''Counting on Frameworks: Mathematics to Aid the Design of Rigid Structures'' is an undergraduate-level book on the mathematics of

structural rigidity In discrete geometry and mechanics, structural rigidity is a combinatorial theory for predicting the flexibility of ensembles formed by rigid bodies connected by flexible linkages or hinges. Definitions Rigidity is the property of a struct ...

. It was written by Jack E. Graver and published in 2001 by the Mathematical Association of America as volume 25 of the Dolciani Mathematical Expositions book series. The Basic Library List Committee of the Mathematical Association of America has recommended its inclusion by undergraduate mathematics libraries.

Topics

The problems considered by ''Counting on Frameworks'' primarily concern systems of rigid rods, connected to each other by flexible joints at their ends; the question is whether these connections fix such a framework into a single position, or whether it can flex continuously through multiple positions. Variations of this problem include the simplest way to add rods to a framework to make it rigid, or the resilience of a framework against the failure of one of its rods. To study this question, Graver has organized ''Counting on Frameworks'' into four chapters. The first chapter studies

square grid In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of meaning it has 4 squares around every vertex. Conway called it a quadrille. The internal angle of th ...

s and methods of

cross bracing In construction, cross bracing is a system utilized to reinforce building structures in which diagonal supports intersect. Cross bracing is usually seen with two diagonal supports placed in an X-shaped manner. Under lateral force (such as wind or ...

the grid to make it rigid, as a way of introducing the notion of the degrees of freedom of a mechanical system. The second chapter provides an introduction to

graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...

, the one-dimensional theory of rigidity through the analysis of the connected components of graphs, and a reformulation of the grid bracing problem in terms of connectivity of an associated bipartite graph. Chapter three concerns two-dimensional rigidity, the concepts of infinitesimal and generic rigidity, the combinatorial and algorithmic aspects of the subject, and the obstacles to extending this theory to three dimensions. A final chapter describes the history of rigidity theory, applications including mechanical linkages,

geodesic dome A geodesic dome is a hemispherical thin-shell structure (lattice-shell) based on a geodesic polyhedron. The triangular elements of the dome are structurally rigid and distribute the structural stress throughout the structure, making geodesic do ...

tensegrity Tensegrity, tensional integrity or floating compression is a structural principle based on a system of isolated components under compression inside a network of continuous tension, and arranged in such a way that the compressed members (usua ...

, the rigidity of molecules in chemistry, and even art. It also discusses open problems for research in this area.

Audience and reception

''Counting on Frameworks'' expects its readers to be familiar with

multivariable calculus Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving several variables, rather ...

, but beyond that level of background material it does not demand much mathematical sophistication. More generally, the editors of ''

Mathematika ''Mathematika'' is a peer-reviewed mathematics journal that publishes both pure and applied mathematical articles. The journal was founded by Harold Davenport in the 1950s. The journal is published by the London Mathematical Society, on behalf of ...

'' recommend it to "Any reader with at least a slight mathematical background". To avoid demanding too much background of its readers, it is unable to present full proofs of some of its results, instead presenting them as intuitive proof sketches. A more advanced and rigorous treatment of the same material can be found in ''Combinatorial Rigidity'' (1993), a graduate textbook co-authored by Graver. It includes exercises for students, making it suitable as an undergraduate textbook. Reviewer Tiong Seng Tay describes it as "an excellent expository book".

References

{{reflist, refs= {{citation , last = Langton , first = Stacy G. , date = May 2002 , work = MAA Reviews , publisher = Mathematical Association of America , title = Review of ''Counting on Frameworks'' , url = https://www.maa.org/press/maa-reviews/counting-on-frameworks-mathematics-to-aid-the-design-of-rigid-structures {{citation , last = Lloyd , first = E. Keith , date = April 2002 , issue = 303 , journal = The London Mathematical Society Newsletter , pages = 9–10 , title = Review of ''Counting on Frameworks'' , url = https://www.lms.ac.uk/sites/lms.ac.uk/files/files/303%20-%20Apr%202002.pdf#page=9 {{citation , date = June 2002 , doi = 10.1112/s0025579300016223 , issue = 1-2 , journal =

, pages = 253–261 , title = Book Reviews , volume = 49 {{citation , last = Schneider , first = Leo , date = May 2002 , issue = 5 , journal =

The Mathematics Teacher Founded in 1920, The National Council of Teachers of Mathematics (NCTM) is a professional organization for schoolteachers of mathematics in the United States. One of its goals is to improve the standards of mathematics in education. NCTM holds an ...

, jstor = 20871062 , page = 392 , title = Review of ''Counting on Frameworks'' , volume = 95 {{citation , last = Servatius , first = Brigitte , author-link = Brigitte Servatius , work =

zbMATH zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Information Infrastruct ...

, title = Review of ''Counting on Frameworks'' , zbl = 0982.52019 {{citation , last = Tay , first = Tiong Seng , work =

MathSciNet MathSciNet is a searchable online bibliographic database created by the American Mathematical Society in 1996. It contains all of the contents of the journal ''Mathematical Reviews'' (MR) since 1940 along with an extensive author database, links ...

, mr = 1843781 , title = Review of ''Counting on Frameworks'' , year = 2002 Mathematics books 2001 non-fiction books Mathematics of rigidity