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

by John D. Barrow

  96

  Some Benefits of Irrationality

  Mysticism might be characterised as the study of those propositions which are equivalent to their own negations. The Western point of view is that the class of all such propositions is empty. The Eastern point of view is that this class is empty if and only if it isn’t.

  Raymond Smullyan

  There is more to photocopying than meets the eye. If you do it here in Europe, then you will soon appreciate a feature so nice that you will have taken it for granted. Put two sheets of A4 paper side by side face-down on the copier and you will be able to reduce them so that they print out, side by side, on a single piece of A4 paper. The fit of the reduced copy to the paper is exact and there are no awkward extra margins on the page in the final copy. Try to do that in the United States with two pieces of standard US letter-sized paper and you will get a very different outcome. So what is going on here, and what has it got to do with mathematics and irrationality?

  The International Standard (I.S.O.) paper sizes, of which A4 is one, derive from a simple observation first made by the German physicist, Georg Lichtenberg, in 1786. Each paper size in the so called A-series has half the area of the next biggest sheet because it is half as wide but just as long. So putting two sheets side by side creates a sheet of the next size up: for example, two A4 sheets make one A3 sheet. If the length is L and the width is W, this means that they must be chosen so that L/W = 2W/L. This requires L2 = 2W2, and so the lengths of the sides are in proportion to the square root of 2, an irrational number that is approximately equal to 1.41: L/W = √2.

  This irrational ratio between the length and the width of every paper size, called the ‘aspect ratio’ of the paper, is the defining feature of the A paper series. The largest sheet, called A0, is defined to have an area of 1 square metre, so its dimensions are L(A0) = 2¼ m and W(A0) = 2-¼ m, respectively. The aspect ratio means that a sheet of A1 has length 2-¼ and width 2-¾, so its area is just ½ square metre. Continuing this pattern, you might like to check that the dimensions of a piece of AN paper, where N = 0, 1, 2, 3, 4, 5, . . . etc., will be

  L(AN) = 2¼–N/2 and W(AN) = 2-¼–N/2

  The area of a single sheet of AN paper will therefore be equal to the width times the length, which is 2-N square metres.

  All sorts of aspect ratios other than √2 could have been chosen. If you were that way inclined you might have gone for the Golden Ratio, so beloved by artists and architects in ancient times. This choice would correspond to picking paper sizes with L/W = (L+W)/L, so L/W = (1+√5)/2, but it would not be a wise choice in practice.

  The beauty of the √2 aspect ratio becomes most obvious if we return to the photocopier. It means that you can reduce one side of A3, or two sheets of A4 side by side, down to a single sheet of A4 without leaving a gap on the final printed page. You will notice that the control panel on your copier offers you a 70% (or 71% if it is a more pedantic make) reduction of A3 to A4. The reason is that 0.71 is approximately equal to 1/√2 and is just right for reducing one A3 or two A4 sheets to one of A4. Two dimensions, L and W, are reduced to L√2 and W√2, so that the area LW is reduced to LW√2, as required if we want to reduce a sheet of any AN size down to the size below. Likewise, for enlargements, the number that appears on the control panel is 140% (or 141% on some photocopiers) because √2 = 1.41 approximately. Another consequence of this constant aspect ratio for all reductions and magnifications is that diagrams retain the same relative shapes: squares do not become rectangles and the circles do not become ellipses when their sizes are changed between A series papers.

  Things are usually different in America and Canada. The American National Standards Institute (ANSI) paper sizes in use there, in inches because that is how they were defined, are A or Letter (8.5 in × 11.0 in), B or Legal (11 in × 17 in), C or Executive (17 in × 22 in), D Ledger (22 in × 34 in), and then E Ledger (34 in × 44 in). They have two different aspect ratios: alternately 17/11 and 22/17. So, if you want to keep the same aspect ratio when merging paper sizes you need to jump two paper sizes rather than one. As a result, you cannot reduce or magnify two sheets of one size down to one sheet of the size below or above without leaving some empty margin on the copy. When you want to make reduced or enlarged copies on a US photocopier you have to change the paper trays around in order to accommodate papers with two aspect ratios rather than using the one √2 factor that we do in the rest of the world. Sometimes a little bit of irrationality helps.

  97

  Strange Formulae

  Decision plus action times planning equals productivity minus delay squared.

  Armando Iannucci

  Mathematics has become such a status symbol in some quarters that there is a rush to use it without thought as to its appropriateness. Just because you can use symbols to re-express some words does not necessarily add to our knowledge. Saying ‘Three Little Pigs’ is more helpful than defining the set of all pigs, the set of all triplets and the set of all little animals and taking the intersection common to all three overlapping sets. An interesting venture in this direction was first made by the Scottish philosopher Francis Hutcheson in 1725, and he became a successful professor of philosophy at Glasgow University on the strength of it. He wanted to compute the moral goodness of individual actions. We see here something of the impact of Newton’s success in describing the physical world using mathematics: his methodology was something to copy and admire in all sorts of other domains. Hutcheson proposed a universal formula to evaluate the virtue, or degree of benevolence, of our actions:

  Hutcheson’s formula for the moral arithmetic has a number of pleasing features. If two people have the same natural ability to do good, then the greatest one that produces the largest public good is the more virtuous. Similarly, if two people produce the same level of public good then the one of lesser natural ability is the more virtuous.

  The other ingredient in Hutcheson’s formula, Private Interest, can contribute positively or negatively (±). If a person’s action benefits the public but harms themselves (for example, they do charitable work for no pay instead of taking paid employment), the Virtue is boosted by Public Good + Private Interest. But if their actions help the public and also themselves (for example, campaigning to stop an unsightly property development that blights their own property as well as their neighbours’) then the Virtue of that action is diminished by the factor Public Good – Private Interest.

  Hutcheson didn’t attribute numerical values to the quantities in his formula but was prepared to adopt them if needed. The moral formula doesn’t really help you because it reveals nothing new. All the information it contains has been plugged in to create it in the first place. Any attempt to calibrate the units of Virtue, Self-Interest and Natural Ability would be entirely subjective and no measurable prediction could ever be made. None the less, the formula is a handy shorthand for a lot of words.

  Something strangely reminiscent of Hutcheson flight of rationalistic fantasy appeared 200 years later in a fascinating project embarked upon by the famous American mathematician George Birkhoff, who was intrigued by the problem of quantifying aesthetic appreciation. He devoted a long period of his career to the search for a way of quantifying what appeals to us in music, art and design. His studies gathered examples from many cultures and his book Aesthetic Measure makes fascinating reading still. Remarkably, he boils it all down to a single formula that reminds me of Hutcheson’s. He believed that aesthetic quality is determined by a measure that is determined by the ratio of order to complexity:

  Aesthetic Measure = Order/Complexity

  He sets about devising ways to calculate the Order and Complexity of particular patterns and shapes in an objective way and applies them to all sorts of vase shapes, tiling patterns, friezes and designs. Of course, as in any aesthetic evaluation, it does not make sense to compare vases with paintings: you have to stay within a particular medium and form for this to make any sense. In the case of polygonal shapes Birkhoff’s measure of
Order adds up scores for the presence or absence of four different symmetries that can be present and subtracts a penalty (of 1 or 2) for certain unsatisfactory ingredients (for example, if the distances between vertices are too small, or the interior angles too close to 0 or 180 degrees, or there is a lack of symmetry). The result is a number that can never exceed 7. The Complexity is defined to be the number of straight lines that contain at least one side of the polygon. So, for a square it is 4, but for a Roman Cross (like the one shown here) it is 8 (4 horizontals plus 4 verticals):

  Birkhoff’s formula has the merit of using numbers to score the aesthetic elements, but unfortunately, aesthetic complexity is too broad for such a simple formula to encompass and, unlike Hutcheson’s cruder attempt, it fails to create a measure that many would agree on. If one applies his formula to modern fractal patterns that appeal to so many (not just mathematicians) with their repeating patterns on smaller and smaller scales, then their order can score no more than 7 but their complexity becomes larger and larger as the pattern explores smaller and smaller scales and its Aesthetic Measure tends rapidly to zero.

  98

  Chaos

  Other countries have unpredictable futures but Russia is a country with an unpredictable past.

  Yuri Afanasiev

  Chaos is extreme sensitivity to ignorance. It arises in situations where a little bit of ignorance about the current state of affairs grows rapidly as time passes, not merely increasing in proportion to the number of time-steps taken but roughly doubling at each step. The famous example of this sort is the weather. We fail to predict the weather in England with high accuracy on many occasions, not because we are not good at predicting or because there is some special unknown secret of meteorology that physicists have failed to discover, but because of our imperfect knowledge of the state of the weather now. We have weather stations every 100 kilometres over parts of the country, fewer over the sea, which take periodic measurements. However, this still leaves scope for considerable variations to exist between the weather stations. The Met Office computer has to extrapolate using a likely form of in-between-the-weather-stations weather. Unfortunately, little differences in that extrapolation often lead to very different future weather conditions.

  This type of sensitivity of the future to the present began to be studied extensively in the 1970s when inexpensive personal computers started to be available to scientists. It was dubbed ‘chaos’ to reflect the unexpected outcomes that could soon result from seemingly innocuous starting conditions because of the rapid growth in the effects of any small level of uncertainty. The movie business latched on to this with the film Jurassic Park, where a small mistake, which led to cross-breeding with dinosaurs, and a broken test tube led to a disaster, as things rapidly went from bad to worse and a small amount of uncertainty ballooned into almost total ignorance. There was even a ‘chaologist‘ on hand to explain it all as we watched the problems snowball out of control.

  There is one interesting aspect of chaos that resonates with entirely non-mathematical experiences that we have of books, music and drama. How do we evaluate whether a book or a play or a piece of music is ‘good’ or better than some other? Why do we think that The Tempest is better than Waiting for Godot, or Beethoven’s 5th Symphony superior to John Cage’s 4’ 33”, which consists of 4 minutes and 33 secondsfn1 of silence?

  One way is to argue that good books are the ones we want to read again, good plays make us want to see them again, and good music creates a desire to hear another performance of the same work. We seek to do that because they possess a little bit of chaotic unpredictability. A little change in the direction and cast of The Tempest, a different orchestra and conductor or a different state of mind in us the reader, will result in a very different overall experience of the play, the music or the book. Run of the mill art lacks that quality. Changes of circumstance produce much the same overall experience. There is no need to have that experience again. Chaos is not just something to be avoided or controlled.

  Some people think that the possibility of chaos is the end of science. There must always be some level of ignorance about everything in the world – we don’t have perfect instruments. How can we hope to predict or understand anything if there is fast growth in these uncertainties. Fortunately, even though individual atoms and molecules move chaotically in the room where I am sitting, their average movement as a whole is entirely predictable. Many chaotic systems have this nice property and we actually use some of those average quantities to measure what is happening. Temperature, for example, is a measure of the average speed of the molecules in the room. Even though the individual molecules have a history that would be impossible to predict after a few collisions with their neighbours and other denser objects in the room, these collisions keep the average fairly steady and eminently predictable. Chaos is not the end of the world.

  fn1 I have always been surprised to discover that my ‘revelation’ as to why it is 4 minutes and 33 seconds (and no other number) is news to all the musicians who have talked to me about it. In fact, the interval of time of 273 seconds was chosen by Cage for his Absolute Zero of sound by analogy with the minus 273 degrees Celsius which is the Absolute Zero of temperature where all classical molecular motion ceases. Remarkably, Cage once claimed this was his most important work.

  99

  All Aboard

  I am not a cloud bunny, I am not an aerosexual. I don’t like aeroplanes. I never wanted to be a pilot like those other platoons of goons who populate the air industry.

  Michael O’Leary, boss of Ryanair

  If you have spent as much time as I have queuing to get on aeroplanes, then you know all the bad plans that can be made. The budget airlines, like Ryanair, just don’t care: it’s a free for all with no seat reservations. Then they realised there was actually an incentive to make it as bad as possible for a while, so that they could sell you the right to ‘priority board’ ahead of the rest. There is no special priority for those with small children or mobility problems, so these passengers slow the general boarding even more. What happens when everyone elects to priority board? I don’t know, but I suspect it is the ultimate aim of the idea.

  Commercial airlines have a variety of methods to alleviate the stress and reduce the delay for economy passengers. Everyone has an assigned seat, and children and those needing extra time get to board first. Some airlines will board by seat number, so that when there is a single entrance at the front, those seated at the back get on first and don’t obstruct other passengers needing to pass them. It all sounds fine, on paper, but in practice somebody blocks the aisle as they try to get their oversize luggage into the overhead rack, people sitting in the aisle seats have to keep getting up to let in those sitting by the windows, and everyone is in somebody’s way. There has got to be a better system.

  A young German astrophysicist, Jason Steffen, based at Fermi Lab near Chicago, thought the same, and started to explore the efficiency of different passenger loading strategies using a simple computer simulation that could accommodate changes in boarding strategy and add in many random variations that perturb the best-laid plans. His virtual aircraft has 120 seats, accommodating 6 passengers per row divided by a central aisle and has no business or first class section. All the virtual passengers have hand baggage to stow.

  It was easy to find the worst loading policy to adopt for a plane with a single entrance at the front: load by seat number starting from the front. All passengers have to fight their way past the passengers already on board and busy packing their hand baggage in order to get to their seats. This is fairly obvious, but it led airlines to conclude that the best strategy is simply the opposite of the worst one: load by seat number from the back. Remarkably, Steffen’s investigation found that this was actually the second slowest method of boarding! Only boarding from the front was worse. Even boarding completely at random, irrespective of seat number, did much better. But the best method was more structured. Passengers should board so that those
sitting at the windows go ahead of those sitting in the middle or on the aisles and they should board in a way that distributes the number of people trying to stow luggage at the same time along the whole length of the plane rather than congregate all in one area.

  If passengers in all even-numbered rows with window seats board first, they have a clear row of aisle space in front and behind and don’t get in each others’ way while stowing bags. Everyone can load bags at the same time. If anyone needs to pass, then there are spare aisles to step into. By beginning from the back, the need to pass other passengers is avoided again. Those sitting in the middle and aisle seats follow on. Then the passengers in the odd-numbered rows follow.

  Not everyone can easily follow this strategy to the letter. Small children need to stay with their parents, but they can still board first. However, the time gained by making it the basic strategy could be considerable. The computer model showed that over hundreds of trials with different small variations (‘problem passengers’), this method was on the average about seven times quicker at loading passengers than the standard load-from-the-back-method. Steffen has patented it!