100 Essential Things You Didn't Know You Didn't Know Page 11
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.
This is not the end of the story. There are some further interesting things that happen (as magicians know only too well) if you have two flat mirrors. Place them at right angles to make an L-shape and look towards the corner of the L. This is something that you can do with a dressing-table mirror with adjustable side mirrors.
Look at yourself, or the pages of a book in this pair of right-angled mirrors and you will find that the image does not get swapped left–right. Your toothbrush appears to be in your right hand if it really is in your right hand. Indeed, to use such a mirror system for shaving or combing our hair is rather confusing because the brain automatically makes the left–right switch in practice. If you change the angle between the mirrors, gradually reducing it below 90 degrees, then something odd happens when you reach 60 degrees. The image looks just as it would when you look into a single flat mirror and is left–right reversed.
The 60-degree inclination of the mirrors ensures that a beam shone at one mirror will return along exactly the same path and create the same type of virtual image as you would see in a single plane mirror.
49
The Most Infamous Mathematician
He is the organiser of half that is evil and of nearly all that is undetected in [London]. He is a genius, a philosopher, an abstract thinker. He has a brain of the first order.
Sherlock Holmes in ‘The Final Problem’
There was a time – and it may still be the time for some – when the most well-known mathematician among the general public was a fictional character. Professor James Moriarty was one of Arthur Conan Doyle’s most memorable supporting characters for Sherlock Holmes. The ‘Napoleon of crime’ was a worthy adversary for Holmes and even required Mycroft Holmes’s talents to be brought into play on occasion to thwart his grand designs. He appears in person only in two of the Holmes stories, ‘The Final Problem’ and ‘The Valley of Fear’, but is often lurking behind the scenes, as in the case of ‘The Red-Headed League’ where he plans an ingenious deception in order for his accomplices, led by John Clay, to tunnel into a bank vault from the basement of a neighbouring pawnshop.
We know a little of Moriarty’s career from Holmes’s descriptions of him. He tells us that
He is a man of good birth and excellent education, endowed by nature with a phenomenal mathematical faculty. At the age of twenty-one he wrote a treatise upon the binomial theorem, which has had a European vogue. On the strength of it he won the mathematical chair at one of our smaller universities, and had, to all appearances, a most brilliant career before him.
Professor James Moriarty
But the man had hereditary tendencies of the most diabolical kind. A criminal strain ran in his blood, which, instead of being modified, was increased and rendered infinitely more dangerous by his extraordinary mental powers. Dark rumours gathered round him in the University town, and eventually he was compelled to resign his chair and come down to London . . .
Later, in The Valley of Fear, Holmes reveals a little more about Moriarty’s academic career and his versatility. Whereas his early work was devoted to the problems of mathematical series, twenty-four years later we see him active in the advanced study of dynamical astronomy,
Is he not the celebrated author of The Dynamics of an Asteroid, a book which ascends to such rarefied heights of pure mathematics that it is said that there was no man in the scientific press capable of criticizing it?
Conan Doyle made careful use of real events and locations in setting his stories, and it is possible to make a good guess as to the real villain on whom Professor Moriarty was styled. The prime candidate is one Adam Worth (1844–1902), a German gentleman who spent his early life in America and specialised in audacious and ingenious crimes. In fact, a Scotland Yard detective of his day, Robert Anderson, did call him ‘the Napoleon of the criminal world’. After starting out as a pickpocket and small-time thief, he graduated to organising robberies in New York. He was caught and imprisoned, but soon escaped and resumed business as usual, expanding its scope to include bank robberies and freeing the safe-breaker Charley Bullard from White Plains jail using a tunnel. Tellingly, for readers of ‘The Red-Headed League’, in November 1869, with Bullard’s help he robbed the Boylston National Bank in Boston by tunnelling into the bank vault from a nearby shop. In order to escape the Pinkerton agents, Worth and Bullard fled to England and were soon carrying out robberies there and in Paris, where they moved in 1871. Worth bought several impressive properties in London and established a wide-ranging criminal network to ensure that he was always at arm’s length from his robberies. His agents never even knew his name (he often used the assumed name of Henry Raymond), but it was impressed upon them that they should not use any violence in the perpetration of their crimes on his behalf. In the end, Worth was caught while visiting Bullard in prison and jailed for seven years in Leuven, Belgium, but was released in 1897 for good behaviour. He immediately stole jewellery to fund his return to nor
mal life and, through the good offices of the Pinkerton detective agency in Chicago, arranged for the return of a painting, The Duchess of Devonshire, to the Agnew & Sons gallery in London in return for a ‘reward’ of $25,000. Worth then returned to London and lived there with his family until his death in 1902. His grave can be found in High-gate cemetery under the name of Henry J. Raymond.
In fact, Worth had stolen this painting of Georgiana Spencer (a great beauty and it appears, a relative through the Spencer family of Princess Diana) by Gainsboroughfn1 from Agnew’s London gallery in 1876 and carried it around with him for many years, rather than sell it. It provides the key clue in establishing that Professor James Moriarty and Adam Worth were one and the same.
In The Valley of Fear Moriarty is interviewed by the police in his house. Hanging on the wall is a picture entitled ‘La Jeune a l’agneau – the young one has the lamb’ – a pun on the ‘Agnew’ gallery that had lost the painting, although no one could ever prove that Worth had stolen it. But alas, as far as I can tell, Worth never wrote a treatise on the binomial theorem or a monograph on the dynamics of asteroids.
fn1 The picture is now in the National Gallery of Art, Washington, D.C., and can be seen online at http://commons.wikimedia.org/wiki/Image:Thomas_Gainsboroguh_Georgiana_Duchess_of_Devonshire_1783.jpg
50
Roller Coasters and Motorway Junctions
What goes up must come down.
Anon.
There was a been on one of those ‘tear drop’ roller coasters that take you up into a loop, over the top and back down? You might have thought that the curved path traces a circular arc, but that’s almost never the case, because, if the riders are to reach the top with enough speed to avoid falling out of the cars at the top (or at least to avoid being supported only by their safety straps), then the maximum g-forces experienced by the riders when the ride returns to the bottom would become dangerously high.
Let’s see what happens if the loop is circular and has a radius r and the fully loaded car has a mass m. The car will be gently started at a height h (which is bigger than r) above the ground and then descend steeply to the base of the loop. If we ignore any friction or air resistance effects on the motion of the car, then it will reach the bottom of the loop with a speed Vb =√2gh. It will then ascend to the top of the loop. If it arrives there with speed Vt, it will need an amount of energy equal to 2mgr + ½mVt2 in order to overcome the force of gravity and ascend a vertical height 2r to the top of the loop and arrive there with a speed Vt. Since the total energy of motion cannot be created or destroyed, we must have (the mass of the car m cancels out of every term)
gh = ½ Vb2 = 2gr + ½ Vt2
At the top of the circular loop the net force on the rider pushing upwards, and stopping him falling out of the car, is the force from the motion in a circle of radius r pushing upwards minus his weight pulling down; so, if the rider’s mass is M, the
Net upwards force at the top = M Vt2/r – Mg
This must be positive to stop him falling out, and so we must have Vt2 > gr.
Looking back at the equations on pp.140–1, this tells us that we must have h >2.5r. So if you just roll away from the start with the pull of gravity alone, you have got to start out at least 2.5 times higher than the top of the loop in order to get to the top with enough speed to avoid falling out of your seat. But this is a big problem. If you start that high up you will reach the bottom of the loop with a speed Vb = √(2gh), which will be larger than √2g(2.5r) = √5gr. As you start to move in a circular arc at the bottom you will therefore feel a downward force equal to your weight plus the outward circular motion force, and this is equal to
Net downwards force at the bottom = Mg + MVb2/r > Mg + 5Mg
Therefore, the net downward force on the riders at the bottom will exceed six times their weight (an acceleration of 6-g). Most riders, unless they were off-duty astronauts or high-performance pilots wearing g-suits, would be rendered unconscious by this force. There would be no oxygen supply getting through to the brain at all. Typically, fairground rides with child riders aim to keep accelerations below 2-g, and those for adults experience at most 4-g.
Circular roller coaster rides seem to be a practical impossibility under this model, but if we look more closely at the two constraints – feel enough upward force at the top to avoid falling out but avoid experiencing lethal downward forces at the bottom – is there a way to change the roller coaster shape to meet both constraints?
When you move in a circle of radius R at speed V you feel an outward acceleration of V2/R. The larger the radius of the circle and so the gentler the curve, the smaller the acceleration you will feel. On the roller coaster the Vt2/r acceleration at the top is what is stopping us falling out, by overcoming our weight Mg acting downwards, so we want that to be big, which means r should be small at the top. On the other hand, when we are at the bottom the circular force is what is creating the extra 5-g of acceleration, and so we could reduce that by moving in a gentler circle with a larger radius. This can be achieved by making the roller coaster shape taller than it is wide, so it looks a bit like two parts of different circles, the one forming the top half with a smaller radius than the one forming the bottom half. The favourite curve that looks like this is called a ‘clothoid’ whose curvature decreases as you move along it in proportion to the distance moved. It was first introduced into roller coaster design in 1976 by the German engineer Werner Stengel for the ‘Revolution’ ride at Six Flags Magic Mountain in California.
Clothoids have another nice feature that has led to their incorporation into the design of complex motorway junction exits or railway lines. If a car is being driven along a curving motorway exit road, then as long as the driver keeps to a constant speed you can simply move the steering wheel with a constant rotation rate. If the bend were a different shape, then you would need to keep adjusting the rate of movement of the steering wheel or the speed of the car.
51
A Taylor-made Explosion
I do not know with what weapons World War III will be fought, but World War IV will be fought with sticks and stones.
Albert Einstein
The first atomic bomb exploded at the Trinity test in New Mexico, USA,210 miles south of Los Alamos, on 16 July 1945. It was a watershed in human history. The creation of this device gave human beings the ability to destroy all human life and produce deadly long-term consequences. Subsequently, an arms race saw an escalation in the energetic yield of these bombs as the United States and the Soviet Union sought to demonstrate their ability to produce increasingly devastating explosions. Although only two of these devices were ever used in conflict,fn1 the ecological and medical consequences of this era of tests in the atmosphere, on the ground, below ground and underwater, are still with us.
The explosions were much photographed at the time and produced a characteristic fireball and a canopy of debris that came to symbolise the consequences of nuclear war. The familiar mushroom cloudfn2 forms for a reason. A huge volume of very hot gas with a low density is created at high pressure near ground level – atomic and nuclear bombs were generally detonated above ground to maximise the effects of the blast wave in all directions. Just like the bubbles rising in boiling water, the gas accelerates up into the denser air above, creating turbulent eddies curving downwards at their edges while additional debris and smoke streams up the centre in a rising column. The material at the core of the detonation is vaporised and heated to tens of millions of degrees, producing copious x-rays, which collide with and energise the atoms and molecules of the air above, creating a flash of white light whose duration depends on the magnitude of the initial explosion. As the front of the column rises, it spins like a tornado and draws in material from the ground to form the ‘stem’ of the growing mushroom shape; its density falls as it spreads, and eventually it finds itself with the same density as the air above it. At that moment, it stops rising and disperses sideways, so that all the material drawn up from the ground bounces
backwards and descends to create a wide expanse of radioactive fall-out.
Ordinary explosions of TNT, or other non-nuclear weapons, have a rather different appearance because of the lower temperatures created at the start of a purely chemical explosion. This results in a turbulent mix of exploding gases rather than the organised stem and umbrella-like mushroom cloud.
One of the pioneers of the studies of the shape and character of large explosions was the remarkable Cambridge mathematician Geoffrey (G.I.) Taylor. Taylor wrote the classified report on the expected character of an atomic bomb explosion in June 1941. He became well known to a wider public after Life magazine in the USA published a sequence of time-lapsed photographs of the 1945 Trinity test in New Mexico. The energy yield from this and other American atomic-bomb explosions was still top secret, but Taylor showed how a few lines of algebra enabled him (and hence anyone else with a smattering of simple mathematics) to work out the approximate energy of an explosion just by looking at the photographs.
Taylor was able to work out the expected distance to the edge of the explosion at any time after its detonation by noting that it can depend significantly on only two things: the energy of the explosion and the density of the surrounding air that it is ploughing through. There is only one way that such a dependence can look,13 so approximately:
The published photos showed the explosion at different times since detonation and those times were printed down the side of each photo with a distance scale running along the bottom of the photo to gauge the size. From the first frame of the photo Taylor noted that after 0.006 seconds the blast wave of the explosion had a radius of roughly 80 metres. We know that the density of air is
1.2 kilograms per cubic metre, and so the equation then tells us that the energy released was about 1014 Joules, which is equivalent to about 25,000 tons of TNT. For comparison, we know that the 2004 Indian earthquake released the energy equivalent to 475 million tons of TNT.