Ian Stewart Page 3

  5×100 + 2×10 + 5×1

  The symbol ‘5’ at the right-hand end stands for ‘five’; the same symbol at the left-hand end stands for ‘five hundred’. A positional number system like this needs a symbol for zero, otherwise it can’t distinguish between numbers like 12, 102, and 1,020.

  Our number system is said to be base 10 or decimal, because the value of a digit is multiplied by 10 every time it moves one place to the left. There’s no particular mathematical reason for using 10: base 7 or base 42 will work just as well. In fact, any whole number (greater than 1) can be used as a base, though bases greater than 10 require new symbols for the extra digits.

  The Mayan civilisation, which goes back to 2000 BC, flourished in Central America from about AD 250 to 900, and then declined, used base 20. So to them, the symbols 5-2-14 meant

  5×202 + 2×20 + 14×1

  which is 2,054 in our notation. They wrote a dot for 1, a horizontal line for 5, and combined these to get all numbers from 1 to 19. From 36 BC onwards they used a strange oval shape for 0. Then they stacked these 20 ‘digits’ vertically to show successive base-20 digits.

  Left: the numbers 0-29 in Mayan; right: Mayan for 5 × 202 + 2 × 20 + 14×1

  It is often suggested that the Mayans employed base 20 because they counted on their toes as well as their fingers. An alternative explanation occurred to me while I was writing this item. Maybe they counted on fingers and thumbs, with a thumb representing 5. Then each dot is a finger, each bar a thumb, and it can all be done on two hands. Admittedly, we don’t have three thumbs, but there are easy ways round this with hands and it’s not an issue for symbols. As for the oval shape for zero: don’t you agree that it looks a bit like a clenched fist? Meaning no fingers and no thumbs.

  This is wild speculation, but I quite like it.

  Much earlier, around 3100 BC, the Babylonians had been even more ambitious, using base 60. Babylon is almost a fabled land, with biblical stories of the Tower of Babel and Shadrach in Nebuchadnezzar’s furnace, and romantic legends of the Hanging Gardens. But Babylon was a real place, and many of its archaeological remains still survive in Iraq. The word ‘Babylonian’ is often used interchangeably for several different social groupings that came and went in the area between the Tigris and Euphrates rivers, who shared many aspects of their cultures.

  We know a lot about the Babylonians because they wrote on clay tablets, and more than a million of these have survived, often because they were in a building that caught fire and baked the clay rock-hard. The Babylonian scribes used short sticks with shaped ends to make triangular marks, known as cuneiform, in the clay. The surviving clay tablets include everything from household accounts to astronomical tables, and some date back to 3000 BC or earlier.

  The Babylonian symbols for numerals were introduced around 3000 BC, and employ two distinct signs for 1 and 10, which were combined in groups to obtain all integers up to 59.

  Babylonian numerals from 1 to 59.

  The 59 groups act as individual digits in base-60 notation, otherwise known as the sexagesimal system. To save my printer having kittens, I’ll do what archaeologists do and write Babylonian numerals like this:

  5,38,4 = 5×60×60 + 38×60 + 4 = 20,284 in decimal

  notation

  The Babylonians didn’t (until the late period) have a symbol to play the role of our zero, so there was a degree of ambiguity in their system, usually sorted out by the context in which the number showed up. For high precision, they also had a symbol equivalent to our decimal point, a ‘sexagesimal point’, indicating that the numbers to its right are multiples of , × = , and so on. Archaeologists represent this symbol by a semicolon (;). For example,

  in decimal (to a close approximation).

  About 2,000 astronomical tablets have been found, mainly routine tables, eclipse predictions, and so on. Of these, 300 are more ambitious - observations of the motion of Mercury, Mars, Jupiter, and Saturn, for instance. The Babylonians were excellent observers, and their figure for the orbital period of Mars was 12,59;57,17 days - roughly 779.955 days, as we’ve just seen. The modern figure is 779.936 days.

  Traces of sexagesimal arithmetic still linger in our culture. We divide an hour into 60 minutes and a minute into 60 seconds. In angular measure, we divide a degree into 60 minutes and a minute into 60 seconds, too - same words, different context. We use 360 degrees for a full circle, and 360 = 6×60. In astronomical work, the Babylonians often interpreted the numeral that would usually be multiplied by 60×60 as being multiplied by 6×60 instead. The number 360 may have been a convenient approximation to the number of days in a year, but the Babylonians knew that 365 and a bit was much closer, and they knew how big that bit was.

  Nobody really knows why the Babylonians used base 60. The standard explanation is that 60 is the smallest number divisible by 1, 2, 3, 4, 5 and 6. There is no shortage of alternative theories, but little hard evidence. We do know that base-60 originated with the Sumerians, who lived in the same region and sometimes controlled it, but that doesn’t help a lot. To find out more, good sites to start from are:

  en.wikipedia.org/wiki/Babylonian_numerals

  www.gap-system.org/~history/HistTopics/Babylonian_numerals.html

  Magic Hexagons

  You’ve probably heard of magic squares - grids of numbers that add up to the same total when read horizontally, vertically or diagonally. Magic hexagons are similar, but now the grid is a honeycomb, and the three natural directions to read the numbers are at 120° to each other. In Cabinet (page 270) I told you that there are only two possible magic hexagons, ignoring symmetrically related ones: a silly one of size 1 and a sensible one of size 3.

  The only possible normal magic hexagons, of size 1 and 3, and an abnormal hexagon of size 7.

  That’s true for ‘normal’ magic hexagons, where the numbers are consecutive integers starting 1, 2, 3, . . . . But it turns out that there are more possibilities if you allow ‘abnormal’ ones, where the numbers remain consecutive but start further along, say 3, 4, 5, . . . . The largest known abnormal magic hexagon was found by Zahray Arsen in 2006. It has size 7, the numbers run from 2 to 128, and the magic constant - the sum of the numbers in any row or slanting line - is 635. Arsen has also discovered abnormal magic hexagons of size 4 and 5. See en.wikipedia.org/wiki/Magic_hexagon

  The Collatz-Syracuse-Ulam Problem

  Simple questions need not be easy to answer. Here’s a famous example. You can explore it with pencil and paper, or a calculator, but what it does in general baffles even the world’s greatest mathematicians. They think they know the answer, but no one can prove it. It goes like this.

  Think of a number. Now apply the following rules over and over again:

  • If the number is even, divide it by 2.

  • If the number is odd, multiply it by 3 and add 1.

  What happens?

  I thought of 11. This is odd, so the next number is 3×11 + 1 = 34. That’s even, so I divide by 2 to get 17. This is odd, and leads to 52. After that the numbers go 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. From there, we get 4, 2, 1, 4, 2, 1 indefinitely. So usually we add a third rule:

  • If you reach 1, stop.

  In 1937, Lothar Collatz asked whether this procedure always reaches 1, no matter what number you start with. More than seventy years later, we still don’t know the answer. There are several other names for this problem: the Syracuse problem, the 3n + 1 problem, the Ulam problem. It is often posed as a conjecture which states that the answer is yes, and that’s what most mathematicians expect.

  The fates of the numbers 1-20, and anything else they lead to.

  One thing that makes the Collatz-Syracuse-Ulam problem or conjecture hard is that the numbers don’t always get smaller as you proceed. The chain starting with 15 gets up to 160 before eventually subsiding. Little old 27 positively explodes:

  It takes 111 steps to get to 1. But it does get there eventually.

  This kind of thing makes you wonder whether there might be s
ome particular number for which the process is even more explosive, and heads off to infinity. The numbers will go up and down a lot, of course. Any odd number leads to an increase, but the number can’t increase twice in succession: when n is odd, 3n + 1 is even, so the next step after that is to divide by 2. But the result at that stage is still bigger than n; in fact, it’s (3n + 1). However, if this is also even, we get to something smaller than n, namely (3n + 1). So what happens is quite delicate.

  If no number explodes to infinity, the other possibility is that there might be some other cycle which some numbers hit instead of 4 → 2 → 1. It has been proved that any such cycle must contain at least 35,400 terms.

  Up to 100 million, the number that takes longest to reach 1 is 63,728,127, which requires 949 steps.

  Computer calculations show that every starting number up to at least 19×258 ≈ 5.48×1018 eventually hits 1. This is impressively large, and a lot of theoretical input has to go into the computation - you don’t just check the numbers one by one. But the example of Skewes’ number (see page 46) shows that 1018 isn’t really very big, as such things go, so the computer evidence isn’t as convincing as it might seem. Everything we know about the question conspires to indicate that if there is an exceptional number that doesn’t hit 1, it will be absolutely gigantic.

  Probability calculations suggest that the probability that some number heads off to infinity is zero. However, such calculations are not rigorous, because the numbers arising are not truly random. Exceptions might still occur anyway, and even if the argument were rigorous it would not rule out running into a different cycle.

  If the process is extended so that it can start with zero or negative integers, four other cycles appear. They all involve numbers bigger than -20, so you might like to search for them (see the answer on page 279). The conjecture now becomes: these five cycles are all that can happen.

  There are also connections with chaotic dynamics and fractal geometry, which lead to some beautiful ideas and pictures, but don’t solve the problem either. There’s a lot of information about this problem on the internet, for example:

  en.wikipedia.org/wiki/Collatz_conjecture

  mathworld.wolfram.com/CollatzProblem.html

  www.numbertheory.org/3x+1/

  The Jeweller’s Dilemma

  Rattler’s Jewellers had promised Mrs Jones that they would fit her nine pieces of gold chain together to make a necklace, an endless loop of chain. It would cost them £1 to cut each link, and £2 to rejoin it - a total of £3 per link. If they cut one link at the end of each separate piece, linking the pieces one at a time, the total cost would be £27. However, they had promised to do this for less than the cost of a new chain, which was £26. Help Rattler’s avoid losing money - and, more importantly, make the cost to Mrs Jones as small as possible - by finding a better way to fit the pieces of chain together.

  Nine lengths of chain.

  Answer on page 279

  What Seamus Didn’t Know

  Our first cat, who rejoiced in the name Seamus Android, was possibly one of the few cats on earth that did not always land on its feet. Seamus didn’t have a clue. He would come down the stairs one step at a time, head first. At one point, Avril tried to train him to land on his feet by holding him upside down over a thick cushion and letting go. He liked the game but made no effort to turn in mid-air.

  Oops . . . What do I do now?

  There is a mathematical issue here. Associated with any moving body is a quantity called angular momentum, which, roughly speaking, is the mass multiplied by rate of spin about a suitable axis. Newton’s laws of motion imply that the angular momentum of any moving body is conserved, that is, does not change. So how can a falling cat turn over without touching anything?

  Answer on page 279

  Why Toast Always Falls Buttered-Side Down

  A cat is not the only proverbial falling object. Toast is another. It always lands buttered-side down. If not, you must have buttered the wrong side.

  Curiously, there is some truth to this adage. Robert Matthews has analysed the dynamics of falling toast, which does in fact have a propensity to land in a way that gets butter (or in my case marmalade) all over the carpet and ruins the toast. This lends support to Murphy’s law: Anything that can go wrong, will go wrong.

  Matthews applied some basic mechanics to explain why toast tends to land buttered-side down. It turns out that tables are just the right height for the toast to make one half turn before it hits the floor. This may not be an accident, because the height of tables is related to the height of humans, and if we were much taller then the force of gravity would smash our skulls if we tripped. Matthews thus traces the trajectory of toast to a universal feature of the fundamental constants of the universe in relation to intelligent life forms. To my mind, this is probably the most convincing example of ‘cosmological fine-tuning’.

  The Buttered Cat Paradox

  Suppose we put the previous two pieces of folklore together:

  • Cats always land on their feet.

  • Toast always lands buttered-side down.

  Therefore . . . what? The buttered cat paradox takes these statements as given, and asks what would happen to a cat, dropped from a considerable height, to whose back is firmly attached a slice of buttered toast - buttered-side outwards from the cat, of course.4

  At the time of writing, the favoured answer is that, as the cat nears the ground, some kind of antigravity effect kicks in, and the cat hovers just off the ground while spinning madly over and over.

  However, this argument has some logical loopholes, and it ignores basic mechanics. We’ve just seen that the mathematics of falling cats, and falling toast, lends scientific support to both adages. So what does the same mathematics say about a buttered cat?

  What happens depends on how massive the toast is compared with the cat. If the toast is an ordinary slice, the cat has no difficulty in coping with the small amount of extra angular momentum that the toast contributes, and still lands on its feet. The toast doesn’t land at all.

  However, if the toast is made of some kind of incredibly dense bread,5 so that its mass is much larger than that of the cat, then Matthews’s analysis applies and the toast lands buttered-side down with the cat upside down waving its paws frantically in the air.

  What happens for intermediate masses? The simplest possibility is that there is a critical cat-to-toast mass ratio [C : T]crit, below which the toast wins and above which the cat wins. But it wouldn’t surprise me to find a range of mass ratios for which the cat lands on its side or, indeed, exhibits more complex transitional behaviour. Chaos cannot be ruled out, as any cat owner knows.

  Lincoln’s Dog

  Abraham Lincoln once asked: ‘How many legs will a dog have if you call its tail a leg?’

  OK, how many?

  Discussion on page 281

  Whodunni’s Dice

  Grumpelina, the Great Whodunni’s beautiful assistant, placed a blindfold over the eyes of the famous stage magician. A member of the audience then rolled three dice.

  ‘Multiply the number on the first dice by 2 and add 5,’ said Whodunni. ‘Then multiply the result by 5 and add the number on the second dice. Finally, multiply the result by 10 and add the number on the third dice.’

  As he spoke, Grumpelina chalked up the sums on a blackboard which was turned to face the audience so that Whodunni could not have seen it, even if the blindfold had been transparent.

  ‘What do you get?’ Whodunni asked.

  ‘Seven hundred and sixty-three,’ said Grumpelina.

  Whodunni made strange passes in the air. ‘Then the dice were—’

  What? And how did he do it?

  Answer on page 282

  A Flexible Polyhedron

  A polyhedron is a solid whose faces are polygons. It has been known since 1813 that a convex polyhedron (one with no indentations) is rigid: it cannot flex without changing the shapes of its faces. This was proved by Augustin-Louis Cauchy. For a lo
ng time, no one could decide whether a non-convex polyhedron must also be rigid, but in 1977 Robert Connelly discovered a flexible polyhedron with 18 faces. His construction was gradually simplified by various mathematicians, and Klaus Steffen improved it to a flexible polyhedron with 14 triangular faces. This is known to be the smallest possible number of triangular faces in a flexible polyhedron. You can watch it flex on:

  demonstrations.wolfram.com/SteffensFlexiblePolyhedron/uk.youtube.com/watch?v=OH2kg8zjcqk

  You can make one by cutting the diagram from thin card, folding it, and joining the edges marked with the same letters. You can add flaps to do this, or use sticky tape. The dark lines show ‘hill’ folds, the grey ones ‘valley’ folds.

  Cut out and fold: dark lines are ‘hill’ folds, grey lines are ‘valley’ folds.

  Join the edges as marked to get Steffen’s flexible polyhedron.

  But What About Concertinas?

  Hang on a mo - isn’t there an obvious way to make a flexible polyhedron? What about the bellows used by blacksmiths to blow air into a fire? Or for that matter, what about a concertina? That has a flexible series of zigzag flaps. If you replace the two big pieces on the ends by flat-sided boxes, which they almost are anyway, then it’s a polyhedron. And it’s flexible. So what’s the big deal?

  Although a concertina is a polyhedron, and flexible, it is not a flexible polyhedron. Remember, the shapes of its faces are not permitted to change. They start out flat, so they have to stay flat, which means they can’t bend. Not even the tiniest bit. But when you play a concertina, and the flexible bit opens up, the faces do bend. Very slightly.