100 Essential Things You Didn't Know You Didn't Know
Contents
Cover
About the Book
About the Author
Also by John Barrow
Dedication
Title Page
Epigraph
Preface
1. Pylon of the Month
2. A Sense of Balance
3. Monkey Business
4. Independence Day
5. Rugby and Relativity
6. Wagons Roll
7. A Sense of Proportion
8. Why Does the Other Queue Always Move Faster?
9. Two’s Company, Three’s a Crowd
10. It’s a Small World After All
11. Bridging That Gap
12. On the Cards
13. Tally Ho
14. Relationships
15. Racing Certainties
16. High Jumping
17. Superficiality
18. VAT in Eternity
19. Living in a Simulation
20. Emergence
21. How to Push a Car
22. Positive Feedback
23. The Drunkard’s Walk
24. Faking It
25. The Flaw of Averages
26. The Origami of the Universe
27. Easy and Hard Problems
28. Is This a Record?
29. A Do-It-Yourself Lottery
30. I Do Not Believe It!
31. Flash Fires
32. The Secretary Problem
33. Fair Divorce Settlements: the Win–Win Solution
34. Many Happy Returns
35. Tilting at Windmills
36. Verbal Conjuring
37. Financial Investment with Time Travellers
38. A Thought for Your Pennies
39. Breaking the Law of Averages
40. How Long are Things Likely to Survive?
41. A President who Preferred the Triangle to the Pentagon
42. Secret Codes in Your Pocket
43. I’ve Got a Terrible Memory for Names
44. Calculus Makes You Live Longer
45. Getting in a Flap
46. Your Number’s Up
47. Double Your Money
48. Some Reflections on Faces
49. The Most Infamous Mathematician
50. Roller Coasters and Motorway Junctions
51. A Taylor-made Explosion
52. Walk Please, Don’t Run!
53. Mind-reading Tricks
54. The Planet of the Deceivers
55. How to Win the Lottery
56. A Truly Weird Football Match
57. An Arch Problem
58. Counting in Eights
59. Getting a Mandate
60. The Two-headed League
61. Creating Something out of Nothing
62. How to Rig An Election
63. The Swing of the Pendulum
64. A Bike with Square Wheels
65. How Many Guards Does an Art Gallery Need?
66. . . . and What About a Prison?
67. A Snooker Trick Shot
68. Brothers and Sisters
69. Playing Fair with a Biased Coin
70. The Wonders of Tautology
71. What a Racket
72. Packing Your Stuff
73. Sent Packing Again
74. Crouching Tiger
75. How the Leopard Got His Spots
76. The Madness of Crowds
77. Diamond Geezer
78. The Three Laws of Robotics
79. Thinking Outside the Box
80. Googling in the Caribbean – The Power of the Matrix
81. Loss Aversion
82. The Lead in Your Pencil
83. Testing Spaghetti to Destruction
84. The Gherkin
85. Being Mean with the Price Index
86. Omniscience can be a Liability
87. Why People aren’t Cleverer
88. The Man from Underground
89. There are No Uninteresting Numbers
90. Incognito
91. The Ice Skating Paradox
92. The Rule of Two
93. Segregation and Micromotives
94. Not Going with the Flow
95. Venn Vill They Ever Learn
96. Some Benefits of Irrationality
97. Strange Formulae
98. Chaos
99. All Aboard
100. The Global Village
Notes
Copyright
About the Book
‘If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.’ John von Neumann
Mathematics can tell you things about the world that can’t be learned in any other way. This hugely informative and wonderfully entertaining little book answers one hundred essential questions about existence. It unravels the knotty, clarifies the conundrums and sheds light into dark corners. From winning the lottery, placing bets at the races and escaping from bears to sports, Shakepeare, Google, game theory, drunks, divorce settlements and dodgy accounting; from chaos to infinity and everything in between, 100 Essential Things You Didn’t Know You Didn’t Know has all the answers!
About the Author
John D. Barrow is Professor of Mathematical Sciences and Director of the Millenium Mathematics Project at Cambridge University, Fellow of Clare Hall, Cambridge, a Fellow of the Royal Society, and the current Gresham Professor of Geometry at Gresham College, London. His previous books include The Origin of the Universe; The Universe that Discovered Itself; The Book of Nothing; The Constants of Nature: From Alpha to Omega; The Infinite Book: A Short Guide to the Boundless, Timeless and Endless; The Artful Universe Expanded; New Theories of Everything and, most recently, Cosmic Imagery: Key Images in the History of Science. He is also the author of the award-winning play Infinities.
Also by John Barrow
Theories of Everything
The Left Hand of Creation
(with Joseph Silk)
L’Homme et le Cosmos
(with Frank J. Tipler)
The Anthropic Cosmological Principle
(with Frank J. Tipler)
The World within the World
The Artful Universe
Pi in the Sky
Perchè il mondo è matematico?
Impossibility
The Origin of the Universe
Between Inner Space and Outer Space
The Universe that Discovered Itself
The Book of Nothing
The Constants of Nature:
From Alpha to Omega
The Infinite Book:
A Short Guide to the Boundless,
Timeless and Endless
Cosmic Imagery:
Key Images in the History of Science
100 Essential Things You Didn’t Know
You Didn’t Know About Sport
The Book of Universes
To David and Emma
I continued to do arithmetic with my father, passing proudly through fractions to decimals. I eventually arrived at the point where so many cows ate so much grass, and tanks filled with water in so many hours. I found it quite enthralling.
Agatha Christie
Preface
This is a little book of bits and pieces – bits about off-beat applications of mathematics to everyday life, and pieces about a few other things not so very far away from it. There are a hundred to choose from, in no particular order, with no hidden agenda and no invisible thread. Sometimes you will find only words, but sometimes you will find some numbers as well, and very occasionally a few further notes that show you some of the formulae behind the appearances. Maths is
interesting and important because it can tell you things about the world that you can’t learn in any other way. When it comes to the depths of fundamental physics or the breadth of the astronomical universe we have almost come to expect that. But I hope that here you will see how simple ideas can shed new light on all sorts of things that might otherwise seem boringly familiar or just pass by unnoticed.
Lots of the examples contained in the pages to follow were stimulated by the goals of the Millennium Mathematics Projectfn1, which I came to Cambridge to direct in 1999. The challenge of showing how mathematics has something to tell about most things in the world around us is one that, when met, can play an important part in motivating and informing people, young and old, to appreciate and understand the place of mathematics at the root of our understanding of the world.
I would like to thank Steve Brams, Marianne Freiberger, Jenny Gage, John Haigh, Jörg Hensgen, Helen Joyce, Tom Körner, Imre Leader, Drummond Moir, Robert Osserman, Jenny Piggott, David Spiegelhalter, Will Sulkin, Rachel Thomas, John H. Webb, Marc West, and Robin Wilson for helpful discussions, encouragement, and other practical inputs that contributed to the final collection of essential things you now find before you.
Finally, thanks to Elizabeth, David, Roger and Louise for their unnervingly close interest in this book. Several of these family members now often tell me why pylons are made of triangles and tightrope walkers carry long poles. Soon you will know too.
John D. Barrow
August 2008, Cambridge
fn1 www.mmp.maths.org
1
Pylon of the Month
Like Moses parting the waves, National Grid Company PLC’s 4YG8 leads his fellow pylons through this Oxfordshire housing estate towards the ‘promised land’ of Didcot Power Station.
The December 1999 Pylon of the Month
There are some fascinating websites about, but none was more beguiling than the iconic Pylon of the Month,fn1 once devoted to providing monthly pin-ups of the world’s most exciting and seductive electricity pylons. The ones shown on the website below are from Scotland. Alas, Pylon of the Month now seems to have become a cobweb site, but there is still something to learn from it, since for the mathematician every pylon tells a story. It is about something so prominent and ubiquitous that, like gravity, it goes almost unnoticed.
Next time you go on a train journey, look carefully at the pylons as they pass swiftly by the windows. Each is made of a network of metal struts that make use of a single recurring polygonal shape. That shape is the triangle. There are big triangles and smaller ones nested within them. Even apparent squares and rectangles are merely separate pairs of triangles. The reason forms a small part of an interesting mathematical story that began in the early nineteenth century with the work of the French mathematician Augustin-Louis Cauchy.
Of all the polygonal shapes that we could make by bolting together straight struts of metal, the triangle is special. It is the only one that is rigid. If they were hinged at their corners, all the others can be flexed gradually into a different shape without bending the metal. A square or a rectangular frame provides a simple example: we see that it can be deformed into a parallelogram without any buckling. This is an important consideration if you aim to maintain structural stability in the face of winds and temperature changes. It is why pylons seem to be great totems to the god of all triangles.
If we move on to three-dimensional shapes then the situation is quite different: Cauchy showed that every convex polyhedron (i.e. in which the faces all point outwards) with rigid faces, and hinged along its edges, is rigid. And, in fact, the same is true for convex polyhedra in spaces with four or more dimensions as well.
What about the non-convex polyhedra, where some of the faces can point inwards? They look much more squashable. Here, the question remained open until 1978 when Robert Connelly found an example with non-convex faces that is not rigid and then showed that in all such cases the possible flexible shifts keep the total volume of the polyhedron the same. However, the non-convex polyhedral examples that exist, or that may be found in the future, seem to be of no immediate practical interest to structural engineers because they are special in the sense that they require a perfectly accurate construction, like balancing a needle on its point. Any deviation from it at all just gives a rigid example, and so mathematicians say that ‘almost every’ polyhedron is rigid. This all seems to make structural stability easy to achieve – but pylons do buckle and fall down. I’m sure you can see why.
fn1 http://www.drookitagain.co.uk/coppermine/thumbnails.php?album=34
2
A Sense of Balance
Despite my privileged upbringing, I’m quite well-balanced. I have a chip on both shoulders.
Russell Crowe in A Beautiful Mind
Whatever you do in life, there will be times when you feel you are walking a tightrope between success and failure, trying to balance one thing against another or to avoid one activity gobbling up every free moment of your time. But what about the people who really are walking a tightrope. The other day I was watching some old newsreel film of a now familiar sight: a crazy tightrope walker making a death-defying walk high above a ravine and a rushing river. One slip and he would have become just another victim of Newton’s law of gravity.
We have all tried to balance on steps or planks of wood at times, and we know from experience that some things help to keep you balanced and upright: don’t lean away from the centre, stand up straight, keep your centre of gravity low. All the things they teach you in circus school. But those tightrope walkers always seem to carry very long poles in their hands. Sometimes the poles flop down at the ends because of their weight, sometimes they even have heavy buckets attached. Why do you think the funambulists do that?
The key idea you need to understand why the tightrope walker carries a long pole to aid balance is inertia. The larger your inertia, the slower you move when a force is applied. It has nothing to do with centre of gravity. The farther away from the centre that mass is distributed, the higher a body’s inertia is, and the harder it is to move it. Take two spheres of different materials that have the same diameter and mass, one solid and one hollow, and it will be the hollow one with all its mass far away at its surface that will be slower to move or to roll down a slope. Similarly, carrying the long pole increases the tightrope walker’s inertia by placing mass far away from the body’s centre line – inertia has units of mass times distance squared. As a result, any small wobbles about the equilibrium position happen more slowly. They have a longer time period of oscillation, and the walker has more time to respond to the wobbles and restore his balance. Compare how much easier it is to balance a one-metre stick on your finger compared with a 10-centimetre one.
3
Monkey Business
I have a spelling chequer
It came with my pee sea
It plainly marques four my revue
Miss takes I cannot see
I’ve run this poem threw it
I’m shore yaw pleased to no
It’s letter perfect in its weigh
My chequer told me sew . . .
Barri Haynes
The legendary image of an army of monkeys typing letters at random and eventually producing the works of Shakespeare seems to have emerged gradually over a long period of time. In Gulliver’s Travels, written in 1726, Jonathan Swift tells of a mythical Professor of the Grand Academy of Lagado who aims to generate a catalogue of all scientific knowledge by having his students continuously generate random strings of letters by means of a mechanical printing device. The first mechanical typewriter had been patented in 1714. After several eighteenth- and nineteenth-century French mathematicians used the example of a great book being composed by a random deluge of letters from a printing works as an example of extreme improbability, the monkeys appear first in 1909, when the French mathematician Émile Borel suggested that randomly typing monkeys would eventually produce every book in France’s Bibliothèque Nationale.
Arthur Eddington took up the analogy in his famous book The Nature of the Physical World in 1928, where he anglicised the library: ‘If I let my fingers wander idly over the keys of a typewriter it might happen that my screed made an intelligible sentence. If an army of monkeys were strumming on typewriters they might write all the books in the British Museum.’
Eventually this oft-repeated example picked the ‘Complete Works of Shakespeare’ as the prime candidate for random recreation. Intriguingly, there was a website that once simulated an ongoing random striking of typewriter keys and then did pattern searches against the ‘Complete Works of Shakespeare’ to identify matching character strings. This simulation of the monkeys’ actions began on 1 July 2003 with 100 monkeys, and the population of monkeys was effectively doubled every few days until recently. In that time they produced more than 1035 pages, each requiring 2,000 keystrokes.
A running record was kept of daily and all-time record strings until the Monkey Shakespeare Simulator Project site stopped updating in 2007. The daily records are fairly stable, around the 18- or 19-character-string range, and the all-time record inches steadily upwards. For example, one of the 18-character strings that the monkeys have generated is contained in the snatch:
. . . Theseus. Now faire UWfIlaNWSK2d6L;wb . . .
The first 18 characters match part of an extract from A Midsummer Night’s Dream that reads
. . . us. Now faire Hippolita, our nuptiall houre . . .
For a while the record string was 21-characters long, with
. . . KING. Let fame, that wtIA’”yh!”VYONOvwsFOsbhzkLH . . .
which matches 21 letters from Love’s Labour’s Lost
KING. Let fame, that all hunt after in their lives,
Live regist’red upon our brazen tombs,
And then grace us in the disgrace of death; . . .
In December 2004 the record reached 23 characters with
Poet. Good day Sir FhlOiX5a]OM,MLGtUGSxX4IfeHQbktQ . . .
which matched part of Timon of Athens
Poet. Good day Sir
Pain. I am glad y’are well
Poet. I haue not seene you long, how goes the World?
Pain. It weares sir, as it growes . . .