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

by John Bryant, Chris Sangwin

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.

Product Details

ISBN-13:	9780691149929
Publisher:	Princeton University Press
Publication date:	03/20/2011
Pages:	320
Product dimensions:	6.00(w) x 9.20(h) x 1.00(d)

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
Hard Lines     1
Cutting Lines     5
The Pythagorean Theorem     6
Broad Lines     10
Cutting Lines     12
Trial by Trials     15
How to Draw a Straight Line     17
Approximate-Straight-Line Linkages     22
Exact-Straight-Line Linkages     33
Hart's Exact-Straight-Line Mechanism     38
Guide Linkages     39
Other Ways to Draw a Straight Line     41
Four-Bar Variations     46
Making Linkages     49
The Pantograph     51
The Crossed Parallelogram     54
Four-Bar Linkages     56
The Triple Generation Theorem     59
How to Draw a Big Circle     60
Chebyshev's Paradoxical Mechanism     62
Building the World's First Ruler     65
Standards of Length     66
Dividing the Unit by Geometry     69
Building the World's First Ruler     73
Ruler Markings     75
Reading Scales Accurately     81
Similar Triangles and the Sector     84
Dividing the Circle     89
Units of Angular Measurement     92
Constructing Base Angles via Polygons     95
Constructing a Regular Pentagon     98
Building the World's First Protractor     100
Approximately Trisecting an Angle     102
Trisecting an Angle by Other Means     105
Trisection of an Arbitrary Angle     106
Origami     110
Falling Apart     112
Adding Up Sequences of Integers     112
Duijvestijn's Dissection     114
Packing     117
Plane Dissections     118
Ripping Paper     120
A Homely Dissection     123
Something More Solid     125
Follow My Leader     127
In Pursuit of Coat-Hangers     138
What Is Area?     141
Practical Measurement of Areas     149
Areas Swept Out by a Line     151
The Linear Planimeter     153
The Polar Planimeter of Amsler     158
The Hatchet Planimeter of Prytz     161
The Return of the Bent Coat-Hanger     165
Other Mathematical Integrators     170
All Approximations Are Rational     172
Laying Pipes under a Tiled Floor      173
Cogs and Millwrights     178
Cutting a Metric Screw     180
The Binary Calendar     182
The Harmonograph     184
A Little Nonsense!     187
How Round Is Your Circle?     188
Families of Shapes of Constant Width     191
Other Shapes of Constant Width     193
Three-Dimensional Shapes of Constant Width     196
Applications     197
Making Shapes of Constant Width     202
Roundness     204
The British Standard Summit Tests of BS3730     206
Three-Point Tests     210
Shapes via an Envelope of Lines     213
Rotors of Triangles with Rational Angles     218
Examples of Rotors of Triangles     220
Modern and Accurate Roundness Methods     224
Plenty of Slide Rule     227
The Logarithmic Slide Rule     229
The Invention of Slide Rules     233
Other Calculations and Scales     237
Circular and Cylindrical Slide Rules     240
Slide Rules for Special Purposes     241
The Magnameta Oil Tonnage Calculator     245
Non-Logarithmic Slide Rules     247
Nomograms      249
Oughtred and Delamain's Views on Education     251
All a Matter of Balance     255
Stacking Up     255
The Divergence of the Harmonic Series     259
Building the Stack of Dominos     261
The Leaning Pencil and Reaching the Stars     265
Spiralling Out of Control     267
Escaping from Danger     269
Leaning Both Ways!     270
Self-Righting Stacks     271
Two-Tip Polyhedra     273
Uni-Stable Polyhedra     274
Finding Some Equilibrium     277
Rolling Uphill     277
Perpendicular Rolling Discs     279
Ellipses     287
Slotted Ellipses     291
The Super-Egg     292
Epilogue     296
References     297
Index     303

What People are Saying About This

David Richeson

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

Nahin

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"

John Mason

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

From the Publisher

"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

"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

"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

Editorial Reviews

"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 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.

Mathematics Teacher - Tim Erickson

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.

LMS Newsletter - John Sharp

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.

Journal of the Society of Model and Experimental Engineers - Norman Billingham

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.

Plus Magazine - Owen Smith

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 - Matthew Killeya

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.

American Scientist - Stan Wagon

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

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 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 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

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

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."
—New Scientist

Matthew Killeya

"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

From the Publisher

"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