How Round Is Your Circle?
Where Engineering and Mathematics Meet
1st Edition
Published on 28. February 2011
320 pages
978-1-4008-3795-3 (ISBN)
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.
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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Persons
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!
Content
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
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
