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

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Overview

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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Editorial Reviews

EMS Newsletter
The book is very nicely printed and contains many nice figures and photographs of physical models, as well as an extensive bibliography. It can be recommended as a formal or recreational lecture both for mathematicians and engineers.
American Scientist - Stan Wagon
There are many books that include ideas or instructions for making mathematical models. What is special about this one is the emphasis on the relation of model- or tool-building with the physical world. The authors have devoted themselves to making wood or metal models of most of the constructions presented; 33 color plates nicely show off their success in this area.
New Scientist - Matthew Killeya
The question posed by this book turns out to be a real toughie, but nevertheless the authors urge you to answer it. This gem of a book tackles several such questions, revealing why they are crucial to engineering and to our understanding of our everyday world. With a nice emphasis on practical experiments, the authors do a refreshing job of bringing out the mathematics you learned in school but sadly never knew why. And they show just how intuitive it can be.
Plus Magazine - Owen Smith
This is a great book for engineers and mathematicians, as well as the interested lay person. Although some of the theoretical mathematics may not be familiar, you can skip it without losing the point. For school teachers and lecturers seeking to inspire, this is a fantastic resource.
Journal of the Society of Model and Experimental Engineers - Norman Billingham
This book is very clearly written and beautifully illustrated, with line drawings and a collection of photographs of practical models. I can strongly recommend it to anyone with a bit of math knowledge and an interest in engineering problems—a terrific book.
LMS Newsletter - John Sharp
This book has many gems and rainbows. . . . The book will appeal to all recreational mathematicians . . . not just because of the way it is written, but also because of the way puzzles, plane dissections and packing and the odd paper folding or origami task are used to bring a point home. . . . More than one copy of this book should be in every school library. . . . It should help to inspire a new generation into mathematics or engineering as well as be accessible to the general reader to show how much mathematics has made the modern world.
Mathematics Teacher - Tim Erickson
This book can be dense, but it is great for dipping into, a rich resource of interesting thinking and project ideas. Bryant and Sangwin, the engineer and the mathematician, must have had a great time putting this book together. Their enthusiasm and humor shine through.
From the Publisher

"There are many books that include ideas or instructions for making mathematical models. What is special about this one is the emphasis on the relation of model- or tool-building with the physical world. The authors have devoted themselves to making wood or metal models of most of the constructions presented; 33 color plates nicely show off their success in this area."--Stan Wagon, American Scientist

"The question posed by this book turns out to be a real toughie, but nevertheless the authors urge you to answer it. This gem of a book tackles several such questions, revealing why they are crucial to engineering and to our understanding of our everyday world. With a nice emphasis on practical experiments, the authors do a refreshing job of bringing out the mathematics you learned in school but sadly never knew why. And they show just how intuitive it can be."--Matthew Killeya, New Scientist

"Mathematics teachers and Sudoku addicts will simply be unable to put the book down. . . . Part magic show, part history lesson, and all about geometry, How Round Is Your Circle? is an eloquent testimonial to the authors' passion for numbers. Perhaps it will spark a similar interest in some young numerophile-to-be."--Civil Engineering

"This is a great book for engineers and mathematicians, as well as the interested lay person. Although some of the theoretical mathematics may not be familiar, you can skip it without losing the point. For school teachers and lecturers seeking to inspire, this is a fantastic resource."--Owen Smith, Plus Magazine

"This book is very clearly written and beautifully illustrated, with line drawings and a collection of photographs of practical models. I can strongly recommend it to anyone with a bit of math knowledge and an interest in engineering problems--a terrific book."--Norman Billingham, Journal of the Society of Model and Experimental Engineers

"This book has many gems and rainbows. . . . The book will appeal to all recreational mathematicians . . . not just because of the way it is written, but also because of the way puzzles, plane dissections and packing and the odd paper folding or origami task are used to bring a point home. . . . More than one copy of this book should be in every school library. . . . It should help to inspire a new generation into mathematics or engineering as well as be accessible to the general reader to show how much mathematics has made the modern world."--John Sharp, LMS Newsletter

"This book can be dense, but it is great for dipping into, a rich resource of interesting thinking and project ideas. Bryant and Sangwin, the engineer and the mathematician, must have had a great time putting this book together. Their enthusiasm and humor shine through."--Tim Erickson, Mathematics Teacher

"The book is very nicely printed and contains many nice figures and photographs of physical models, as well as an extensive bibliography. It can be recommended as a formal or recreational lecture both for mathematicians and engineers."--EMS Newsletter

American Scientist
There are many books that include ideas or instructions for making mathematical models. What is special about this one is the emphasis on the relation of model- or tool-building with the physical world. The authors have devoted themselves to making wood or metal models of most of the constructions presented; 33 color plates nicely show off their success in this area.
— Stan Wagon
New Scientist
The question posed by this book turns out to be a real toughie, but nevertheless the authors urge you to answer it. This gem of a book tackles several such questions, revealing why they are crucial to engineering and to our understanding of our everyday world. With a nice emphasis on practical experiments, the authors do a refreshing job of bringing out the mathematics you learned in school but sadly never knew why. And they show just how intuitive it can be.
— Matthew Killeya
LMS Newsletter
This book has many gems and rainbows. . . . The book will appeal to all recreational mathematicians . . . not just because of the way it is written, but also because of the way puzzles, plane dissections and packing and the odd paper folding or origami task are used to bring a point home. . . . More than one copy of this book should be in every school library. . . . It should help to inspire a new generation into mathematics or engineering as well as be accessible to the general reader to show how much mathematics has made the modern world.
— John Sharp
Plus Magazine
This is a great book for engineers and mathematicians, as well as the interested lay person. Although some of the theoretical mathematics may not be familiar, you can skip it without losing the point. For school teachers and lecturers seeking to inspire, this is a fantastic resource.
— Owen Smith
Journal of the Society of Model and Experimental Engineers
This book is very clearly written and beautifully illustrated, with line drawings and a collection of photographs of practical models. I can strongly recommend it to anyone with a bit of math knowledge and an interest in engineering problems—a terrific book.
— Norman Billingham
Civil Engineering
Mathematics teachers and Sudoku addicts will simply be unable to put the book down. . . . Part magic show, part history lesson, and all about geometry, How Round Is Your Circle? is an eloquent testimonial to the authors' passion for numbers. Perhaps it will spark a similar interest in some young numerophile-to-be.
Mathematics Teacher
This book can be dense, but it is great for dipping into, a rich resource of interesting thinking and project ideas. Bryant and Sangwin, the engineer and the mathematician, must have had a great time putting this book together. Their enthusiasm and humor shine through.
— Tim Erickson
Matthew Killeya
"The question posed by this book turns out to be a real toughie, but nevertheless the authors urge you to answer it. This gem of a book tackles several such questions, revealing why they are crucial to engineering and to our understanding of our everyday world. With a nice emphasis on practical experiments, the authors do a refreshing job of bringing out the mathematics you learned in school but sadly never knew why. And they show just how intuitive it can be."
—New Scientist
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Product Details

  • ISBN-13: 9780691149929
  • Publisher: Princeton University Press
  • Publication date: 3/26/2011
  • Pages: 352
  • Sales rank: 777,649
  • Product dimensions: 6.00 (w) x 9.20 (h) x 1.00 (d)

Meet the Author

John Bryant is a retired chemical engineer. He was lecturer in engineering at the University of Exeter until 1994. Chris Sangwin is lecturer in mathematics at the University of Birmingham. He is the coauthor of "Mathematics Galore!"

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Table of Contents

Preface xiii Acknowledgements xix

Chapter 1: Hard Lines 1
1.1 Cutting Lines 5
1.2 The Pythagorean Theorem 6
1.3 Broad Lines 10
1.4 Cutting Lines 12 1.5 Trial by Trials 15

Chapter 2: How to Draw a Straight Line 17
2.1 Approximate-Straight-Line Linkages 22
2.2 Exact-Straight-Line Linkages 33
2.3 Hart’s Exact-Straight-Line Mechanism 38
2.4 Guide Linkages 39
2.5 Other Ways to Draw a Straight Line 41

Chapter 3: Four-Bar Variations 46
3.1 Making Linkages 49
3.2 The Pantograph 51
3.3 The Crossed Parallelogram 54
3.4 Four-Bar Linkages 56
3.5 The Triple Generation Theorem 59
3.6 How to Draw a Big Circle 60
3.7 Chebyshev’s Paradoxical Mechanism 62

Chapter 4: Building the World’s First Ruler 65
4.1 Standards of Length 66
4.2 Dividing the Unit by Geometry 69
4.3 Building the World’s First Ruler 73
4.4 Ruler Markings 75
4.5 Reading Scales Accurately 81
4.6 Similar Triangles and the Sector 84

Chapter 5: Dividing the Circle 89
5.1 Units of Angular Measurement 92
5.2 Constructing Base Angles via Polygons 95
5.3 Constructing a Regular Pentagon 98
5.4 Building the World’s First Protractor 100
5.5 Approximately Trisecting an Angle 102
5.6 Trisecting an Angle by Other Means 105
5.7 Trisection of an Arbitrary Angle 106
5.8 Origami 110

Chapter 6: Falling Apart 112
6.1 Adding Up Sequences of Integers 112
6.2 Duijvestijn’s Dissection 114
6.3 Packing 117
6.4 Plane Dissections 118
6.5 Ripping Paper 120
6.6 A Homely Dissection 123
6.7 Something More Solid 125
Chapter 7: Follow My Leader 127

Chapter 8: In Pursuit of Coat-Hangers 138
8.1 What Is Area? 141
8.2 Practical Measurement of Areas 149
8.3 Areas Swept Out by a Line 151
8.4 The Linear Planimeter 153
8.5 The Polar Planimeter of Amsler 158
8.6 The Hatchet Planimeter of Prytz 161
8.7 The Return of the Bent Coat-Hanger 165
8.8 Other Mathematical Integrators 170

Chapter 9: All Approximations Are Rational 172
9.1 Laying Pipes under a Tiled Floor 173
9.2 Cogs and Millwrights 178
9.3 Cutting a Metric Screw 180
9.4 The Binary Calendar 182
9.5 The Harmonograph 184
9.6 A Little Nonsense! 187

Chapter 10: How Round Is Your Circle? 188
10.1 Families of Shapes of Constant Width 191
10.2 Other Shapes of Constant Width 193
10.3 Three-Dimensional Shapes of Constant Width 196
10.4 Applications 197
10.5 Making Shapes of Constant Width 202
10.6 Roundness 204
10.7 The British Standard Summit Tests of BS3730 206
10.8 Three-Point Tests 210
10.9 Shapes via an Envelope of Lines 213
10.10 Rotors of Triangles with Rational Angles 218
10.11 Examples of Rotors of Triangles 220
10.12 Modern and Accurate Roundness Methods 224

Chapter 11: Plenty of Slide Rule 227
11.1 The Logarithmic Slide Rule 229
11.2 The Invention of Slide Rules 233
11.3 Other Calculations and Scales 237
11.4 Circular and Cylindrical Slide Rules 240
11.5 Slide Rules for Special Purposes 241
11.6 The Magnameta Oil Tonnage Calculator 245
11.7 Non-Logarithmic Slide Rules 247
11.8 Nomograms 249
11.9 Oughtred and Delamain’s Views on Education 251

Chapter 12: All a Matter of Balance 255
12.1 Stacking Up 255
12.2 The Divergence of the Harmonic Series 259
12.3 Building the Stack of Dominos 261
12.4 The Leaning Pencil and Reaching the Stars 265
12.5 Spiralling Out of Control 267
12.6 Escaping from Danger 269
12.7 Leaning Both Ways! 270
12.8 Self-Righting Stacks 271
12.9 Two-Tip Polyhedra 273
12.10 Uni-Stable Polyhedra 274

Chapter 13: Finding Some Equilibrium 277
13.1 Rolling Uphill 277
13.2 Perpendicular Rolling Discs 279
13.3 Ellipses 287
13.4 Slotted Ellipses 291
13.5 The Super-Egg 292

Epilogue 296
References 297
Index 303

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