''Convex Polyhedra'' is a book on the mathematics of convex polyhedra, written by Soviet mathematician

Aleksandr Danilovich Aleksandrov Aleksandr Danilovich Aleksandrov (russian: Алекса́ндр Дани́лович Алекса́ндров, alternative transliterations: ''Alexandr'' or ''Alexander'' (first name), and ''Alexandrov'' (last name)) (4 August 1912 – 27 July 19 ...

, and originally published in Russian in 1950, under the title ''Выпуклые многогранники''. It was translated into German by

Wilhelm Süss Wilhelm Süss (7 March 1895 – 21 May 1958) was a German mathematician. He was founder and first director of the Oberwolfach Research Institute for Mathematics. Biography He was born in Frankfurt, Germany, and died in Freiburg im Breisgau, Germ ...

as ''Konvexe Polyeder'' in 1958. An updated edition, translated into English by Nurlan S. Dairbekov, Semën Samsonovich Kutateladze and Alexei B. Sossinsky, with added material by

Victor Zalgaller Victor (Viktor) Abramovich Zalgaller ( he, ויקטור אבּרמוביץ' זלגלר; russian: Виктор Абрамович Залгаллер; 25 December 1920 – 2 October 2020) was a Russian-Israeli mathematician in the fields of ge ...

, L. A. Shor, and Yu. A. Volkov, was published as ''Convex Polyhedra'' by Springer-Verlag in 2005.

Topics

The main focus of the book is on the specification of geometric data that will determine uniquely the shape of a three-dimensional convex polyhedron, up to some class of geometric transformations such as congruence or similarity. It considers both bounded polyhedra ( convex hulls of finite sets of points) and unbounded polyhedra (intersections of finitely many half-spaces). The 1950 Russian edition of the book included 11 chapters. The first chapter covers the basic topological properties of polyhedra, including their topological equivalence to spheres (in the bounded case) and Euler's polyhedral formula. After a lemma of

Augustin Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...

on the impossibility of labeling the edges of a polyhedron by positive and negative signs so that each vertex has at least four sign changes, the remainder of chapter 2 outlines the content of the remaining book. Chapters 3 and 4 prove

Alexandrov's uniqueness theorem The Alexandrov uniqueness theorem is a rigidity theorem in mathematics, describing three-dimensional convex polyhedra in terms of the distances between points on their surfaces. It implies that convex polyhedra with distinct shapes from each oth ...

, characterizing the surface geometry of polyhedra as being exactly the

metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...

s that are topologically spherical locally like the Euclidean plane except at a finite set of points of positive

angular defect In geometry, the (angular) defect (or deficit or deficiency) means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the Euclidean plane would. The opposite notion is the excess. Classically the defe ...

, obeying Descartes' theorem on total angular defect that the total angular defect should be

4\pi

. Chapter 5 considers the metric spaces defined in the same way that are topologically a disk rather than a sphere, and studies the flexible polyhedral surfaces that result. Chapters 6 through 8 of the book are related to a theorem of Hermann Minkowski that a convex polyhedron is uniquely determined by the areas and directions of its faces, with a new proof based on

invariance of domain Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space \R^n. It states: :If U is an open subset of \R^n and f : U \rarr \R^n is an injective continuous map, then V := f(U) is open in \R^n and f is a homeomorph ...

. A generalization of this theorem implies that the same is true for the perimeters and directions of the faces. Chapter 9 concerns the reconstruction of three-dimensional polyhedra from a two-dimensional perspective view, by constraining the vertices of the polyhedron to lie on rays through the point of view. The original Russian edition of the book concludes with two chapters, 10 and 11, related to Cauchy's theorem that polyhedra with flat faces form rigid structures, and describing the differences between the rigidity and infinitesimal rigidity of polyhedra, as developed analogously to Cauchy's rigidity theorem by

Max Dehn Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German mathematician most famous for his work in geometry, topology and geometric group theory. Born to a Jewish family in Germany, Dehn's early life and career took place in Germany. ...

. The 2005 English edition adds comments and bibliographic information regarding many problems that were posed as open in the 1950 edition but subsequently solved. It also includes in a chapter of supplementary material the translations of three related articles by Volkov and Shor, including a simplified proof of Pogorelov's theorems generalizing Alexandrov's uniqueness theorem to non-polyhedral convex surfaces.

Audience and reception

Robert Connelly Robert Connelly (born July 15, 1942) is a mathematician specializing in discrete geometry and rigidity theory. Connelly received his Ph.D. from University of Michigan in 1969. He is currently a professor at Cornell University. Connelly is best ...

writes that, for a work describing significant developments in the theory of convex polyhedra that was however hard to access in the west, the English translation of ''Convex Polyhedra'' was long overdue. He calls the material on Alexandrov's uniqueness theorem "the star result in the book", and he writes that the book "had a great influence on countless Russian mathematicians". Nevertheless, he complains about the book's small number of exercises, and about an inconsistent level presentation that fails to distinguish important and basic results from specialized technicalities. Although intended for a broad mathematical audience, ''Convex Polyhedra'' assumes a significant level of background knowledge in material including

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

, and

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...

. Reviewer Vasyl Gorkaviy recommends ''Convex Polyhedra'' to students and professional mathematicians as an introduction to the mathematics of convex polyhedra. He also writes that, over 50 years after its original publication, "it still remains of great interest for specialists", after being updated to include many new developments and to list new open problems in the area.

References

{{reflist, refs= {{citation , last = Busemann , first = H. , author-link = Herbert Busemann , journal = Mathematical Reviews , mr = 0040677 , title = Review of ''Выпуклые многогранники'' {{citation , last = Connelly , first = Robert , authorlink = Robert Connelly , date = March 2006 , doi = 10.1137/SIREAD000048000001000149000001 , issue = 1 , journal = SIAM Review , jstor = 20453762 , pages = 157–160 , title = Review of ''Convex Polyhedra'' , url = http://pi.math.cornell.edu/~connelly/alexandrov.pdf, archive-url=https://web.archive.org/web/20170830001838/http://www.math.cornell.edu/~connelly/alexandrov.pdf, archive-date=2017-08-30, url-status=dead , volume = 48 {{citation , last = Gorkaviy , first = Vasyl , journal = zbMATH , title = Review of ''Convex Polyhedra'' , zbl = 1067.52011 {{citation , last = Kaloujnine , first = L. , authorlink = Lev Kaluznin , journal = zbMATH , language = German , title = Review of ''Выпуклые многогранники'' , zbl = 0041.50901 {{zbl, 0079.16303 {{citation , last = Ruane , first = P. N. , date = November 2006 , doi = 10.1017/S002555720018074X , issue = 519 , journal = The Mathematical Gazette , jstor = 40378241 , pages = 557–558 , title = Review of ''Convex Polyhedra'' , volume = 90 Polyhedra Mathematics books 1950 non-fiction books 2005 non-fiction books

Topics

Audience and reception

See also

References