How Round Is Your Circle Where Engineering and Mathematics Meet 1st Edition by John Bryant 1400837952 9781400837953

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ISBN 10: 1400837952 

ISBN 13: 9781400837953

Author: John Bryant

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:

Chapter 1 Hard Lines
1.1 Cutting Lines
1.2 The Pythagorean Theorem
1.3 Broad Lines
1.4 Cutting Lines
1.5 Trial by Trials

Chapter 2 How to Draw a Straight Line
2.1 Approximate-Straight-Line Linkages
2.2 Exact-Straight-Line Linkages
2.3 Hart’s Exact-Straight-Line Mechanism
2.4 Guide Linkages
2.5 Other Ways to Draw a Straight Line

Chapter 3 Four-Bar Variations
3.1 Making Linkages
3.2 The Pantograph
3.3 The Crossed Parallelogram
3.4 Four-Bar Linkages
3.5 The Triple Generation Theorem
3.6 How to Draw a Big Circle
3.7 Chebyshev’s Paradoxical Mechanism

Chapter 4 Building the World’s First Ruler
4.1 Standards of Length
4.2 Dividing the Unit by Geometry
4.3 Building the World’s First Ruler
4.4 Ruler Markings
4.5 Reading Scales Accurately
4.6 Similar Triangles and the Sector

Chapter 5 Dividing the Circle
5.1 Units of Angular Measurement
5.2 Constructing Base Angles via Polygons
5.3 Constructing a Regular Pentagon
5.4 Building the World’s First Protractor
5.5 Approximately Trisecting an Angle
5.6 Trisecting an Angle by Other Means
5.7 Trisection of an Arbitrary Angle
5.8 Origami

Chapter 6 Falling Apart
6.1 Adding Up Sequences of Integers
6.2 Duijvestijn’s Dissection
6.3 Packing
6.4 Plane Dissections
6.5 Ripping Paper
6.6 A Homely Dissection
6.7 Something More Solid

Chapter 7 Follow My Leader

Chapter 8 In Pursuit of Coat-Hangers
8.1 What Is Area?
8.2 Practical Measurement of Areas
8.3 Areas Swept Out by a Line
8.4 The Linear Planimeter
8.5 The Polar Planimeter of Amsler
8.6 The Hatchet Planimeter of Prytz
8.7 The Return of the Bent Coat-Hanger
8.8 Other Mathematical Integrators

Chapter 9 All Approximations Are Rational
9.1 Laying Pipes under a Tiled Floor
9.2 Cogs and Millwrights
9.3 Cutting a Metric Screw
9.4 The Binary Calendar
9.5 The Harmonograph
9.6 A Little Nonsense!

Chapter 10 How Round Is Your Circle?
10.1 Families of Shapes of Constant Width
10.2 Other Shapes of Constant Width
10.3 Three-Dimensional Shapes of Constant Width
10.4 Applications
10.5 Making Shapes of Constant Width
10.6 Roundness
10.7 The British Standard Summit Tests of BS3730
10.8 Three-Point Tests
10.9 Shapes via an Envelope of Lines
10.10 Rotors of Triangles with Rational Angles
10.11 Examples of Rotors of Triangles
10.12 Modern and Accurate Roundness Methods

Chapter 11 Plenty of Slide Rule
11.1 The Logarithmic Slide Rule
11.2 The Invention of Slide Rules
11.3 Other Calculations and Scales
11.4 Circular and Cylindrical Slide Rules
11.5 Slide Rules for Special Purposes
11.6 The Magnameta Oil Tonnage Calculator
11.7 Non-Logarithmic Slide Rules
11.8 Nomograms
11.9 Oughtred and Delamain’s Views on Education

Chapter 12 All a Matter of Balance
12.1 Stacking Up
12.2 The Divergence of the Harmonic Series
12.3 Building the Stack of Dominos
12.4 The Leaning Pencil and Reaching the Stars
12.5 Spiralling Out of Control
12.6 Escaping from Danger
12.7 Leaning Both Ways!
12.8 Self-Righting Stacks
12.9 Two-Tip Polyhedra
12.10 Uni-Stable Polyhedra

Chapter 13 Finding Some Equilibrium
13.1 Rolling Uphill
13.2 Perpendicular Rolling Discs
13.3 Ellipses
13.4 Slotted Ellipses
13.5 The Super-Egg

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Tags: John Bryant, Round, Circle, Engineering, Mathematics