100 Essential Things You Didn't Know You Didn't Know Page 3

  Sasha Azevedo

  How many people do you know? Let’s take a round number, like 100, as a good average guess. If your 100 acquaintances each know another 100 different people then you are connected by one step to 10,000 people: you are actually better connected than you thought. After n of these steps you are connected to 102(n+1) people. As of this month the population of the world is estimated to be 6.65 billion, which is 109.8, and so when 2(n+1) is bigger than 9.8 you have more connections than the whole population of the world. This happens when n is bigger than 3.9, so just 4 steps would do it.

  This is a rather remarkable conclusion. It makes a lot of simple assumptions that are not quite true, like the one about all your friend’s friends being different from each other. But if you do the counting more carefully to take that into account, it doesn’t make much difference. Just six steps is enough to put you in contact with just about anyone on Earth. Try it – you will be surprised how often fewer than 6 steps links you to famous people.

  There is a hidden assumption in this that is not well tested by thinking about your links to the Prime Minister, David Beckham or the Pope. You will be surprisingly close to these famous people because they have many links with others. But try linking to an Amazonian Indian tribe member or a Mongolian herdsman and you will find that the chain of links is much longer. You might not even be able to close it. These individuals live in ‘cliques’, which have very simple links once you look beyond their immediate close-knit communities.

  If your network of connections is a chain or a circle, then you are only connected by one person on either side and overall connectedness is poor. However, if you are in a ring of these connections with some other random links added, then you can get from one point on the ring to any other very quickly.

  In recent years we have begun to appreciate how dramatic the effects of a few long-range connections can be for the overall connectivity. A group of hubs that produce lots of connections to nearby places can be linked up very effectively by just adding a few long-range connections between them.

  These insights are important when it comes to figuring out how much coverage you need in order for mobile phone links between all users to be possible, or how a few infected individuals can spread a disease by their interactions with members of a population. When airlines try to plan how they arrange hubs and routes so as to minimise journey times, or maximise the number of cities they can connect with a single change of planes or minimise costs, they need to understand the unexpected properties of these ‘small world’ networks.

  The study of connectedness shows us that the ‘world’ exists at many levels: transport links, phone lines, email paths all create networks of interconnections that bind us together in unlikely ways. Everything is rather closer than we thought.

  11

  Bridging That Gap

  Like a bridge over troubled water.

  Paul Simon and Art Garfunkel

  One of the greatest human engineering achievements has been the construction of bridges to span rivers and gorges that would otherwise be impassable. These vast construction projects often have an aesthetic quality about them that places them in the first rank of modern wonders of the world. The elegant Golden Gate Bridge, Brunel’s remarkable Clifton Suspension Bridge and the Ponte Hercilio Luz in Brazil have spectacular shapes that look smooth and similar. What are they?

  There are two interesting shapes that appear when weights and chains are suspended and they are often confused or simply assumed to be the same. The oldest of these problems was that of describing the shape that is taken up by a hanging chain or rope whose ends are fixed at two points on the same horizontal level. You can see the shape easily for yourself. The first person to claim they knew what this shape would be was Galileo, who in 1638 maintained that a chain hanging like this under gravity would take up the shape of a parabola (this has the graph y2 = Ax where A is any positive number). But in 1669 Joachim Jungius, a German mathematician who had special interests in the applications of mathematics to physical problems, showed him to be wrong. The determination of the actual equation for the hanging chain was finally calculated by Gottfried Leibniz, Christiaan Huygens, David Gregory and Johann Bernoulli in 1691 after the problem had been publicly announced as a challenge by Johann Bernoulli a year earlier. The curve was first called the catenaria by Huygens in a letter to Leibniz, and it was derived from the Latin word catena for ‘chain’, but the introduction of the anglicised equivalent ‘catenary’ seems to have been due to the US President Thomas Jefferson in a letter to Thomas Paine, dated 15 September 1788, about the design of a bridge. Sometimes the shape was also known as the chainette or funicular curve.

  The shape of a catenary reflects the fact that its tension supports the weight of the chain itself and the total weight born at any point is therefore proportional to the total length of chain between that point and the lowest point of the chain. The equation for the hanging chain has the form y = Bcosh(x/B) where B is the constant tension of the chain divided by its weight per unit length.1 If you hold two ends of a piece of hanging chain and move them towards each other, or apart, then the shape of the string will continue to be described by this formula but with a different value of B for each position. This curve can also be derived by asking for the shape that makes the centre of gravity of the suspended chain as low as possible.

  Another spectacular example of a man-made catenary can be seen in St Louis, Missouri, where the Gateway Arch is an upside-down catenary (see here). This is the optimal shape for a self-supporting arch, which minimises the shear stresses because the stress is always directed along the line of the arch towards the ground. Its exact mathematical formula is written inside the arch. For these reasons, catenary arches are often used by architects to optimise the strength and stability of structures; a notable example is in the soaring high arches of Antoni Gaudí’s unfinished Sagrada Familia Church in Barcelona.

  Another beautiful example is provided by the Rotunda building designed by John Nash in 1819 to be the Museum of Artillery, located on the edge of Woolwich Common in London. Its distinctive tent-like roof, influenced by the shape of soldiers’ bell tents, has the shape of one half of a catenary curve.

  John Nash’s Rotunda building

  There is, however, a big difference between a hanging chain and a suspension bridge like the Clifton or the Golden Gate. Suspension bridges don’t only have to support the weight of their own cables or chains. The vast bulk of the weight to be supported by the bridge cable is the deck of the bridge. If the deck is horizontal with a constant density and cross-sectional area all the way along it, then the equation for the shape of the supporting cable is now a parabola y = x2/2B, where B is (as for the hanging chain equation) a constant equal to the tension divided by the weight per unit length of the bridge deck.

  One of the most remarkable is the Clifton Suspension Bridge in Bristol, designed by Isambard Kingdom Brunel in 1829 but completed only in 1865, three years after his death. Its beautiful parabolic form remains a fitting monument to the greatest engineer since Archimedes.

  The St Louis Gateway Arch

  12

  On the Cards

  Why don’t children collect things anymore? Whatever happened to those meticulously kept stamp albums . . . ?

  Woman’s Hour, BBC Radio 4

  Last weekend, hidden between books in the back of my bookcase, I came across two sets of cards that I had collected as a young child. Each set contained fifty high-quality colour pictures of classic motor cars with a rather detailed description of their design and mechanical specification on the reverse. Collecting sets of cards was once all the rage. There were collections of wartime aircraft, animals, flowers, ships and sportsmen – since these collections all seemed to be aimed at boys – to be amassed from buying lots of packets of bubble gum, breakfast cereals or packets of tea. Of the sports cards, just as with today’s Panini ‘stickers’, the favoured game was football (in the US it was baseball), and I always had
my suspicions about the assumption that all the players’ cards were produced in equal numbers. Somehow everyone seemed to be trying to get the final ‘Bobby Charlton’ card that was needed to complete the set. All the other cards could be acquired by swopping duplicates with your friends, but everyone lacked this essential one.

  It was a relief to discover that even my own children engaged in similar acquisitive practices. The things collected might change but the basic idea was the same. So what has mathematics got to do with it? The interesting question is to ask how many cards we should expect to have to buy in order to complete the set, if we assume that each of them is produced in equal numbers and so has an equal chance of being found in the next packet that you open. The motor car sets I came across each contained 50 cards. The first card I get will always be one I haven’t got but what about the second card? There is a 49/50 chance that I haven’t already got it. Next time it will be a 48/50 chance and so on.

  After you have acquired 40 different cards there will be a 10/50 chance that the next one will be one you haven’t already got. So on the average you will have to buy another 50/10, or 5 more cards to have a better than evens chance of getting another new one that you need for the set. Therefore, the total number of cards you will need to buy on average to get the whole set of 50 will be the sum of 50 terms:

  50/50 + 50/49 + 50/48 + . . . + 50/3 + 50/2 + 50/1

  where the first term is the certain case of the first card you get and each successive term tells you how many extra cards you need to buy to get the 2nd, 3rd and so on missing members of the set of 50 cards.

  As there can be collections with all sorts of different numbers of cards in them, let’s consider acquiring a set with any number of cards in it, that we will call N. Then the same logic tells us that on the average we will have to buy a total of

  (N/N) + (N/N-1) + (N/N-2) + . . . + N/2 + N/1 cards

  Taking out the common factor N in the numerators of each term, this is just

  N(1 + 1/2 + 1/3 + . . . + 1/N).

  The sum of terms in the brackets is the famous ‘harmonic’ series. When N becomes large it is well approximated by 0.58 + ln(N) where ln(N) is the natural logarithm of N. So as N gets realistically large we see that the number of cards we need to buy on the average to complete our set is about

  Cards needed ≈ N × [0.58 + ln(N)]

  For my sets of 50 motor car cards the answer is 224.5, and I should have expected to have to have bought on average about 225 cards to make up my set of 50. Incidentally, our calculation shows how much harder it gets to complete the second half of the collection than the first half. The number of cards that you need to buy in order to collect N/2 cards for half a set is

  (N/N) + (N/N-1) + (N/N-2) + . . . + N/(½N+1)

  which is the difference between N times the harmonic series summed to N and summed to N/2 terms, so

  Cards needed for half a set ≈ N × [ln(N) + 0.58 – ln(N/2) – 0.58] = Nln(2) = 0.7N

  Or just 35 to get the first half of my set of 50.

  I wonder if the original manufacturers performed such calculations. They should have, because they enable you to work out the maximum possible profit you could expect to gain in the long run from marketing a particular size set of cards. It is likely to be a maximum possible profit only because collectors will trade cards and be able to acquire new cards by swopping rather than buying new ones.

  What impact can friends make by swopping duplicates with you?

  Suppose that you have F friends and you all pool cards in order to build up F+1 sets so that you have one each. How many cards would you need to do this? On the average, when the number of cards N is large, and you share cards, the answer approaches

  N × [ln(N) + F ln(lnN) + 0.58]

  On the other hand, if you had each collected a set without swopping, you would have needed about (F+1)N[ln(N) + 0.58] cards to complete F+1 separate sets. For N = 50, the number of card purchases saved would be 156F. Even with F = 1 this is a considerable economy.

  If you know a little statistics you might like to show that the deviation that can be expected on the N × [0.58 + ln(N)] result is close to 1.3N. This is quite significant in practice because it means that you have a 66% chance of needing to collect 1.3 N more or less than the average. For our 50-card set this uncertainty in the expected number of purchases is 65. There was a story a few years ago that a consortium was targeting suitable national lotteries by calculating the average number of tickets that needed to be bought in order to have a good chance of collecting all the possible numbers – and so including the winning one. The members neglected to include the likely variance away from the average result but were very lucky to find that they did have a winning ticket among the millions they had bought.

  If the probability of each card appearing is not the same, then the problem becomes harder but is still soluble. In that case it is more like the problem of coin collecting where you try to collect a coin with each available year date on. You don’t know whether equal numbers were minted in each year (almost certainly they weren’t) or how many may have been withdrawn later, so you can’t rely on there being an equal chance of collecting an 1840 penny or an 1890 one. But if you do find a 1933 English penny (of which only 7 were made and 6 are accounted for) then be sure to let me know.

  13

  Tally Ho

  It is a profoundly erroneous truism . . . that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them.

  Alfred North Whitehead

  The hopeless prisoner is incarcerated in a dark, dank, forgotten cell. The days, months and years are passing slowly. There are many more to come. The scene is familiar from the movies. But there is usually an interesting mathematical sub-text. The prisoner has been keeping track of the days by systems of marks on the cell wall. The oldest human artefacts with records of counting in Europe and Africa go back more than 30,000 years and show similar elaborate groups of tally marks following the days of the month and the accompanying phases of the Moon.

  The standard European pattern of tallying goes back to finger counting and results in sets of 4 vertical(ish) lines | | | | being completed by a cross slash to signal a score of five. On to the next set of five and so on. The vertical bars plus slash denoting each counted item show ways of counting that predate our formal counting systems and led to the adoption of the Roman numerals I, II and III or the simple Chinese rod numeral systems. They are closely linked to simple methods of finger counting and make use of groups of 5 and 10 as bases for the accumulated sets of marks. Ancient tallying systems recorded marks on bone or were notches carved in wood. Single notches recorded the first few, but a half-cross notch V was then used to mark the 5, with a complete cross X for the ten, hence the V and X Roman numerals; 4 was made either by addition as IIII, or by subtraction as IV. Tallying remained a serious official business in England until as late as 1826, with the Treasury using great wooden tally sticks to keep records of large sums entering and leaving the Exchequer. This use is also the source of the multiple meanings of the word ‘score’, which means to make a mark and to keep count, as well as the quantity 20. The word tally comes from the word for cut, as still in ‘tailor’. When a debt was owed to the Treasury, the tally stick with its scored marks was cleft down the middle and the debtor given one half, the Treasury the other. When the debt was settled the two pieces were joined to check that they ‘tallied’.

  Counting these tally marks is rather laborious, especially if large totals arise. Individual marks have to be mentally counted and then the number of sets has to be totalled as well. In South America we find the occasional use of a memorable system that uses a gradual build-up of numbers by lines around a square, completed by the two diagonals .

  We are familiar with a variant of this square frame counting when we keep score in a cricket match by making six marks in three rows o
f two – a ‘dot’ if no run is scored, or the number scored, or a ‘w’ if a wicket falls. Other symbols denote wides, byes, no-balls and leg byes. If no runs are scored off all six balls, the six dots are joined up to create the sides of an M, denoting a ‘maiden’ over; if no runs are scored and a wicket is taken, they are joined to form a W, to indicate a ‘wicket maiden’ over. In this way a glance at the score book reveals the pattern and number of runless overs.

  It is tempting to combine the insights from South American games with those of the cricket scorer to create an ideal aide-memoire for anyone wanting to tally in tens and not have to count up to ten vertical marks to ‘see’ where the running total has reached. First count from 1 to 4 by placing dots at the four corners of a square, keep on counting from 5 to 8 by adding the four sides, and count 9 and 10 by adding the two diagonals. Each set of 10 is represented by the completed square of four dots and 6 lines. For the next 10 move on to a new square. Completion of a ten is obvious at a glance.

  14

  Relationships

  Relationship: The civilised conversationalist uses this word in public only to describe a seafaring vessel carrying members of his family.

  Cleveland Amory

  Most magazines have endless articles and correspondence about relationships. Why? Answer: Relationships are complicated, sometimes interesting and can often appear unpredictable. This is just the type of situation that mathematics can help you with.

  The simplest relationships between things have a property that we call ‘transitivity’ and it makes life simple. Being ‘taller than’ is one of these transitive relationships. So if Ali is taller than Bob and Bob is taller than Carla, then Ali is necessarily taller than Carla. This relationship is a property of heights. But not all relations are like this. Ali might like Bob and Bob might like Carla, but that does not mean that Ali likes Carla. These ‘intransitive’ relations can create very unusual situations when it comes to deciding what you should do when everyone does not agree.