''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 structure ...
. It was written by Jack E. Graver and published in 2001 by the
Mathematical Association of America
The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure a ...
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 the s ...
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
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
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 conne ...
, the one-dimensional theory of rigidity through the analysis of the
connected components of graphs, and a reformulation of the
grid bracing In the mathematics of structural rigidity, grid bracing is a problem of adding cross bracing to a square grid to make it into a rigid structure. It can be solved optimally by translating it into a problem in graph theory on the connectivity of bipar ...
problem in terms of connectivity of an associated
bipartite graph
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V are ...
. 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 dom ...
s,
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 (usually ...
, 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 th ...
, 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
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[{{citation
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, date = May 2002
, work = MAA Reviews
, publisher = ]Mathematical Association of America
The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure a ...
, 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
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''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 ...
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, title = Review of ''Counting on Frameworks''
, zbl = 0982.52019
[{{citation
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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