Counting on frameworks. Mathematics to aid the design of rigid structures (Q2743935)

The rough sketch of a proposed structure, the first among the many ingredients necessary to design and build a structure of rigid rods and ball joints, is compared to the mathematics presented here, namely the use of the combinatorial information about the object, i.e. the number of rods and the instructions on how these rods are to be combined. NEWLINENEWLINENEWLINEIn chapter 1, Counting on Frameworks, the reader is quickly drawn into the problem by the example of square grids, an easily understood problem occurring naturally in dimension 2. Exercises are placed strategically within the text to make the reader pause and reflect so that no subtlety is left unnoticed. Many important concepts, such as rigidity, motions, and degrees of freedom of a framework are painlessly introduced here. NEWLINENEWLINENEWLINETo reach a higher level of sophistication, Counting on Frameworks is followed by a chapter on graph theory, which introduces all necessary tools in a fashion targeted toward the mission of the book. Basic definitions are presented in an unusual context, which makes reading this chapter interesting even to people well versed in graph theory. NEWLINENEWLINENEWLINEIn the third chapter the combinatorial and geometric aspects of the theory are separated and the concepts of infinitesimal and generic rigidity are introduced. All major theorems are proved rigorously but in an easy to digest fashion, and their algorithmic implications are discussed. The reader experiences here how the theory and the body of knowledge change with the dimension of the ambient space and will be delighted to find himself at the forefront with a clear understanding of open research problems in dimension 3. NEWLINENEWLINENEWLINEThe book ends with a chapter on history and many interesting applications ranging from chemistry to modern art.