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.
|Publisher:||Princeton University Press|
|Product dimensions:||6.00(w) x 9.25(h) x (d)|
|Age Range:||8 - 12 Years|
About 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!
Table of Contents
Preface xiii Acknowledgements xix
Chapter 1: Hard Lines 11.1 Cutting Lines 51.2 The Pythagorean Theorem 61.3 Broad Lines 101.4 Cutting Lines 12 1.5 Trial by Trials 15
Chapter 2: How to Draw a Straight Line 172.1 Approximate-Straight-Line Linkages 222.2 Exact-Straight-Line Linkages 332.3 Hart s Exact-Straight-Line Mechanism 382.4 Guide Linkages 392.5 Other Ways to Draw a Straight Line 41
Chapter 3: Four-Bar Variations 463.1 Making Linkages 493.2 The Pantograph 513.3 The Crossed Parallelogram 543.4 Four-Bar Linkages 563.5 The Triple Generation Theorem 593.6 How to Draw a Big Circle 603.7 Chebyshev s Paradoxical Mechanism 62
Chapter 4: Building the World s First Ruler 654.1 Standards of Length 664.2 Dividing the Unit by Geometry 694.3 Building the World s First Ruler 734.4 Ruler Markings 754.5 Reading Scales Accurately 814.6 Similar Triangles and the Sector 84
Chapter 5: Dividing the Circle 895.1 Units of Angular Measurement 925.2 Constructing Base Angles via Polygons 955.3 Constructing a Regular Pentagon 985.4 Building the World s First Protractor 1005.5 Approximately Trisecting an Angle 1025.6 Trisecting an Angle by Other Means 1055.7 Trisection of an Arbitrary Angle 1065.8 Origami 110
Chapter 6: Falling Apart 1126.1 Adding Up Sequences of Integers 1126.2 Duijvestijn s Dissection 1146.3 Packing 1176.4 Plane Dissections 1186.5 Ripping Paper 1206.6 A Homely Dissection 1236.7 Something More Solid 125Chapter 7: Follow My Leader 127
Chapter 8: In Pursuit of Coat-Hangers 1388.1 What Is Area? 1418.2 Practical Measurement of Areas 1498.3 Areas Swept Out by a Line 1518.4 The Linear Planimeter 1538.5 The Polar Planimeter of Amsler 1588.6 The Hatchet Planimeter of Prytz 1618.7 The Return of the Bent Coat-Hanger 1658.8 Other Mathematical Integrators 170
Chapter 9: All Approximations Are Rational 1729.1 Laying Pipes under a Tiled Floor 1739.2 Cogs and Millwrights 1789.3 Cutting a Metric Screw 1809.4 The Binary Calendar 1829.5 The Harmonograph 1849.6 A Little Nonsense! 187
Chapter 10: How Round Is Your Circle? 18810.1 Families of Shapes of Constant Width 19110.2 Other Shapes of Constant Width 19310.3 Three-Dimensional Shapes of Constant Width 19610.4 Applications 19710.5 Making Shapes of Constant Width 20210.6 Roundness 20410.7 The British Standard Summit Tests of BS3730 20610.8 Three-Point Tests 21010.9 Shapes via an Envelope of Lines 21310.10 Rotors of Triangles with Rational Angles 21810.11 Examples of Rotors of Triangles 22010.12 Modern and Accurate Roundness Methods 224
Chapter 11: Plenty of Slide Rule 22711.1 The Logarithmic Slide Rule 22911.2 The Invention of Slide Rules 23311.3 Other Calculations and Scales 23711.4 Circular and Cylindrical Slide Rules 24011.5 Slide Rules for Special Purposes 24111.6 The Magnameta Oil Tonnage Calculator 24511.7 Non-Logarithmic Slide Rules 24711.8 Nomograms 24911.9 Oughtred and Delamain s Views on Education 251
Chapter 12: All a Matter of Balance 25512.1 Stacking Up 25512.2 The Divergence of the Harmonic Series 25912.3 Building the Stack of Dominos 26112.4 The Leaning Pencil and Reaching the Stars 26512.5 Spiralling Out of Control 26712.6 Escaping from Danger 26912.7 Leaning Both Ways! 27012.8 Self-Righting Stacks 27112.9 Two-Tip Polyhedra 27312.10 Uni-Stable Polyhedra 274
Chapter 13: Finding Some Equilibrium 27713.1 Rolling Uphill 27713.2 Perpendicular Rolling Discs 27913.3 Ellipses 28713.4 Slotted Ellipses 29113.5 The Super-Egg 292
Epilogue 296References 297Index 303
What People are Saying About This
I learned a lot from this book. I think it will have wide appeal, including with those readers who are interested in mathematics and those who are interested in building models. I was up until midnight the other night making a hatchet planimeter out of a coat hanger and washers!
David Richeson, Dickinson College
Truly impressive. This book builds a bridge across the ordinarily huge chasm separating how engineers and mathematicians view the world. Its innovative approach will be refreshing to readers with an engineering bent, and an eye-opener for many mathematicians. The audience for this book includes just about anyone who has any curiosity at all about how mathematics helps in explaining the world.
Paul J. Nahin, author of "An Imaginary Tale"
This book is a mine of exploration and information. I would recommend it to anyone with an interest in how things work and in how mathematics can help make sense of the world. Budding engineers and mathematicians will find it an inspiration.
John Mason, The Open University