What happens when you multiply it by 17?
Answer on page 305
Which is Bigger?
Which is bigger: eπ or πe?
They are surprisingly close together. Recall that e ≈ 2.71828 and π ≈ 3.14159.
Answer on page 306
Sums That Go On For Ever
They sound like a childhood nightmare, but sums where you never get to the end are among the most important mathematical inventions. Of course, you don’t work them out by doing an infinitely long calculation, but, conceptually, they open up very powerful practical ways to calculate things that mathematicians and scientists want to know.
Back in the 18th century, mathematicians were coming to grips with - or often not coming to grips with - the paradoxical behaviour of infinite sums (or series). They were happy to use sums like
(where the . . . means that the series never stops) and they were also happy that this particular sum is exactly equal to 2. Indeed, if the sum is s, then
so s = 2.
However, the innocuous-looking series
1 - 1 + 1 - 1 + 1 - 1 + . . .
is a different matter. Bracketed like this:
(1 - 1) + (1 - 1) + (1 - 1) + . . .
it reduces to 0 + 0 + 0 + . . ., which surely must be 0. But bracketed like this:
1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + . . .
it becomes 1 + 0 + 0 + 0 + . . ., which surely must be 1. (The extra + signs in front of the brackets are there because the minus sign does double duty: both as an instruction to subtract, and to denote a negative number.) No less a figure than the great Euler tried the same sort of trick that we used to sum the first series, letting s be the sum and manipulating the series to get an equation for s. He observed that
s = 1 - 1 + 1 - 1 + 1 - 1 + . . .
= 1 - (1 - 1 + 1 - 1 + 1 - 1 + . . .) = 1 - s
and argued that s = .
This is a nice compromise between the conflicting values 0 and 1 that we’ve just found, but at the time Euler’s suggestion just muddied the waters further. And they were already fairly murky. The first satisfactory answer was to distinguish between convergent series, which settle down closer and closer to some specific number, and divergent ones, which don’t. For instance, successive steps in the first series give the numbers
which get ever closer to 2 (and only to 2). So this series converges, and its sum is defined to be 2. However, the second series leads to successive sums
1, 0, 1, 0, 1, . . .
which hop to and fro, but never settle down near any specific number. So that series is divergent. Divergent series were declared taboo, because they couldn’t safely be manipulated using the standard rules of algebra. Convergent series were better behaved, but even those sometimes had to be handled with care.
Much, much later it turned out that there are clever ‘summation methods’ that can assign a meaningful sum to certain divergent series, in such a way that appropriate versions of the standard rules of algebra still work. The key to these methods lies in the interpretation placed on the series, and I don’t want to dig into the rather technical ideas involved, except to record that Euler’s controversial can be justified in such a setting. In astronomy, another approach led to a theory of ‘asymptotic series’ that can be used to calculate positions of planets and so on, even though the series diverge. These ideas proved useful in several other areas of science.
The first message here is that, whenever a traditional concept in mathematics is extended into a new realm, it is worth asking whether the expected features persist, and often the answer is ‘some do, some don’t’. The second message is: don’t give up on a good idea, just because it doesn’t work.
The Most Outrageous Proof
The Great Whodunni, with the assistance of Grumpelina, produces a length of soft rope from thin air and ties a knot in it. A little further along, he ties a second knot. Holding the two free ends in each hand, he gives the rope a shake - and the knots disappear.
Mathematically, it’s obvious, of course. The second knot must be the anti-knot of the first one. You just tie it so that all the twists and turns cancel out. Right?
Wrong. Topologists know that there is no such thing as an anti-knot.
To be sure, there are very complicated knots that turn out not to be knotted at all. But that’s a different issue. What you can’t do is tie two genuine (un-unknottable) knots in the same piece of rope, clearly separated from each other, and then deform the whole thing into an unknotted piece of rope. Not if the ends of the rope are glued together or otherwise pinned down so that the knots can’t escape.
Not only do topologists know that: they can prove it. The first proofs were really complicated, but eventually someone found a very simple proof. Which is completely outrageous. You probably won’t believe it when I show it to you. Especially not when we’ve just been exposed to the paradoxical properties of infinite series.
A mathematician’s knot is a closed curve in space, and it is genuinely knotted if it can’t be continuously deformed into a circle - the archetypal unknotted closed curve. Real knots are tied in lengths of string that have ends, and the only reason we can tie them at all is because the ends can poke through loops to create the knot. However, the topology of such ‘knots’ isn’t very interesting, because they can all be unknotted. So mathematicians have to redefine knots to stop them falling off the ends of the string. Gluing the ends into a circle is one method, but there’s another one: put the knot inside a box and glue the ends to the walls of the box. If the string stays inside the box, the knot can’t escape over the ends. (The box can be any size and shape provided it is topologically equivalent to a rectangle; in fact, any polygon whose edges don’t cross is acceptable.) The two approaches are equivalent, but the second is more convenient for present purposes.
Two knots tied in boxes . . .
. . . and how to add them.
If you tie two knots K and L in turn along two separate strings, then they can be ‘added’ by joining the ends together. Call the result K + L. The unknot, a straight string without a knot, can sensibly be denoted by 0, because K + 0 is topologically equivalent to K, which we can write as K + 0 = K by employing the equals sign to indicate topological equivalence. The usual algebraic rules
K + L = L + K, K + (L + M ) = (K + L) + M
can also be proved; the second one is easy, the first requires more thought.
Now we can see why Whodunni’s trick must, indeed, be a trick. In effect, he appeared to tie two knots K and K* that cancelled each other out. Now, if two knots K and K* cancel, then
K + K* = 0 = K* + K
I’m tempted to replace K* by -K, because it plays the same role, but the notation gets a bit messy if I do.
The outrageous idea is to consider the infinite knot
K + K* + K + K* + K + K* + . . .
Bracketed like this:
(K + K*) + (K + K*) + (K + K*) + . . .
we get 0 + 0 + 0 +. . ., which in topology as well as arithmetic is equal to 0. But bracketed like this:
K + (K* + K) + (K* + K) + (K* + K) + . . .
we get K + 0 + 0 + 0 + . . ., which in topology as well as arithmetic is equal to K. Therefore, 0 = K, so K was not a genuine knot to begin with.
In the previous item, we saw that this argument is not legitimate for numbers, and that’s what makes the proof seem outrageous. However, with some technical effort it turns out that it is legitimate for knots. You just have to define the infinite ‘sum’ of knots using ever-smaller boxes. If you do that, the sum converges to a well-defined knot. The manipulations with brackets are correct. I don’t claim that’s obvious, but if you’re a topologist it pretty much is.
Tying a wild knot inside a triangle formed from an infinite sequence of shrinking trapezoidal boxes.
Infinite knots like this are called wild knots, and as the name suggests they should be handled with care. A mathematician called Raymond Wilder invented an especially unruly class of knots. You can guess what th
ose are called.
Colorado Smith and the Solar Temple
Smith and Brunnhilde had penetrated to the inner sanctum of the Solar Temple of Psyttakosis IV, overcoming various minor obstacles on the way, such as the Pit of Everlasting Flame, the Creepy Crocodile Corridor, and the Valley of Vicious Venomous Vipers. Now, panting slightly from their exertions, they stood at the edge of the Temple Plaza - a square array of 64 slabs, four of which were decorated with a golden sun-disc. Behind them, the only entrance had been closed by a shining disc of solid gold with the weight of a dozen elephants.
But that was only to be expected. As Smith said, ‘We’ll just have to think our way out.’
Location of the sun-discs.
For once, Brunnhilde did not find this entirely reassuring. Maybe it was the earthquake and the puffs of dust thickening the air around them. Or was it the roar of approaching water? The carpet of scorpions on the floor, scuttling from cracks in the stonework? Or just the spikes round all the walls, which even now were extending towards them?
‘What do we have to do this time?’ she asked, having been in this position so often that she knew the script by heart.
‘According to the Lost Papyrus of Bentnosy, we must choose four non-overlapping connected regions, each composed of 16 slabs, so that each includes one slab with a sun-disc,’ replied Smith. ‘Then the secret exit will open and let us into the adjoining treasure chamber - the one with those caskets of diamonds and emeralds I told you about. From there we just have to get through the underground maze that leads to the—’
‘That seems easy enough,’ said Brunnhilde, quickly sketching a solution. She caught his eye. ‘What’s the catch, Smith?’
Not like this!
‘Ah . . . Well, according to an obscure inscription on the Oxyrhincus Ostracon of Djamm-Ta’art, which is a Late Period commentary on Bentnosy’s papyrus, each of the four regions must be the same shape.’
‘Ah. That makes it harder.’ Brunnhilde smiled a hopeful smile, and tore up her sketch. ‘I suppose the answer is in Bentnosy’s papyrus?’
‘Apparently not,’ said Smith. ‘It’s not on the Ostracon, either - front or back.’
‘Oh. Well, do you think we’ll be able to work it out before that huge block of granite squashes us to the thickness of gold leaf?’
‘What block of granite?’
‘The one hanging over our heads on burning ropes.’
‘Oh, that block of granite. Strange, Bentnosy didn’t mention anything like that.’
Help Smith and Brunnhilde escape their dire predicament.
Answer on page 308
Why Can’t I Add Fractions Like I Multiply Them?
Well, you can if you wish - it’s a free country. Allegedly. But you won’t get the right answer.
At school we are taught an easy way to multiply fractions: just multiply the numbers on the top, and those on the bottom, like this:
But the rule for adding them is much messier: ‘Put them over the same denominator (bottom), then add the numerators (tops).’ Why can’t we add them in a similar way? Why is
wrong? And what should we do instead?
Answer on page 308
Farey, Farey, Quite Contrary
As soon as you say that some mathematical idea makes no sense, it turns out to be really useful and perfectly sensible. Although the rule
is not the correct way to add fractions, it is still a possible way to combine them, as the geologist John Farey, Sr, suggested in 1816 in the Philosophical Magazine. He hit on the idea of writing all fractions a/b whose denominator b is less than or equal to some specific number, in numerical order. Only fractions whose numerical values lie between 0 and 1 (inclusive) are allowed, so 0 ≤ a ≤ b. To avoid repetitions, he also required the fraction to be in ‘lowest terms’, which means that a and b do not have a common factor (bigger than 1). That is, a fraction like is disallowed, because 4 and 6 both have the common factor 2. It should be replaced by , which has the same numerical value but doesn’t involve common factors.
The resulting sequences of fractions are called Farey sequences. Here are the first few:
Farey noticed - but could not prove - that in any such sequence, the fraction immediately between a/b and c/d is the ‘forbidden sum’ (a + b)/(c + d). For instance, between and we find , which is (1 + 2)/(2 + 3). Augustin-Louis Cauchy supplied a proof in his Exercises de Mathématique, crediting Farey with the idea. Actually, it had all been published by C. Haros in 1802, but nobody had noticed.
So, although you can’t add two fractions this way, the formula has its uses, and we can define the mediant
provided that the fractions are in lowest terms. One problem with them not being in lowest terms is that different versions of the same fraction can lead to different results. For example,
which is different.
Farey sequences are widely used in number theory, and also show up in non-linear dynamics - ‘chaos theory’.
Pooling Resources
Alice and Betty owned adjacent market stalls, and both were selling cheap plastic bracelets. Each had 30 bracelets. Alice had decided to price hers at 2 for £10, while Betty was thinking of charging 3 for £20. So together they would make £150 + £200 = £350, provided that they both sold all their bracelets.
Worried that the competition might destabilise the market, they decided to pool their resources, and reasoned that 2 for £10 and 3 for £20 combine to give 5 for £30. At that price, if they sold all 60 bracelets, then their total income would be £360, which was £10 better.
Just across the way, Christine and Daphne were also selling bracelets, and also had 30 each to sell. Christine was thinking of selling hers at 2 for £10, while Daphne was thinking of undercutting the competition severely by selling hers at 3 for £10. When they got wind of what Alice and Betty were doing, they too decided to pool their resources, and sell their 60 bracelets at 5 for £20.
Was this a good idea?
Answer on page 310
Welcome to the Rep-Tile House
A rep-tile is more properly known as a replicating polygon, and it is a shape in the plane that can be dissected into a number of identical copies, each the same shape but smaller. The shapes are allowed to have their boundaries in common, but do not otherwise overlap. If the polygon has s sides and it dissects into c copies, it is called a c-rep s-gon. Several different 4-sided rep-tiles (4-gons) are known. Most are 4-rep, but there are k-rep 4-gons for every k.
Top: replicating 4-gons. If the parallelogram at the bottom has sides 1 and √k, then it is rep-k.
Every triangle (3-gon) is 4-rep. Some special triangles are 3-rep or 5-rep.
Replicating 3-gons. The first can be any shape. The second has sides 1 (vertical) and √3 (horizontal). The third has sides 1 (vertical) and 2 (horizontal).
Only one 5-sided rep-tile has yet been discovered: the sphinx. It requires four copies. There is a unique 5-rep 3-gon (triangle), and exactly three 4-rep 6-gons are known.
The only 4-rep 5-gon, the sphinx, and the three known 4-rep 6-gons.
There are several rep-tiles that stretch ‘polygon’ to the limit. And some go beyond that, having infinitely many sides - but, hey, let’s be broad-minded.
More exotic rep-tiles.
The first 4-rep 4-gon in the first picture is also rep-9. Can you dissect it into nine copies of itself? As far as I am aware, every known rep-4 tile is also rep-9, but this has not been proved in general.
Answer on page 310
Cooking on a Torus
Now, I’m going to set the utilities puzzle (Cabinet, page 199; and Hoard, page 117) for the third time, with a new twist. Metaphorically and literally. Three houses have to be connected to three utility companies - water, gas, electricity. Each house must be connected to all three utilities. Can you do this without the connections crossing? Assume there is no third direction to pass pipes over or under cables, and you are not allowed to pass the connections through a house or a utility company building. Note: connections.
No quibble-cooks (see page 116) allowed!
Connect houses to utilities, on a torus and a Möbius band.
What’s the difference this time? I’m not asking you to work in the plane. Try it on a torus (metaphorical twist) and a Möbius band (literal twist). A torus is a surface with a hole, like a doughnut. A Möbius band is formed by joining the ends of a strip of paper with a half-twist (Cabinet, page 111).
By the way: mathematicians think of a surface like the Möbius band as having zero thickness, so that the utilities, houses and lines connecting them lie in it, not on it. But a real sheet of paper actually has two distinct surfaces, very close together. You can either think of the surface as being transparent, or (better) imagine that the lines are drawn on paper with ink that soaks through, so that everything is visible on both surfaces of the paper.28
If you don’t use this convention, then some of the lines in my answer end up on the back of the strip and don’t join up with the houses or utilities. You are then trying to solve the analogous problem on a cylindrical band with a double-twist. Topologically, this is the same as an ordinary cylindrical band, and in particular it has two distinct sides. Now there is no solution. Why not? A cylinder can be flattened out topologically in the plane, to form an annulus - the region between two circles. So any solution of the puzzle on a cylindrical band also provides a solution in the plane. But no solution in the plane exists without cooking (Cabinet, page 199).
Ian Stewart Page 11