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How Round Is Your Circle?

Book 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.

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Contents
  5. Preface
  6. Acknowledgements
  7. Chapter 1 Hard Lines
    1. 1.1 Cutting Lines
    2. 1.2 The Pythagorean Theorem
    3. 1.3 Broad Lines
    4. 1.4 Cutting Lines
    5. 1.5 Trial by Trials
  8. Chapter 2 How to Draw a Straight Line
    1. 2.1 Approximate-Straight-Line Linkages
    2. 2.2 Exact-Straight-Line Linkages
    3. 2.3 Hart’s Exact-Straight-Line Mechanism
    4. 2.4 Guide Linkages
    5. 2.5 Other Ways to Draw a Straight Line
  9. Chapter 3 Four-Bar Variations
    1. 3.1 Making Linkages
    2. 3.2 The Pantograph
    3. 3.3 The Crossed Parallelogram
    4. 3.4 Four-Bar Linkages
    5. 3.5 The Triple Generation Theorem
    6. 3.6 How to Draw a Big Circle
    7. 3.7 Chebyshev’s Paradoxical Mechanism
  10. Chapter 4 Building the World’s First Ruler
    1. 4.1 Standards of Length
    2. 4.2 Dividing the Unit by Geometry
    3. 4.3 Building the World’s First Ruler
    4. 4.4 Ruler Markings
    5. 4.5 Reading Scales Accurately
    6. 4.6 Similar Triangles and the Sector
  11. Chapter 5 Dividing the Circle
    1. 5.1 Units of Angular Measurement
    2. 5.2 Constructing Base Angles via Polygons
    3. 5.3 Constructing a Regular Pentagon
    4. 5.4 Building the World’s First Protractor
    5. 5.5 Approximately Trisecting an Angle
    6. 5.6 Trisecting an Angle by Other Means
    7. 5.7 Trisection of an Arbitrary Angle
    8. 5.8 Origami
  12. Chapter 6 Falling Apart
    1. 6.1 Adding Up Sequences of Integers
    2. 6.2 Duijvestijn’s Dissection
    3. 6.3 Packing
    4. 6.4 Plane Dissections
    5. 6.5 Ripping Paper
    6. 6.6 A Homely Dissection
    7. 6.7 Something More Solid
  13. Chapter 7 Follow My Leader
  14. Chapter 8 In Pursuit of Coat-Hangers
    1. 8.1 What Is Area?
    2. 8.2 Practical Measurement of Areas
    3. 8.3 Areas Swept Out by a Line
    4. 8.4 The Linear Planimeter
    5. 8.5 The Polar Planimeter of Amsler
    6. 8.6 The Hatchet Planimeter of Prytz
    7. 8.7 The Return of the Bent Coat-Hanger
    8. 8.8 Other Mathematical Integrators
  15. Chapter 9 All Approximations Are Rational
    1. 9.1 Laying Pipes under a Tiled Floor
    2. 9.2 Cogs and Millwrights
    3. 9.3 Cutting a Metric Screw
    4. 9.4 The Binary Calendar
    5. 9.5 The Harmonograph
    6. 9.6 A Little Nonsense!
  16. Chapter 10 How Round Is Your Circle?
    1. 10.1 Families of Shapes of Constant Width
    2. 10.2 Other Shapes of Constant Width
    3. 10.3 Three-Dimensional Shapes of Constant Width
    4. 10.4 Applications
    5. 10.5 Making Shapes of Constant Width
    6. 10.6 Roundness
    7. 10.7 The British Standard Summit Tests of BS3730
    8. 10.8 Three-Point Tests
    9. 10.9 Shapes via an Envelope of Lines
    10. 10.10 Rotors of Triangles with Rational Angles
    11. 10.11 Examples of Rotors of Triangles
    12. 10.12 Modern and Accurate Roundness Methods
  17. Chapter 11 Plenty of Slide Rule
    1. 11.1 The Logarithmic Slide Rule
    2. 11.2 The Invention of Slide Rules
    3. 11.3 Other Calculations and Scales
    4. 11.4 Circular and Cylindrical Slide Rules
    5. 11.5 Slide Rules for Special Purposes
    6. 11.6 The Magnameta Oil Tonnage Calculator
    7. 11.7 Non-Logarithmic Slide Rules
    8. 11.8 Nomograms
    9. 11.9 Oughtred and Delamain’s Views on Education
  18. Chapter 12 All a Matter of Balance
    1. 12.1 Stacking Up
    2. 12.2 The Divergence of the Harmonic Series
    3. 12.3 Building the Stack of Dominos
    4. 12.4 The Leaning Pencil and Reaching the Stars
    5. 12.5 Spiralling Out of Control
    6. 12.6 Escaping from Danger
    7. 12.7 Leaning Both Ways!
    8. 12.8 Self-Righting Stacks
    9. 12.9 Two-Tip Polyhedra
    10. 12.10 Uni-Stable Polyhedra
  19. Chapter 13 Finding Some Equilibrium
    1. 13.1 Rolling Uphill
    2. 13.2 Perpendicular Rolling Discs
    3. 13.3 Ellipses
    4. 13.4 Slotted Ellipses
    5. 13.5 The Super-Egg
  20. Epilogue
  21. References
  22. Index