100 Essential Things You Didn't Know You Didn't Know (15 page)

In the Russian revolutionary period, Tamm was a young professor teaching physics at the University of Odessa in the Ukraine. Food was in short supply in the city, and so he made a trip to a nearby village, which was under the apparent control of the communists, in an attempt to trade some silver spoons for something more edible, like chickens. Suddenly, the village was captured by an anti-communist bandit leader and his militia, armed with rifles and explosives. The bandits were suspicious of Tamm, who was dressed in city clothes, and took him to their leader, who demanded to know who he was and what he did. Tamm tried to explain that he was merely a university professor looking for food.

‘What kind of professor?’ the bandit leader asked.

‘I teach mathematics,’ Tamm replied.

‘Mathematics?’ said the bandit. ‘All right! Then give me an estimate of the error one makes by cutting off Maclaurin’s series at the n
^th
term.
¹¹
Do this and you will go free. Fail, and you will be shot!’

Tamm was not a little astonished. At gunpoint, somewhat nervously, he managed to work out the answer to the problem – a tricky piece of mathematics that students are taught in their first course of calculus in a university degree course of mathematics. He showed it to the bandit leader, who perused it and declared ‘Correct! Go home!’

Tamm never discovered who that strange bandit leader was. He probably ended up in charge of university quality assurance somewhere.

45

Getting in a Flap

In ancient days two aviators procured to themselves wings. Daedalus flew safely through the middle air and was duly honoured on his landing. Icarus soared upwards to the sun till the wax melted which bound his wings and his flight ended in fiasco. The classical authorities tell us, of course, that he was only ‘doing a stunt’; but I prefer to think of him as the man who brought to light a serious constructional defect in the flying-machines of his day.

Arthur S. Eddington

Lots of things get around by flapping about: birds and butterflies with their wings, whales and sharks with their tails, fish with their fins. In all these situations there are three important factors at work that determine the ease and efficiency of movement. First, there is size – larger creatures are stronger and can have larger wings and fins, which act on larger volumes of air or water. Next, there is speed – the speed at which they can fly or swim tells us how rapidly they are engaging with the medium in which they are moving and the drag force that it exerts to slow them down. Third, there is the rate at which they can flap their wings or fins. Is there a common factor that would allow us to consider all the different movements of birds and fish at one swoop?

As you have probably guessed, there is such a factor. When scientists or mathematicians are faced with a diversity of examples
of
a phenomenon, like flight or swimming, that differ in the detail but retain a basic similarity, they often try to classify the different examples by evaluating a quantity that is a pure number. By this I mean that it doesn’t have any units, in the way that a mass or a speed (length per unit time) does. This ensures that it stays the same if the units used to measure the quantities are changed. So, whereas the numerical value of a distance travelled will change from 10,000 to 6¼ if you switch units from metres to miles, the ratio of two distances – like the distance travelled divided by the length of your stride – will not change if you measure the distance and your stride length
in the same units
, because it is just the number of strides that you need to take to cover the distance.

In our case there is one way to combine the three critical factors – the flapping rate per unit of time, f, the size of the flapping strokes, L, and the speed of travel, V – so as to get a quantity that is a pure number.
^fn1
This combination is just fL/V and it is called the ‘Strouhal number’, after Vincenc Strouhal (1850–1922), a Czech physicist from Charles University in Prague.

In 2003 Graham Taylor, Robert Nudds and Adrian Thomas, at Oxford University, showed that if we evaluate the value of this Strouhal number St = fL/V for many varieties of swimming and flying animals at their cruising speeds (rather than in brief bursts when pursuing prey or escaping from attack), then they fall into a fairly narrow range of values that could be said to characterise the results of the very different evolutionary histories that led to these animals. They considered a very large number of different animals, but let’s just pick on a few different ones to see something of this unity in the face of superficial diversity.

For a flying bird, f will be the frequency of wing flaps per second, L will be the overall span of the two flapping wings, and V will be the forward flying speed. A typical kestrel has an f of about 5.6
flaps
per second, a flap extent of about 0.34 metres and a forward speed of about 8 metres per second, giving St(kestrel) = (5.6 × 0.34)/8 = 0.24. A common bat has V = 6 metres per second, a wingspan of 0.26 metres, and a flapping rate of 8 times per second, so it has a Strouhal number of St(bat) = (8 × 0.26)/6 = 0.35. Doing the same calculation for forty-two different birds, bats and flying insects always gave a value of St in the range 0.2–0.4. They found just the same for the marine species they studied as well. More extensive studies were then carried out by Jim Rohr and Frank Fish (!) at San Diego and West Chester, Pennsylvania, to investigate this quantity for fish, sharks, dolphins and whales. Most (44 per cent) were found to lie in the range 0.23 to 0.28, but the overall range spanned 0.2 to 0.4, just like the range of values for the flying animals.

You can try this on humans too. A good male club standard swimmer will swim 100 metres in 60 seconds, so V = 5/3 metres per second, and uses about 54 complete stroke cycles of each arm (so the stroke frequency is 0.9 per second), with an arm-reach in the water of about 0.7 metres. This give St(human swimmer) = (0.9 × 2/3)/5/3 = 0.36, which places us rather closer to the birds and the fishes than we might have suspected. However, arguably the world’s most impressive swimmer has been the Australian long-distance star Shelley Taylor-Smith, who has won the world marathon swimming championships seven times. She completed a 70 km swim in open sea water inside 20 hours with an average stroke rate of 88 strokes per minute. With an effective stroke reach of 1 metre that gives her the remarkable Strouhal number of 1.5, way up there with the mermaids.

^fn1
The unit of frequency f is 1/time, of size L is length, and of speed V is length/time so the combination fL/V has no units: it is a pure dimensionless number.

46

Your Number’s Up

Enter your postcode to view practical joke shops near you.

Practical Jokeshop UK

Life seems to be defined by numbers at every turn. We need to remember PIN numbers, account numbers, pass codes and countless reference numbers for every institution and government department under the Sun, and for several that never see it. Sometimes, I wonder whether we are going to run out of numbers. One of the most familiar numbers that labels us geographically (approximately) is the postcode. Mine is CB3 9LN and, together with my house number, it is sufficient to get all my mail delivered accurately, although we persist with adding road names and towns as back-up information or perhaps because it just sounds more human. My postcode follows an almost universal pattern in the United Kingdom: using four letters and two numbers. The positions of the letters and numbers don’t really matter, although in practice they do because the letters also designate regional sorting and distribution centres (CB is Cambridge). But let’s not worry about this detail – the postal service certainly wouldn’t if it found itself running out of 6-symbol postcodes – and simply ask how many different postcodes of this form could there be? You have 26 choices from A to Z for each of the four slots for letters and 10 choices from 0 to 9 for each of the numerals. If these are each chosen independently then the total number of different postcodes following
the
current pattern is equal to 26×26×10×10×26×26, which equals 45,697,600 or nearly 46 million. Currently, the number of households in the United Kingdom is estimated to be about 26,222,000, or just over 26 million, and is projected to increase to about 28.5 million by the year 2020. So, even our relatively short postcodes have more than enough capacity to deal with the number of households and give each of them a unique identifier if required.

If we want to label individuals uniquely, then the postcode formula is not good enough. In 2006 the population of the United Kingdom was estimated at 60,587,000, about 60.5 million, and far greater than the number of postcodes. The closest thing that we have to an identity number is our National Insurance number, which is used by several agencies to identify us – the possibility of all these agencies coordinating their data by using this number is the step that alarms many civil liberty groups the most. A National Insurance number has a pattern of the form NA 123456 Z, which contains six numerals and 3 letters. As before, we can easily work out how many different National Insurance numbers this recipe permits. It’s

26 × 26 × 10 × 10 × 10 × 10 × 10 × 10 × 26

This is a
big
number – 17,576,000,000 – seventeen billion, five hundred and seventy-six million, and vastly bigger than the population of the United Kingdom (and even than its projected value of 75 million by 2050). In fact, the population of the whole world is currently only about 6.65 billion and projected to reach 9 billion by the year 2050. So there are plenty of numbers – and letters – to go round.

47

Double Your Money

The value of your investments can go down as well as up.

UK consumer financial advice

Recently you will have discovered that the value of your investments can plummet as well as go down. So, suppose you want to play safe and place cash in a straightforward savings account with a fixed, or slowly, changing rate of interest. How long will it take to double your money? Although nothing in this world is so certain as death and taxes (and the version of the latter that goes with the former), let’s forget about them both and work out a handy rule of thumb for the time needed to double your money.

Start out by putting an amount A in a savings account with an annual fractional rate of interest r (so 5% interest corresponds to r = 0.05), then it will have grown to A × (1+r) after one year, to A × (1+r)
²
after two years, to A × (1+r)
³
after three years and so on. After, say, n years your savings will have become an amount equal to A × (1+r)
ⁿ
. This will be equal to twice your original investment, that is 2A, when (1+r)
ⁿ
= 2. If we take natural logarithms of this formula, and note that ln(2) = 0.69 approximately, and ln(1+r) is approximately equal to r when r is much less than 1 (which it always is – typically r is about 0.05 to 0.06 at present in the UK), then the number of years needed for your investment to double is given by the simple formula n = 0.69/r. Let’s round
0
.69 off to 0.7 and think of r as R per cent, so R = 100r, then we have the handy rule that
¹²

n = 70/R

This shows, for example, that when the rate R is 7% we need about ten years to double our money, but if interest rates fall to 3.5% we will need twenty.

48

Some Reflections on Faces

In another moment Alice was through the glass, and had jumped lightly down into the Looking-glass room.

Lewis Carroll

None of us has seen our own face – except in the mirror. Is the image a true one? A simple experiment will tell you. After the bathroom mirror has got steamed up, draw a circle around the image of your face on the glass. Measure its diameter with the span of your finger and thumb, and compare it with the actual size of your face. You will always find that the image on the mirror is exactly one-half of the size of your real face. No matter how far away from the mirror you stand, the image on the glass will always be half size.

How strange this is. We have got so used to our appearance in the mirror when shaving or combing our hair almost every day of our lives that we have become immune to the big difference between reality and perceived reality. There is nothing mysterious about the optics of the situation. When we look at a plane mirror, a ‘virtual’ image of our face is always formed at the same distance ‘behind’ the mirror as we are in front of it. Therefore, the mirror is always located halfway between you and your virtual face. Light does not pass through the mirror to create an image behind the mirror, of course; it simply
appears
to be coming from this location. Walk towards a plane mirror and notice that your image appears to approach you at twice the speed you are walking.

The next odd thing about your image in the mirror is that it has changed handedness. Hold your toothbrush in your right hand and it will appear in your left hand in the mirror image. There is a left–right reversal in the image, but the image is not inverted: there is no up–down reversal: if you are looking at your image in a hand-held mirror and you rotate the mirror clockwise by 90 degrees then your image is unchanged.

Hold up a transparent sheet with writing on it and something different happens. The writing is not reversed by the mirror if we hold the transparency facing us so that we can read it. If we held up a piece of paper facing the same way then we would not be able to read it in the mirror because it is opaque. The mirror enables us to see the back of an object that is not transparent even though we are standing in front of it. But to see the front of it we would need to rotate it. If we rotate it around its vertical axis to face the mirror then we switch the right and left sides over. The left–right reversal of the mirror image is produced by this change of the object. If we rotate a page of a book that we are reading about its horizontal axis so that it faces the mirror then it appears upside down because it really has been inverted and not reversed in the left–right sense. When no mirror is present we don’t get these effects because we can only see the front of the object we are looking at, and after it is rotated we only see the back. The reason the letters on the page of the book are reversed left–right is because we have turned the book about a vertical axis to make it face the mirror. The letters are not turned upside down, but we could make them appear upside down in the mirror if we turned the book about its horizontal axis, from bottom to top instead.