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Convex Polyhedra

A.D. Alexandrov
Publisher: 
Springer Verlag
Publication Date: 
2005
Number of Pages: 
539
Format: 
Hardcover
Series: 
Springer Monographs in Mathematics
Price: 
129.00
ISBN: 
3-540-23158-7
Category: 
Monograph
[Reviewed by
John McCleary
, on
11/10/2005
]

The text under review is a translation and expansion of a classic work on convex polyhedra, first published in Russian in 1950, then translated into German in 1958. It has been updated by V. A. Zalgaller, and fitted with supplements by Yu. A. Volkov and L. A. Shor. It is a testimony to Alexandrov's depth of insight and care in exposition that this monograph is still the basic text in the subject, and that it finally has an English translation (by Dairbekov, Kutateladze, and Sossinsky) more than fifty years after its writing.

What is most compelling about this book is the choice of method — a polyhedron is a geometric object of some simplicity, accessible by ideas from synthetic geometry and elementary topology, and demonstrating anew the power of these methods. The basic question of the book is to establish which data associated to a polyhedron determine it up to Euclidean equivalence. In two dimensions, for example, the conditions a + b > c, a + c > b and b + c > a determine the lengths of the sides of a triangle, which is unique up to isometry of the plane. The first such uniqueness theorem for polyhedra was given by Cauchy in 1813: two closed convex polyhedra composed of the same number of equal similarly situated faces are congruent (via a Euclidean motion of three-space).

The combinatorial data that describe a polyhedron are given by a development. A development is a collection of polygons, together with a prescription for gluing them along their sides. The rules for gluing include the geometric assumptions that sides can be glued only if they have the same lengths, and that a side is glued to at most one other side. Such data also carry topological implications and conditions for convexity follow.

To restore spatial geometry Alexandrov introduced the intrinsic metric of a development. Distances are measured along the faces of the polyhedron and so notions such as geodesics are defined. Viewing a closed, convex polyhedron as a sort of surface, the analogue of the Gauss map is a prescription of face directions. In this setting, Alexandrov develops and proves Minkowski's theorem, which states that two polyhedra with pairwise parallel faces of equal area are, in fact, parallel translates of each other.

The failure of uniqueness for polyhedra with boundary and a given development is the notion of a flexible polyhedron. Alexandrov proved conditions equivalent to the existence of flexible polyhedra from which his rigidity theorems follow.

The original editions of this book included conjectures and problems, many of which were solved in the ensuing decades. Before his death, Alexandrov, with the assistance of Zalgaller, prepared footnotes for this edition bringing it up to date and including an expanded bibliography. The richness and beauty of the mathematics of polyhedra is the main gift of this book. It is bound to influence another fifty years of research on this subject.


John McCleary is Professor of Mathematics at Vassar College.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Content and Purpose of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Order and Character of the Exposition . . . . . . . . . . . . . . . . . . . . . . . 3
Remarks for the Professional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1 Basic Concepts and Simplest Properties of Convex
Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1 Definition of a Convex Polyhedron . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Determining a Polyhedron from the Planes of Its Faces . . . . 16
1.3 Determining a Closed Polyhedron from Its Vertices . . . . . . . 21
1.4 Determining an Unbounded Polyhedron from Its Vertices
and the Limit Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.5 The Spherical Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.6 Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
1.7 Topological Properties of Polyhedra and Developments . . . . 56
1.8 Some Theorems of the Intrinsic Geometry of Developments 72
1.9 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2 Methods and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.1 The Cauchy Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.2 The Mapping Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
2.3 Determining a Polyhedron from a Development (Survey of
Chapters 3, 4, and 5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
2.4 Polyhedra with Prescribed Face Directions (Survey of
Chapters 6, 7, and 8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
2.5 Polyhedra with Vertices on Prescribed Rays (Survey of
Chapter 9) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
2.6 Infinitesimal Rigidity Theorems (Survey of Chapters 10
and 11) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
2.7 Passage from Polyhedra to Curved Surfaces . . . . . . . . . . . . . . 136
2.8 Basic Topological Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
2.9 The Domain Invariance Theorem . . . . . . . . . . . . . . . . . . . . . . . 147
X Contents
3 Uniqueness of Polyhedra with Prescribed Development . 155
3.1 Several Lemmas on Polyhedral Angles . . . . . . . . . . . . . . . . . . . 155
3.2 Equality of Dihedral Angles in Polyhedra with Equal
Planar Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
3.3 Uniqueness of Polyhedra with Prescribed Development . . . . 169
3.4 Unbounded Polyhedra of Curvature Less Than 2p . . . . . . . . 173
3.5 Polyhedra with Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
3.6 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
4 Existence of Polyhedra with Prescribed Development . . . 193
4.1 The Manifold of Developments . . . . . . . . . . . . . . . . . . . . . . . . . 193
4.2 The Manifold of Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
4.3 Existence of Closed Convex Polyhedra with Prescribed
Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
4.4 Existence of Unbounded Convex Polyhedra with
Prescribed Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
4.5 Existence of Unbounded Polyhedra Given the Development
and the Limit Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
5 Gluing and Flexing Polyhedra with Boundary . . . . . . . . . . . 229
5.1 Gluing Polyhedra with Boundary . . . . . . . . . . . . . . . . . . . . . . . 229
5.2 Flexes of Convex Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
5.3 Generalizations of Chapters 4 and 5 . . . . . . . . . . . . . . . . . . . . . 261
6 Conditions for Congruence of Polyhedra with Parallel
Faces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
6.1 Lemmas on Convex Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . 271
6.2 On Linear Combination of Polyhedra . . . . . . . . . . . . . . . . . . . . 281
6.3 Congruence Conditions for Closed Polyhedra . . . . . . . . . . . . . 287
6.4 Congruence Conditions for Unbounded Polyhedra . . . . . . . . . 291
6.5 Another Proof and Generalization of the Theorem on
Unbounded Polyhedra. Polyhedra with Boundary . . . . . . . . . 295
6.6 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
7 Existence Theorems for Polyhedra with Prescribed
Face Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
7.1 Existence of Polyhedra with Prescribed Face Areas . . . . . . . . 311
7.2 Minkowski's Proof of the Existence of Polyhedra with
Prescribed Face Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
7.3 Existence of Unbounded Polyhedra with Prescribed Face
Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
7.4 The General Existence Theorem for Unbounded Polyhedra . 327
7.5 Existence of Convex Polyhedra with Prescribed Support
Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
Contents XI
7.6 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
8 Relationship Between the Congruence Condition for
Polyhedra with Parallel Faces and Other Problems . . . . . 349
8.1 Parallelohedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
8.2 A Polyhedron of Least Area with Fixed Volume . . . . . . . . . . 359
8.3 Mixed Volumes and the Brunn Inequality . . . . . . . . . . . . . . . . 366
9 Polyhedra with Vertices on Prescribed Rays . . . . . . . . . . . . 377
9.1 Closed Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
9.2 Unbounded Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
9.3 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
10 Infinitesimal Rigidity of Convex Polyhedra with
Stationary Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
10.1 Deformation of Polyhedral Angles . . . . . . . . . . . . . . . . . . . . . . 404
10.2 The Strong Cauchy Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
10.3 Stationary Dihedral Angles for Stationary Planar Angles . . 415
10.4 Infinitesimal Rigidity of Polyhedra and Equilibrium of
Hinge Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
10.5 On the Deformation of Developments . . . . . . . . . . . . . . . . . . . 425
10.6 Rigidity of Polyhedra with Stationary Development . . . . . . . 429
10.7 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
11 Infinitesimal Rigidity Conditions for Polyhedra with
Prescribed Face Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
11.1 On Deformations of Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . 439
11.2 Infinitesimal Rigidity Theorems for Polyhedra . . . . . . . . . . . . 445
11.3 Relationship of Infinitesimal Rigidity Theorems with One
Another and with the Theory of Mixed Volumes . . . . . . . . . . 453
11.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
12 Supplements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
12.1 Supplement to Chapter 3: Yu. A. Volkov. An Estimate for
the Deformation of a Convex Surface in Dependence on
the Variation of Its Intrinsic Metric . . . . . . . . . . . . . . . . . . . . . 463
12.2 Supplement to Chapter 4: Yu. A. Volkov. Existence of
Convex Polyhedra with Prescribed Development. I . . . . . . . . 492
12.3 Supplement to Chapter 5: L. A. Shor. On Flexibility of
Convex Polyhedra with Boundary . . . . . . . . . . . . . . . . . . . . . . . 506
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537

Dummy View - NOT TO BE DELETED