Counting On Frameworks: Mathematics to Aid the Design of Rigid Structures by J.E. Graver (Dolciani Mathematical Expositions 25) The Mathematical Association, 2001, 180 pp, £23.95, ISBN 0-88385-331-0.

Scaffolding is used extensively in the building industry. Its design and erection raises a number of interesting questions, both mathematical and engineering. For example, are the rods and planks strong enough to bear the weight of the builders and their equipment and is the scaffolding braced sufficiently to make it rigid? It is the latter question which motivates the material in the book. In real life, of course, nothing is completely rigid, so the frameworks considered by Graver are idealised structures composed of rigid rods joined at their ends by flexible joints. A tetrahedron constructed in this way is clearly rigid, but a cube is not. The dimension of the underlying space is relevant, however: a rectangle with one diagonal is rigid if it lies in a 2-dimensional space, but in three dimensions it could be folded along the diagonal.

The main problems considered in the book are (a) to determine whether a given design will yield a rigid framework and (b) if the framework is non-rigid, what is the minimal number of additional rods needed to brace it and where should they be placed? Current health and safety legislation in the UK places much emphasis on risk assessment and another topic mentioned briefly by the author is relevant here: designing a framework which remains rigid if any one of its rods fails.

The author develops three mathematical models for considering frameworks. The first of these is a degrees of freedom model. This is easy to apply to small structures, but it lacks a rigorous basis and for some special frameworks, it leads to wrong conclusions.

The distance between a pair of joints connected by a rigid rod in a framework remains unchanged no matter how the framework is moved. If the whole framework is non-rigid, however, then the distance between a pair of joints not directly connected by a rod may alter. The author's second model is a standard one in the subject. For each rod, the distance between the coordinates of its endpoints must equal the length of the rod. Each rod, therefore, yields a quadratic equation in the coordinates. Solving a system of quadratic equations is not an easy matter and, when it can be done, the solution set may not be unique. To decide whether the framework itself is rigid, it is necessary to consider the distance between each pair of joints not connected by a rod. If two different solutions yield different distances between one such pair, then the framework is non-rigid.

For his third model, the author introduces a concept of infinitesimal rigidity. This form of rigidity implies ordinary rigidity (but the full proof is beyond the scope of the book). The converse is not true, however, and there is a small number of exceptional frameworks which are rigid but not infinitesimally rigid. This new form of rigidity leads to systems of linear equations which can, of course, be solved easily, so this model leads to correct predictions for ‘generic’ frameworks. But characterising the exceptional frameworks is an unsolved problem.

If the rigidity of the rods in a framework is ignored, and then the resulting object can be regarded as a graph, called the structure graph of the framework. Some elementary graph theory is, therefore, developed in the book.

The final chapter of the book is entitled ‘History and Applications’. One short section deals with linkages, a topic rather unrelated to the main theme of the book. A linkage is a planar framework in which some of the joints are pinned so that they cannot be moved from their initial positions. Other joints can be moved, but the linkage will constrain them to move along certain paths or to lie within certain areas. An important problem is to construct a linkage in which one of the joints moves along a straight line. The solution involves a nice application of geometrical inversion and deserves to be better known. The author bases his account of linkages on A.B. Kempe's classic booklet ‘How to Draw a Straight Line’ first published in 1877.

Other topics considered in the concluding chapter include Geodesic Domes and Tensegrity. In a tensegrity framework, some of the rigid bracings are replaced by wires or cables which must, of course, be under tension in order to serve any purpose in the rigidity of the framework.

Parts of the book can be appreciated with little mathematical knowledge, but for a full understanding of the entire book some knowledge of calculus and linear algebra is required, together with what the author terms ‘comfort with abstract mathematical concepts and with simple proofs’.

The book is generally well written, but in places the style is rather informal and occasionally the informality leads to slight vagueness. For example, in the instructions on p 149 for constructing a particular geodesic dome, the author includes ‘Remove the edges joining two 10-valent vertices’. This is a minor quibble, however, and overall I enjoyed reading the book. It sets out to develop mathematical models for rigidity and it is a useful addition to the literature in that area.

Keith Lloyd

University of Southampton