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

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Princeton University Press, 2008 - Mathematics - 306 pages
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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.

 

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Contents

Hard Lines
1
11 Cutting Lines
5
13 Broad Lines
10
14 Cutting Lines
12
15 Trial by Trials
15
How to Draw a Straight Line
17
21 ApproximateStraightLine Linkages
22
22 ExactStraightLine Linkages
33
87 The Return of the Bent CoatHanger
165
88 Other Mathematical Integrators
170
All Approximations Are Rational
172
91 Laying Pipes under a Tiled Floor
173
92 Cogs and Millwrights
178
93 Cutting a Metric Screw
180
94 The Binary Calendar
182
95 The Harmonograph
184

23 Harts ExactStraightLine Mechanism
38
24 Guide Linkages
39
25 Other Ways to Draw a Straight Line
41
FourBar Variations
46
31 Making Linkages
49
32 The Pantograph
51
33 The Crossed Parallelogram
54
34 FourBar Linkages
56
35 The Triple Generation Theorem
59
36 How to Draw a Big Circle
60
37 Chebyshevs Paradoxical Mechanism
62
Building the Worlds First Ruler
65
41 Standards of Length
66
42 Dividing the Unit by Geometry
69
43 Building the Worlds First Ruler
73
44 Ruler Markings
75
45 Reading Scales Accurately
81
46 Similar Triangles and the Sector
84
Dividing the Circle
89
51 Units of Angular Measurement
92
52 Constructing Base Angles via Polygons
95
53 Constructing a Regular Pentagon
98
54 Building the Worlds First Protractor
100
55 Approximately Trisecting an Angle
102
56 Trisecting an Angle by Other Means
105
57 Trisection of an Arbitrary Angle
106
58 Origami
110
Falling Apart
112
62 Duijvestijns Dissection
114
63 Packing
117
64 Plane Dissections
118
65 Ripping Paper
120
66 A Homely Dissection
123
67 Something More Solid
125
FOLLOW MY LEADER
127
In Pursuit of CoatHangers
138
81 What Is Area?
141
82 Practical Measurement of Areas
149
83 Areas Swept Out by a Line
151
84 The Linear Planimeter
153
85 The Polar Planimeter of Amsler
158
86 The Hatchet Planimeter of Prytz
161
96 A Little Nonsense
187
How Round Is Your Circle?
188
101 Families of Shapes of Constant Width
191
102 Other Shapes of Constant Width
193
103 ThreeDimensional Shapes of Constant Width
196
104 Applications
197
105 Making Shapes of Constant Width
202
106 Roundness
204
107 The British Standard Summit Tests of BS3730
206
108 ThreePoint Tests
210
109 Shapes via an Envelope of Lines
213
1010 Rotors of Triangles with Rational Angles
218
101 1 Examples of Rotors of Triangles
220
Plenty of Slide Rule
227
111 The Logarithmic Slide Rule
229
113 Other Calculations and Scales
237
114 Circular and Cylindrical Slide Rules
240
115 Slide Rules for Special Purposes
241
116 The Magnameta Oil Tonnage Calculator
245
117 NonLogarithmic Slide Rules
247
118 Nomograms
249
119 Oughtred and Delamains Views on Education
251
All a Matter of Balance
255
122 The Divergence of the Harmonic Series
259
123 Building the Stack of Dominos
261
124 The Leaning Pencil and Reaching the Stars
265
125 Spiralling Out of Control
267
126 Escaping from Danger
269
127 Leaning Both Ways
270
128 SelfRighting Stacks
271
129 TwoTip Polyhedra
273
1210 UniStable Polyhedra
274
Finding Some Equilibrium
277
132 Perpendicular Rolling Discs
279
133 Ellipses
287
134 Slotted Ellipses
291
135 The SuperEgg
292
Epilogue
296
References
297
Index
303
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About the author (2008)

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!

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