100 Essential Things You Didn't Know You Didn't Know Read online

  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 . . .