by Wells, David
Because in this case the two numbers are integers, it is easy to spot that they are 3 and 5. This particular transformation turns out to be useful, but many aren't. There is an element of experiment. You make a move (mentally in simpler cases) to see what you get and sometimes, naturally, the experiment fails.
This division throughout by x could be a waste of time or a big step towards a general method for solving all quadratic equations. (It's the latter.) The important point is that it is entirely a matter of algebraic seeing. Whatever area of maths grips you, there will be appropriate ways of seeing and other ways which are useless, which is why Jean Dieudoneé exhorted his students to develop an intuition for the abstract. Since all mathematics is more-or-less abstract, his advice applies to everyone. Fortunately, there is a very enjoyable means of acquiring this intuition: just remember that – like abstract games – ‘maths is not a spectator sport’, and get stuck in!
Archimedes’ lemma and proof by looking
This elegant theorem appears in a Book of Lemmas purportedly written by the great Greek mathematician Archimedes: two perpendicular lines divide the circumference of a circle into four arcs, AB, BC, CD and DA creating a very simple figure (Figure 16.1).
Figure 16.1 Archimedes’ lemma
Archimedes proved that AB + CD = BC + DA. To see why this is so, move the vertical line to the right. AB increases in length by the same amount that CD decreases so the total AB + CD does not change at all. Similarly, BC + DA is unchanged. Now move the horizontal line down a bit. The same is true.
So we can move the perpendicular lines anywhere we like, provided they remain parallel to their originals, and the total lengths, AB+CD and BC + DA will remain unchanged. Fine! We choose to move them so that their intersection is the centre of the circle, then AB + CD = BC + DA = one half of the circumference and so each sum equals half the circumference.
This proof-by-transformation is typically dynamic. We observe how the lines behave when we move them. It is so convincing because it appears to depend only on the symmetry of the circle – and what could be more obvious than that?! As so often the appearance of symmetry is so overwhelming that we are once convinced [Hutchins 1952: 564–5].
Chinese proofs by dissection
What is the radius of the incircle of the right-angled triangle in Figure 16.2? We could calculate it by finding the area of the triangle in two ways, and comparing them.
Figure 16.2 Chinese dissection to show inradius
If the radius is r and the sides are a, b and c, then the area is equal to, ar + br + cr; but it also ab. Putting these two results together,
However, Liu Hui in his third-century book, Commentary on The Nine Chapters of the Mathematical Art, drew a diagram like Figure 16.3 (which we have simplified slightly).
Figure 16.3 Dissected triangle re-arranged
The height of the rectangle is r, and its length is the distance round the triangle [Yan & Shiran 1987: 70–71].
This suggests – by analogy – that other results might be proved by dissection and indeed they were. The ancient Chinese mathematicians appreciated the value of analogy as this dialogue in the 2000-year old Zhoubi suanjing illustrates. Master Chen is explaining to a pupil:
The method of calculation is very simple to explain, but it is of wide application. This is because ‘man has a wisdom of analogy’ that is to say, after understanding a particular line of argument one can infer various kinds of similar reasoning, or in other words, by asking one question one can reach ten thousand things. When one can draw inferences about other cases from one instance and one is able to generalise, then one can say that one really knows how to calculate.
Master Chen continues:
after you have learnt something, beware that what you have learnt is not wide enough and after you have learnt widely, beware that you have not specialised enough.
[Yan & Shiran 1987: 28]
Napoleon's theorem
Napoleon's theorem, named after the French emperor who was also an amateur mathematician, says that if you construct equilateral triangles on the sides of any triangle, the centres form another equilateral triangle (Figure 16.4).
Figure 16.4 Napoleon figure
The same is true if the equilateral triangles are drawn on the opposite sides, facing inwards. The first case can be proved by continuing the figure to create the tessellation (Figure 16.5a) which has rotational symmetry.
Figure 16.5a Napoleon tesselation – 1
We can delete some of the lines and highlight some of the triangles to make the rotational symmetry even more obvious (Figure 16.5b).
Figure 16.5b Napoleon tesselation – 2
Napoleon's theorem might seem just one more idiosyncratic, attractive and intriguing property from elementary Euclidean geometry. However, we can generalise it to show that this is not so. Draw the tessellation in Figure 16.6 – which if continued collapses into a limit point at the top of the figure – and we get a different perspective.
Figure 16.6 Napoleon spiral tesselation
This tessellation ‘works’ with triangles of any two different shapes. Now look carefully (as always) at the bit of the tesselation shown in Figure 16.7.
Figure 16.7 Napoleon with non-central points
According to the tessellation, any three corresponding points in the three outer triangles will form a triangle of the same shape. Make them all equilateral and choose their centres and – ‘Hey presto!’ – Napoleon's theorem. This novel perspective on Napoleon's theorem has many more consequences [Wells 1988: Chapter 6] [Wells 1995: 178–184].
The polygonal numbers
What is the sum of the counting numbers, from 1?
We can find the answer to this question from Figures 16.8 and 16.9 which illustrates the sum 1 to 7.
Figure 16.8 Divided rectangle to show 1 + 2 + 3 ⋯
Figure 16.9 Divided rectangle – second version
The rectangle in Figure 16.8 is 7 by 8 and has been dissected into two identical parts, each with an area of 1 + 2 + 3 + 4 + 5 + 6 + 7, so that total must be 1/2 of 7 × 8 = 28. For the sum 1 to 8 the slightly different arrangement in Figure 16.9 shows that the answer is now 1/2 of 8 × 9.
This idea can be applied to any sequence of the counting numbers from 1, so the answer to our question seems clearly to be N(N+1). The same idea proves-by-looking that the first N odd numbers sum to N2, a square number (Figure 16.10)
Figure 16.10 Square divided into gnomons
Alternatively, you don't need to draw a picture at all, you just lay out some pebbles or counters. By summing these sequences we are doing nothing to help the handyman, surveyor, businessman or housewife. We are simply stepping back and noticing that the odd numbers are interesting. ‘Looking’, as we have already said, doesn't have to mean looking at a diagram. We can ‘look’ at algebraic equations, formulae, or pages of manipulation and ‘see’ what is going on, and ‘see’ possibilities.
Linguists have noted that the commonest use of ‘I see what you mean’ is to mean, ‘I understand you.’ We study a position at chess or a mathematical diagram of equation to ‘see what it means’.
One of the most famous anecdotes in the history of mathematics tells of the young Gauss in first school. Told to add up the numbers from 1 to 100, the rest of the class worked away at their slates while Gauss wrote down the answer, 5050.
So the grand total is 101×50 = 5050. But this also equals 1/2×100×101 matching our previous general rule. Gauss ‘saw’ arithmetically or even algebraically, in this case, rather than geometrically.
The Greeks were as intrigued by number patterns as they were by geometry though they made less progress in understanding them, partly because they had no algebraic notation and partly because they didn't ask the questions that would have forced them to make progress. As Nichomachus (c.100 CE) explained in his Arithmetic, the numbers in this sequence, 1
3
6
10
15
21
28‘and so on
'
are called triangular because, ‘their regular formations, when expressed graphically (Figure 16.11), will be at once triangular and equilateral’.
Figure 16.11 A triangle of dots
Figure 16.12 A square of dots
He then introduced the square numbers (Figure 16.12), and the pentagonal numbers, 1, 5, 12, 22, 35, 51, 70…. followed by the hexagonal, heptagonal and so forth, explaining that, ‘the doctrine of these numbers is to the highest degree in accord with their geometrical representation’ which he illustrated by pointing out that each square is divided diagonally into two triangles and every square number into two consecutive triangular numbers (as shown in Figure 16.13) and that any triangular number added to the ‘next’ square number makes a pentagonal number (Figure 16.14), triangles
1
3
6
10
15
21
28
…
squares
1
4
9
16
25
36
49
…
pentagonals
1
5
12
22
35
51
70
…
Figure 16.13 Square of dots, divided into two triangles
Figure 16.14 Pentagonal array of dots
…and that the same process applies to triangles added to pentagons to get the hexagonal numbers ‘and so on to infinity’ [Hutchins 1952: 835]. There are many other such patterns. Figure 16.15 shows an example.
Figure 16.15 Hexagon divided into six triangles
Every hexagonal number is the sum of six identical triangular numbers, plus 1: Hn = 6Tn−1 + 1. The visual diagram allows any number of such identities to be ‘read off’ from this and similar figures, and ‘proved’ by sight, which would take relatively complex manipulation to prove by algebra – and algebra does not allow them to be ‘read off’ in the first place.
Problems with partitions
The conclusions that Nichomachus drew from dissecting the polygonal numbers are simple examples of combinatorics or The Art of Advanced Counting [Berge 1971]. His trick – compare Liu Hui – was to count the same set of objects in two ways, and compare the results.
Partitions is another topic where this trick works very well – up to a point. The partitions of a number are simply the ways of breaking it up into parts, including the original number as the first part. Here are the partitions of the integers 1 to 4:
And following are the partitions of 8:
Unless we have missed a line, the number of partitions of 8, called p(8), is 22. The sequence of numbers of partitions starts like this: n
1
2
3
4
5
6
7
8
9
10
…
p(n)
1
2
3
5
7
11
15
22
30
42
…
Finding a formula for p(n) is extremely difficult but we can, by merely looking, draw some very powerful conclusions about special types of partitions. Figure 16.16 is a picture of one partition of 26.
Figure 16.16 Partitions diagram and the same, rotated
If we read off the horizontal rows, then this is 26 = 8 + 6 + 6 + 5 + 1. However, if we rotate the diagram through ninety degrees, then we get the second diagram which shows the ‘dual’ partition: 26 = 5 + 4 + 4 + 4 + 4 + 3 + 1 + 1.
The largest number, 8, in the first partition is the number of numbers in the second partition, and the largest number in the second, 5, is the number of numbers in the first partition. This observation allows us to draw a remarkable conclusion:
The number of partitions of N with at most M parts equals the number of partitions of N in which no part exceeds M.
For example, these are the partitions of 5 into at most 3 parts, and into parts no greater than 3:
Why do these two partitions match up so perfectly? The partition diagrams in Figure 16.17 show why. Notice that the fourth line above is symmetrical and the third and fifth lines match ‘in reverse’, so we have only drawn three diagrams.
Figure 16.17 Partitions diagram in three parts
This conclusion is remarkable for two reasons. It is extremely simple and convincing and yet we need have no idea how many partitions there are of 5 with at most 3 parts. We ignore the very difficult problem of actually counting partitions and instead simply match them up, one-to-one. Now that really is Advanced Counting!
We can be confident of all these conclusions because we are dealing with very simple objects and sets of objects. Our conclusions really do seem both obvious and self-evident. According to the philosopher and mathematician Bertrand Russell,
self-evidence is a psychological property and is therefore subjective and variable. It is essential to knowledge, since all knowledge must be either self-evident or deduced from self-evident knowledge.
A few lines later he added, ‘self-evidence has degrees’ [Russell 1913: 492]. That's true, and here it is of the highest degree. As the proverb says, ‘Seeing is believing’. We might add that this psychological quality of being self-evident is especially prominent in abstract games and other formal situations. Scientific theories, for all their power and extraordinary success, are never self-evident, and as for the meaning of poetry and painting, well, critics can and do argue endlessly.
Invented or discovered? (Again)
These patterns seem to be a ‘natural’ feature of the counting numbers. We didn't put them there and nor did the ancient Greeks and if there are other intelligent beings in the universe who have developed beyond the caveman level we can be confident that they will have thought of counting numbers and spotted features which are forced by the very process of counting in sequence, so it is tempting to say that they have been discovered.
It is true that the particular form they are given – all patterns can be represented in many ways – is at the choice of the discoverer and so we could say that, once again, there is an element of invention.
On the other hand, many different systems of counting numbers have been invented and we can be sure that any distant civilisation is sooner or later going to ‘discover’ the idea of counting and so ‘invent’ a counting system. Just as the rules of a game such as Hex or Go are invented but then force us to recognise a rich structure of possibilities that is not initially apparent to the players, so the very simple rules by which we count also force the existence of the rich patterns and relationships that Nichomachus started to sketch and Fermat, Euler, Gauss and many others investigated with such extraordinary imagination and insight.
17 Structure
Mathematics is the art of giving the same name to different things.
Henri Poincaré [Poincaré 1914]
Mathematicians are always and everywhere looking for patterns. If they can't see patterns on the surface, they look underneath the surface. If they cannot spot a pattern in some isolated object they try to put it in a broader context.
Patterns in everyday life can be superficial – mere decoration – but in mathematics they are more significant. Rather than surface decoration, they resemble internal skeletons. They are the structure upon which mathematics is constructed and if you understand the skeleton, then you understand a lot about the object itself.
Patterns have another use: the same patterns – or skeletons – turn up again and again in different situations, allowing insight to be spread from one to the other, so mathematicians are also, like the best scientists, obsessed with analogies:
For above all I love analogies, my most faithful teachers, acquainted with all the secrets of nature.
Kepler [Rigaud 1841]
Spotting an analogy is j
ust spotting a pattern or common structure so it is no surprise that analogy and structure are basic to mathematics or that beauty is so important, since beauty depends on all three features.
The next section looks at Pythagoras' theorem and at coordinates. It is a crucial feature of mathematics that a situation will be easier to understand if you represent it appropriately and mathematicians spend much time and effort trying to find the best representations.
Following Pythagoras, we look at another aspect of structure, the search for similar structures, and for hidden structures below the surface. The primes are very difficult to understand: one promising line of enquiry is to consider which features are peculiar to the primes alone and which features might be shared by other strange sequences – such as the lucky numbers.
If we draw graphs of quadratic functions they appear to sometimes have two roots and sometimes no roots with a ‘boundary’ case in which the parabola touches the axis. With cubic functions there are apparently 1 or 3 roots, with a similar boundary case. Look more deeply, however, and a hidden structure reveals that – when suitably viewed through the lens of complex numbers – they always have the maximum number of roots, (provided multiple roots are correctly counted), a perfect example of hidden structure illuminating the landscape.