibliography
499(6)
Chapter VIII Polygons, polyhedra, polytopes
505(58)
VIII.1 Introduction
505(1)
VIII.2 Basic notions
506(2)
VIII.3 Polygons
508(5)
VIII.4 Polyhedra: combinatorics
513(5)
VIII.5 Regular Euclidean polyhedra
518(6)
VIII.6 Euclidean polyhedra: Cauchy rigidity and Alexandrov existence
524(6)
VIII.7 Isoperimetry for Euclidean Polyhedra
530(2)
VIII.8 Inscribability properties of Euclidean polyhedra; how to encage a sphere (an egg) and the connection with packings of circles
532(5)
VIII.9 Polyhedra: rationality
537(2)
VIII.10 Polytopes (d ≥ 4): combinatorics I
539(5)
VIII.11 Regular polytopes ( d ≥ 4)
544(6)
VIII.12 Polytopes (d ≥ 4): rationality, combinatorics II
550(5)
VIII.13 Brief allusions to subjects not really touched on
555(3)
Bibliography
558(5)
Chapter IX Lattices, packings and tilings in the plane
563(60)
IX.1 Lattices, a line in the standard lattice Z2 and the theory of continued fractions, an immensity of applications
563(4)
IX.2 Three ways of counting the points Z2 in various domains: pick and Ehrhart formulas, circle problem
567(6)
IX.3 Points of Z2 and of other lattices in certain convex sets: Minkowski's theorem and geometric number theory
573(3)
IX.4 Lattices in the Euclidean plane: classification, density, Fourier analysis on lattices, spectra and duality
576(10)
IX.5 Packing circles (disks) of the same radius, finite or infinite in number, in the plane (notion of density). Other criteria
586(7)
IX.6 Packing of squares, (flat) storage boxes, the grid (or beehive) problem
593(3)
IX.7 Tiling the plane with a group (crystallography). Valences, earthquakes
596(7)
IX.8 Tilings in higher dimensions
603(4)
IX.9 Algorithmics and plane tilings: aperiodic tilings and decidability, classification of Penrose tilings
607(10)
IX.10 Hyperbolic tilings and Riemann surfaces
617(3)
Bibliography
620(3)
Chapter X Lattices and packings in higher dimensions
623(52)
X.1 Lattices and packings associated with dimension 3
623(6)
X.2 Optimal packing of balls in dimension 3, Kepler's conjecture at last resolved
629(10)
X.3 A bit of risky epistemology: the four color problem and the Kepler conjecture
639(2)
X.4 Lattices in arbitrary dimension: examples
641(7)
X.5 Lattices in arbitrary dimension: density, laminations
648(6)
X.6 Packings in arbitrary dimension: various options for optimality
654(5)
X.7 Error correcting codes
659(8)
X.8 Duality, theta functions, spectra and isospectrality in lattices
667(6)
Bibliography
673(2)
Chapter XI Geometry and dynamics I: billiards
675(64)
XI.1 Introduction and motivation: description of the motion of two particles of equal mass on the interior of an interval
675(4)
XI.2 Playing billiards in a square
679(10)
XI.3 Particles with different masses: rational and irrational polygons
689(3)
XI.4 Results in the case of rational polygons: first rung
692(4)
XI.5 Results in the rational case: several rungs higher on the ladder
696(9)
XI.6 Results in the case of irrational polygons
705(5)
XI.7 Return to the case of two masses: summary
710(1)
XI.8 Concave billiards, hyperbolic billiards
710(3)
XI.9 Circles and ellipses
713(4)
XI.10 General convex billiards
717(11)
XI.11 Billiards in higher dimensions
728(2)
XI XYZ Concepts and language of dynamical systems
730(5)
Bibliography
735(4)
Chapter XII Geometry and dynamics II: geodesic flow on a surface
739(46)
XII.1 Introduction
739(2)
XII.2 Geodesic flow on a surface: problems
741(2)
XII.3 Some examples for sensing the difficulty of the problem
743(8)
XII.4 Existence of a periodic trajectory
751(6)
XII.5 Existence of more than one, of many periodic trajectories: and can we count them?
757(15)
XII.6 What behavior can be expected for other trajectories? Ergodicity, entropies
772(7)
XII.7 Do the mechanics determine the metric?
779(2)
XII.8 Recapitulation and open questions
781(1)
XII.9 Higher dimensions
781(1)
Bibliography
782(3)
Selected Abbreviations for Journal Titles 785(4)
Name Index 789(6)
Subject Index 795(32)
Symbol Index 827
Marcel Berger is Ancien Professeur of the University of Paris and emeritus director of research at the Centre National de la Recherche Scientifique (CNRS), from 1979 to 1981 he was president of the French Mathematical Society and from 1985 to 1994 director of the Institut des Hautes Études Scientifiques (IHÉS) in Bures-sur-Yvette.
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