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!"

"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

• 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

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!"

"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

• 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

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!"

"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

• 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