Counting on Frameworks: Mathematics to Aid the Design of Rigid Structures by Jack E. Graver, Jack Graver

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Table of Contents

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Counting on Frameworks
1.1 Designing Structures
Think about building something. The process of building frequently starts like this: First, we have a mental image of the object we wish to build; next, we make a rough sketch; finally, after several more intervening steps, the object, or perhaps a prototype, is constructed. At this stage we can test whether or not the object has all of the properties that our mental image possessed. Discovering at this stage that one or more critical properties is not satisfied is somewhat of a disaster, involving a great deal of wasted effort. Careful planning is essential to reducing the probability of such disasters.
 We will restrict our attention to building structures of a special type and will concentrate on the planning stage. To be specific, the structures we will consider are constructed out of rigid rods connected together at their ends. Look at the cube sketched in Figure 1.1. The "rigidity" of this structure depends entirely on the strength of the joints in maintaining the angles between the rods. Under enough pressure the faces will distort and the structure will collapse. The rigidity of the tetrahedron, on the other hand, does not depend on the joints in this way. Its joints can be flexible and the tetrahedron will still be rigid.
 It is in this sense that we will use the term rigid: A structure made from rods will be rigid if it resists distortion even when all of its joints are flexible. And herein lies our problem:
Can we accurately predict from the rough sketch alone whether or not the finished structure will be rigid?
With the cube and tetrahedron, we are confident in our predictions. But having accurately predicted that the unbraced cube is not rigid, suppose that we would like to add sufficient braces to make it rigid. How should we proceed?
 In Figure 1.2, we have drawn three different bracing schemes. We can argue that the left-hand bracing results in a rigid cube: It is made up of five tetrahedral, one in the center whose edges are the six braces and four other tetrahedral, each sharing a face with the central one. In the second proposed bracing, the braces in the top and bottom faces of the cube have been replaced by the opposite diagonals. What can we say now? With no tetrahedral in sight, can we be sure of any prediction we may make? The third proposed bracing is even more problematic. We have to be careful here to be sure that we understand what the picture is intended to portray. Since we are bracing the cube, no further joints have been introduces and the diagonal braces cross one another but are not connected; they would be permitted to slide across each other if this braced cube were not rigid.
Exercise 1.1 Before you read any further, make your prediction for the rigidity of the second and third structures sketched in the figure.
 To make a "practical" problem out of these examples think of such a cube as a base for a glass topped coffee table. The rods and rubber joints can be easily mailed to the consumer who can then put them together, purchasing the glass top locally. In the spirit of this interpretation, let's answer some of our questions by building a model.
 Models of the first two proposed bracings are easy to construct. Using a standard model building kit, we have built these models and photographed them (Figure 1.3). If you were to build such models yourself, you would verify directly that they were rigid.
 It is not so easy to build a model based on the third sketch. Actual rods have thickness, and at the crossing point in the center they must be bent slightly. So it is impossible to build an undistorted model. On the other hand, if you did manage to build a model that was not too badly distorted, it would seem to be rigid. However, it would not seem to have the robust rigidity of the first two braced cubes, and you might conclude that a coffee table of this design is not such a good idea. You might also conclude that there should be a better way to decide rigidity questions than by building models. In fact, rigidity is clearly one of those properties we would like to be sur5e of long before we get to the building stage.
 The main purpose of this book is to explore the mathematical theory of rigidity and to develop methods for predicting rigidity early in the planning stage of a structure. The only mathematical knowledge expected of the reader is that usually covered in a standard calculus sequence and a first course in linear algebra (an introduction to the vector space of the n-tuples of real numbers, to be specific). Because of this limitation on the required background, the reader will be asked to accept a few of the results without proof. But even these missing proofs can be motivated and explained in part with the tools available to us.
 While the rigidity of 3-dimensional structures is the initial stimulus for this study, we will spend a lot of time and effort considering rigidity in lower dimensions. Just as an extensive knowledge of plane geometry is necessary for a good understanding of solid geometry, an extensive knowledge of plane rigidity is necessary for a good understanding of spatial rigidity. In fact, 1-dimensional rigidity will play an important role in our development. In the next section, we start our investigation by considering a very special problem in planar rigidity: the grid bracing  problem.