Preface

Mathematics and engineering

W. M. Fletcher (tutor) conceived an idea (c. 1912) that Engineering students should be taught some ‘real’ mathematics by the mathematical staff—‘contact with great minds’. The hardworked, hardboiled, and lazy devils hated it as much as I did, to whom, as junior, all dirty work then fell. I asked F. J. Dykes (the sole Lecturer in Engineering) what he would like me to select; all he said was ‘Give the buggers plenty of slide-rule’.

Littlewood (1986, p. 142)

The research mathematician, concerned with purely abstract realms, and the mechanical engineer, who grapples with the practical difficulties of the physical world, are unlikely companions. This book is our attempt to illustrate why mathematicians should take the practical problems of engineering seriously for the real mathematical challenges they present. We encourage them to leave their comfortable world of fictitious thin lines, perfect circles and exact numbers, at least occasionally. To the engineer these simply do not exist, and yet as we shall see they cannot be ignored completely. Furthermore, in order to even perform apparently ‘trivial’ tasks, the engineer owes a lot to mathematics. Moreover, this mathematics is far from the dull repetitive practice of routine tasks which is at least one interpretation of Dyke’s ‘plenty of slide-rule’.

Mathematicians and engineers are disinclined to agree about anything in public: should the area of a circle be described using the neat formula πr² or in terms of the more easily measured diameter as πd², for example? However, there will always be mathematicians who are interested in practical problems and engineers with a serious passion for mathematics. This book, we hope, will be a source of ideas for them and for their students.

In this book we look in some detail at a number of apparently trivial problems, specifically including how to draw a straight line and how to check that something is round. Since circles and straight lines are the simplest geometric curves, these naturally generalize to construction problems for other sorts of curve. There are also associated questions of measurement, which involve constructing lengths and angles. How does one build ‘the first’ protractor, or, at an even more basic level, divide lengths and angles into halves, thirds or any specified ratio? Is this even possible? And, if it is impossible in the strict mathematical sense, when does one need to care or when can an engineering approximation suffice? Most of the topics we have selected are surprising, others are clearly well beyond what we imagine to be possible. Rollers which are not round, and stacks of dominoes which lean arbitrarily far without falling over, for example. We promise that examining these seemingly pointless questions will reveal some of the hidden complexities of the apparently simple.

In many of the chapters we explain how to build physical models or undertake experiments to illustrate instances of these problems. One practical demonstration even illustrates that the mathematics ‘works’. Of course, such an illustration is not a proof as the mathematician knows it. Nevertheless, we believe that making such models and other practical activities are beneficial. This is both from our personal experiences of trying to grasp results that our intuition rejects, and also from working with students and teachers. We also discuss the practical limits associated with both the mathematics and the modelling process. How far may one tilt a stack before it collapses? When does the width of the saw blade matter? These questions inevitably lead to a better understanding of, and appreciation for, the mathematics.

Some experiments may be undertaken in minutes with standard drawing instruments—pencil and paper. Models can be constructed out of commercially available toy sets, or materials that are readily to hand. Others require sophisticated machine tools and workshop facilities. Of course, these latter models are less suitable for classroom construction activities.

We have tried to keep the mathematical explanations intuitive and geometrical, resorting to calculus only when absolutely necessary. Even if the finer points of all the mathematics are not fully explained, we sincerely hope you enjoy the activities and develop a deeper appreciation of some apparently simple topics.

We also wanted to alert you to what were commonplace scientific instruments that have now been rendered obsolete in the recent age of digital technology. The majority of these are analogue devices and most are a cunning mix of the physical and the mathematical. Many are surprising, especially when first encountered. So, before you dismiss them as old hat, we ask that you imagine the reaction of the novice, or the student so immersed in virtual reality that their appreciation of the physical has been impoverished. After all, everything fails to be surprising once it is completely understood! Our experiences of using such instruments to motivate mathematics are very positive indeed. Often such objects (especially slide rules, and occasionally Amsler planimeters) may be obtained second-hand as antiques, sometimes at very little expense. This is currently a practical way for those without workshop facilities or machine tools to obtain a small selection of the best quality mathematical models.

Notes

We suggest practical work with pencil and paper and encourage making models on the kitchen table or in a workshop as ways of bringing more life to mathematics. Our final suggestion is that you go out and see engineering in practice. If we take just one example, the stationary steam engine, we can see one of the straight-line linkages for real in a museum or better still on an original engine preserved in situ. In both instances there is the possibility that they will be running, although perhaps not under steam. Models of these can be seen at many model engineering exhibitions. Most towns have their own model engineering societies and they put on annual displays. There are also the larger regional and national exhibitions, where models run on compressed air. It is often easier to understand the working of a linkage in scale model form than it is when looking at a full-sized prototype, where the sheer physical size of these engines makes it difficult to appreciate them properly from a single viewing position. It may need a walk of several metres to move from one end of a beam to the other on some early pumping engines, or a machine might occupy several floors of a building.

Trade stands at exhibitions provide an excellent opportunity to see what is available in terms of plans, raw materials and tools for you to make your own models. The stewards at model engineering society stands will always provide advice and guidance. Two useful sources of information are Hayes (1990) and the fortnightly publication The Model Engineer, which gives a calendar of events in each issue.

There have, of course, been many previous books that have explored physically motivated mathematics, or illustrated mathematical results with physical models. Perhaps the most famous is Cundy and Rollet (1961), and we would like to record our gratitude for the encouragement which we received from the late H. Martyn Cundy in preparing this work. Other excellent books are those of Cadwell (1966), which is quite mathematical in its approach, and Bolt and Hiscocks (1970), which contains many activities for school students. Other books, such as the very well-known book of Rouse Ball (1960), also contain much in common with our work, although again the emphasis is on the mathematics, not on the relationships with the physical world.

We see great value in making physical models as mathematical experiments, and have done this with the majority of the examples contained in this book. Indeed, the photographed models come from our own collections. We also recommend experimenting with dynamic geometry and computer algebra packages. There are many commercial systems available, but the dynamic geometry package GeoGebra (www.geogebra.at) and the computer algebra system Maxima (maxima.sourceforge.net) are freely available and were used extensively in producing the illustrations for this book.