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How Round Is Your Circle?

How Round Is Your Circle?: Where Engineering and Mathematics Meet

John Bryant
Chris Sangwin
Copyright Date: 2008
Pages: 352
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  • Book Info
    How Round Is Your Circle?
    Book Description:

    How do you draw a straight line? How do you determine if a circle is really round? These may sound like simple or even trivial mathematical problems, but to an engineer the answers can mean the difference between success and failure.How Round Is Your Circle?invites readers to explore many of the same fundamental questions that working engineers deal with every day--it's challenging, hands-on, and fun.

    John Bryant and Chris Sangwin illustrate how physical models are created from abstract mathematical ones. Using elementary geometry and trigonometry, they guide readers through paper-and-pencil reconstructions of mathematical problems and show them how to construct actual physical models themselves--directions included. It's an effective and entertaining way to explain how applied mathematics and engineering work together to solve problems, everything from keeping a piston aligned in its cylinder to ensuring that automotive driveshafts rotate smoothly. Intriguingly, checking the roundness of a manufactured object is trickier than one might think. When does the width of a saw blade affect an engineer's calculations--or, for that matter, the width of a physical line? When does a measurement need to be exact and when will an approximation suffice? Bryant and Sangwin tackle questions like these and enliven their discussions with many fascinating highlights from engineering history. Generously illustrated,How Round Is Your Circle?reveals some of the hidden complexities in everyday things.

    eISBN: 978-1-4008-3795-3
    Subjects: Mathematics, Technology

Table of Contents

  1. Front Matter
    (pp. i-viii)
  2. Table of Contents
    (pp. ix-xii)
  3. Preface
    (pp. xiii-xviii)
  4. Acknowledgements
    (pp. xix-xxii)
  5. Chapter 1 HARD LINES
    (pp. 1-16)

    There are many convincing ways to justify a result. A scientist gathersevidenceby undertaking a systematic experiment. One can undertake mathematical experiments, such as a sequence of calculations. Another kind of experiment is to draw a picture, be it on paper or sketched in the sand with a stick. Few, if any, mathematicians would now accept a picture as a validproofbut sketches do provide us with the simplest and most direct form ofmathematical experiment. When undertaking such an experiment we ask you to think of it as representing a whole class of similar ones. What can...

    (pp. 17-45)

    This chapter heading comes from a book by A. B. Kempe,How to Draw a Straight Line: A Lecture on Linkages(Kempe 1877). The charm and gentle tone of the book suggest that the original lectures must have been a delightful experience. If you ask most people what a straight line is their answer will probably involve ‘the shortest distance between two points’. Of course they do not always mean this since the shortest distance to the Antipodes from Yorkshire is downwards through the centre of the Earth: these two places are at opposite ends of a diameter. What they...

    (pp. 46-64)

    Linkages are such an important topic that we cannot restrict our attention solely to drawing a straight line. The simplest linkage mechanism contains only three obvious bars, and a fourth if you count the fixed base. These found an application in generating an approximate straight line. Of course, applications of four-bar variations are not restricted to generation of approximate straight lines. We shall examine these linkages in more detail and consider the curves that four-bar linkages actually generate.

    It is not possible to attempt to list all the other uses here, so instead we shall take a small sample for...

    (pp. 65-88)

    In chapter 2 we considered how to draw a straight line. This turned out to be a little more difficult than one might assume. What we shall do in this chapter is turn this straight edge into a graduated ruler, so that we can actually begin to measure, and in order for this ruler to be useful we should ensure that the graduations are equally spaced. Just as we cannot use a straight edge to draw the first straight line, so we shall not be able to use an existing ruler to mark out the graduations on our first ruler....

    (pp. 89-111)

    Length is a measure of the extent of a one-dimensional line. In defining length in chapter 4 we had considerable freedom in taking an arbitrary unit length. It was then a geometric task to divide this into fractions and to build the world’s first ruler.

    In this chapter we measure the extent ofrotation, and here we have a natural base unit of one complete rotation. The choice now is how to divide the circle. The quotation from Geoffrey Chaucer (1343–1400) above, from the first scientific book published in England in the vernacular rather than in Latin, describes dividing...

  10. Chapter 6 FALLING APART
    (pp. 112-126)

    In this chapter we discuss mathematical jigsaws based on plane dissections and other solid models. Not only are these fun, but they also illustrate some important basic mathematical results. Mathematical dissections such as these all rely on the concepts of area and volume. We are concerned here with simple geometric shapes but we shall address the more general problem of measuring the area of an irregular shape in chapter 8.

    There is a story about the famous mathematician Carl Friedrich Gauss (1777–1855) that while at primary school he was able to add the numbers from 1 to 100 in...

  11. Chapter 7 FOLLOW MY LEADER
    (pp. 127-137)

    The principal characters in this chapter are two closely related transcendental curves, the tractrix and the catenary, with a strong supporting role being played by a conic section, the parabola. The ways in which the curves are formed are described by the Latin roots of their names:tractare, to pull or draw, andcatena, a chain. The tractrix can be demonstrated most conveniently by pulling a weight across a horizontal table by means of a thread. The only way thread can exert any force on the weight is by traction, and in figure 7.1 if the weight starts from A₀...

    (pp. 138-171)

    In previous chapters we have examined how to draw a straight line and the problems concerned with marking a ruler. In each case we discovered some subtle features associated with these simple ideas. Now we turn our attention to the problem of ascertaining the area of a plane shape.

    Area can certainly be measured with complex scientific instruments, such as those shown in figure 8.19 or in figure 8.25. Seeing the undoubted precision involved in their manufacture it may come as a surprise that we are able to begin this description of ahatchet planimeterwith details of how to...

    (pp. 172-187)

    Imagine that you have a secret so special, so shocking in its implications, that the members of an organization to which you belong are prepared to kill to keep it. This was exactly the situation in Greece in the fifth century b.c.e., approximately a hundred yearsbeforeEuclid wrote hisElements. The organization was the Pythagorean brotherhood, founded around 540 b.c.e. by Pythagoras of Samos (580–500 b.c.e.). They were a monastic organization dedicated to mathematics and number worship who studied numbers and their properties. Their starting point was a view of the world which derived from concrete whole numbers,...

  14. Chapter 10 HOW ROUND IS YOUR CIRCLE?
    (pp. 188-226)

    Imagine you wish to make a circular or spherical object but cannot simply use compasses to mark one out. Alternatively, you might have already made something round and wish to check the accuracy of your construction. We all know what a circle is, so here we discuss the problem of determining if what is supposed to be circular is really circular, and within what limits. This matters a lot in engineering applications of all sorts but particularly with rotating shafts and their bearings.

    So let us begin with an experiment for which you need a United Kingdom 50p (or 20p)...

  15. Chapter 11 PLENTY OF SLIDE RULE
    (pp. 227-254)

    We want to ask you to imagine you are the captain of a supertanker that is docked in port and being loading with oil. Your tanker has a number of separate tanks, perhaps of different volumes, and oil is being pumped aboard. If you decide to fill one tank at a time then the ship will capsize, so you need to balance the ship by part-filling tanks around the vessel. Oil varies in density, and so you cannot rely on a set of pre-prepared tables since you may not know how heavy the oil is until you start the job....

  16. Chapter 12 ALL A MATTER OF BALANCE
    (pp. 255-276)

    This chapter starts our discussion ofbalance. We begin by balancing identical objects one on top of the other in such a way that they do not fall over. Then we move on to other problems in a similar vein. Rather than asking how to make something that balances, these ask how we can make a solid object which will not tip over when placed on a table.

    To begin the chapter we propose another experiment: for this you will need a large supply of identical block-like solid objects—dominoes are ideal and CD cases also work very well. You...

    (pp. 277-295)

    The words of the song are true enough though eventually the coin will lose its rotational energy, fall to one side and stop. It rolls, as does any circular disc, cylinder or ball, because its centre of gravity remains a constant height above a horizontal surface independent of position.

    Following on from the last chapter, which was all about balance, we consider a number of related rolling problems. The first involves two joined cones which appear to roll up a slope. Secondly we show a way of joining two identical discs so that they will roll with no preferred position....

  18. Epilogue
    (pp. 296-296)

    In this book we have tried to illustrate why mathematicians should take the practical problems of engineering seriously. We have also tried to illustrate some mathematics by suggesting experiments and the construction of mathematical models.

    We have not provided a systematic treatment of the Platonic solids: this is a topic that is so comprehensively covered in Cundy and Rollet (1961), and in many other books, that we have little if anything new to add. Similarly, we do not include anything about sundials, which are an excellent source of basic mathematical activities. However, this is extensive enough a field to deserve...

  19. References
    (pp. 297-302)
  20. Index
    (pp. 303-306)