100 Essential Things You Didn't Know You Didn't Know (16 page)

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 normal 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 Gainsborough
^fn1
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 V
_b
=√2gh. It will then ascend to the top of the loop. If it arrives there with speed V
_t
, it will need an amount of energy equal to 2mgr + ½mV
_t
²
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 V
_t
. 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 V
_b
²
= 2gr + ½ V
_t
²

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 V
_t
²
/r – Mg

This must be positive to stop him falling out, and so we must have V
_t
²
> 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 V
_b
= √(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 + MV
_b
²
/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 V
²
/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 V
_t
²
/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.