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

Princeton University Press, 2008 - Mathematics - 306 pages

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.

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Contents

 Hard Lines 1 11 Cutting Lines 5 13 Broad Lines 10 14 Cutting Lines 12 15 Trial by Trials 15 How to Draw a Straight Line 17 21 ApproximateStraightLine Linkages 22 22 ExactStraightLine Linkages 33
 87 The Return of the Bent CoatHanger 165 88 Other Mathematical Integrators 170 All Approximations Are Rational 172 91 Laying Pipes under a Tiled Floor 173 92 Cogs and Millwrights 178 93 Cutting a Metric Screw 180 94 The Binary Calendar 182 95 The Harmonograph 184

 23 Harts ExactStraightLine Mechanism 38 24 Guide Linkages 39 25 Other Ways to Draw a Straight Line 41 FourBar Variations 46 31 Making Linkages 49 32 The Pantograph 51 33 The Crossed Parallelogram 54 34 FourBar Linkages 56 35 The Triple Generation Theorem 59 36 How to Draw a Big Circle 60 37 Chebyshevs Paradoxical Mechanism 62 Building the Worlds First Ruler 65 41 Standards of Length 66 42 Dividing the Unit by Geometry 69 43 Building the Worlds First Ruler 73 44 Ruler Markings 75 45 Reading Scales Accurately 81 46 Similar Triangles and the Sector 84 Dividing the Circle 89 51 Units of Angular Measurement 92 52 Constructing Base Angles via Polygons 95 53 Constructing a Regular Pentagon 98 54 Building the Worlds First Protractor 100 55 Approximately Trisecting an Angle 102 56 Trisecting an Angle by Other Means 105 57 Trisection of an Arbitrary Angle 106 58 Origami 110 Falling Apart 112 62 Duijvestijns Dissection 114 63 Packing 117 64 Plane Dissections 118 65 Ripping Paper 120 66 A Homely Dissection 123 67 Something More Solid 125 FOLLOW MY LEADER 127 In Pursuit of CoatHangers 138 81 What Is Area? 141 82 Practical Measurement of Areas 149 83 Areas Swept Out by a Line 151 84 The Linear Planimeter 153 85 The Polar Planimeter of Amsler 158 86 The Hatchet Planimeter of Prytz 161
 96 A Little Nonsense 187 How Round Is Your Circle? 188 101 Families of Shapes of Constant Width 191 102 Other Shapes of Constant Width 193 103 ThreeDimensional Shapes of Constant Width 196 104 Applications 197 105 Making Shapes of Constant Width 202 106 Roundness 204 107 The British Standard Summit Tests of BS3730 206 108 ThreePoint Tests 210 109 Shapes via an Envelope of Lines 213 1010 Rotors of Triangles with Rational Angles 218 101 1 Examples of Rotors of Triangles 220 Plenty of Slide Rule 227 111 The Logarithmic Slide Rule 229 113 Other Calculations and Scales 237 114 Circular and Cylindrical Slide Rules 240 115 Slide Rules for Special Purposes 241 116 The Magnameta Oil Tonnage Calculator 245 117 NonLogarithmic Slide Rules 247 118 Nomograms 249 119 Oughtred and Delamains Views on Education 251 All a Matter of Balance 255 122 The Divergence of the Harmonic Series 259 123 Building the Stack of Dominos 261 124 The Leaning Pencil and Reaching the Stars 265 125 Spiralling Out of Control 267 126 Escaping from Danger 269 127 Leaning Both Ways 270 128 SelfRighting Stacks 271 129 TwoTip Polyhedra 273 1210 UniStable Polyhedra 274 Finding Some Equilibrium 277 132 Perpendicular Rolling Discs 279 133 Ellipses 287 134 Slotted Ellipses 291 135 The SuperEgg 292 Epilogue 296 References 297 Index 303 Copyright