Proofs from THE BOOK, by Martin Aigner and Giinter M. Ziegler. Pp.199. £19. 1999. ISBN 3 540 63698 6 (Springer-Verlag). I would like to begin with the authors' words from the Preface of this book: Paul Erd6s liked to talk about The Book, in which God maintains the perfect proofs for mathematical theorems, following the dictum of G. H. Hardy that there is no permanent place for ugly mathematics. ErdSs also said that 'you need not believe in God but, as a mathematician, you should believe in The Book'. This excellent book is dedicated to the memory of Paul ErdSs. Paul himself wrote several pages but he is not listed as a co-author because of his death in the summer of 1996. The idea of the book is to present beautiful proofs of some of the most exciting and historically important mathematical theorems. The book focuses on human endeavour to make the proofs more and more simple and striking.

Proofs from THE BOOK , by Martin Aigner and Günter M. Ziegler. Pp. 199. £19. 1999. ISBN 3 540 63698 6 (Springer-Verlag).

IN the preface to his "Elliptic Functions", written in 1892, Prof. A. G. Greenhill complained that, although the subject was then nearly seventy years old, it had not made its way into the ordinary curriculum of mathematical study in Great Britain. Fifty years later, in a lecture to the London Mathematical Society, Prof. E. H. Neville declared that, while his contemporaries at Cambridge thirty-five years ago would have regarded the elements of the theory of Jacobian elliptic functions as a subject which every undergraduate should study, the time had now come when the subject was largely neglected. He claimed that the study of Jacobian elliptic functions was being killed by the unnatural way in which the functions were introduced, and that every general principle was stifled by the lack of symmetry and the multitude of special formulae. The book under review is his attempt to restore the Jacobian functions to their proper place in a university curriculum. Jacobian Elliptic Functions By Eric Harold Neville. Pp. xv + 332. (Oxford: Clarendon Press; London: Oxford University Press, 1944.) 25s. net.

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Jacobian Elliptic Functions

We extend the mathematical theory of rigidity of frameworks (graphs embedded in $d$-dimensional space) to consider nonlocal rigidity and flexibility properties. We provide conditions on a framework under which (I) as the framework flexes continuously it must remain inside a small ball, a property we call "almost-rigidity"; (II) any other framework with the same edge lengths must lie outside a much larger ball; (III) if the framework deforms by some given amount, its edge lengths change by a minimum amount; (IV) there is a nearby framework that is prestress stable, and thus rigid. The conditions can be tested efficiently using semidefinite programming. The test is a slight extension of the test for prestress stability of a framework, and gives analytic expressions for the radii of the balls and the edge length changes. Examples illustrate how the theory may be applied in practice, and we provide an algorithm to test for rigidity or almost-rigidity. We briefly discuss how the theory may be applied to tensegrities.

Almost-rigidity of frameworks

Snappability and singularity-distance of pin-jointed body-bar frameworks

Polyhedra called Siamese dipyramids are known to be non-flexible, however their physical models behave like physical models of flexible polyhedra. We discuss a simple mathematical method for explaining the model flexibility of the Siamese dipyramids.

On the model flexibility of Siamese dipyramids

The Jessen orthogonal icosahedron is known to be non-flexible however its physical models behave like physical models of flexible polyhedra. Adapting ideas of Milka, we propose two mathematical approaches for explaining the flexibility of physical models of the Jessen orthogonal icosahedron.

On model flexibility of the Jessen orthogonal icosahedron

We distinguish two particular classes of lightlike surfaces in the Minkowski space M n , which may be viewed as natural analogues of space- and timelike minimal surfaces.

On minimal lightlike surfaces in Minkowski space–time

We describe pseudo-spherical submanifolds with degenerate Bianchi transformation in constant curvature spaces.

Vasyl Gorkavyy

Papers

On model flexibility of the Jessen orthogonal icosahedron

On the model flexibility of Siamese dipyramids

On minimal lightlike surfaces in Minkowski space–time

Pseudo-spherical Submanifolds with Degenerate Bianchi Transformation

On inflating closed mylar shells