by Wells, David
Sylvester added that, ‘Induction and analogy are the special characteristics of modern mathematics’ [Midonik 1968 v.2: 370–1].
Whew! But Sylvester was right. Mathematicians, like game players, construct their own miniature worlds whose landscapes they explore in a scientific and game-like manner. In these new worlds are a wealth of objects, properties, and relationships which the mathematician observes and which – so often – appear to have a life of their own. We say that a series or sequence ‘goes on for ever’ or ‘tends to infinity’ or that a one shape is ‘transformed’ into another. We observe not merely static objects, lying as it were in museum cases, but dynamic scenarios which move and change. The fact that so many of these objects were constructed and named by mathematicians in the first place does not make them any less able to be studied by the methods of science: it just means that you hope to go one step further and mathematically prove your conclusions.
Empirical scientists can look, explore, conjecture and test, but not prove, so we might say that, yes, mathematics is also empirical but it is empirical-plus and it's plus because it is game-like, the crucial point.
The empirical but game-like nature of maths appears especially when we use physical drawings or models as aids to thinking. Archimedes who was killed by a Roman soldier as he drew a geometrical diagram in the sand did not mistake the diagram for mathematical reality. It was a representation and a rather crude one, forced by the lack of cheap writing materials. A mere child could produce a more accurate representation today, with endless supplies of paper and accurate drawing instruments – and maybe a computer and a geometrical drawing package.
Such representations are essential because however well you can visualise it will at best be difficult and usually be impossible to do many experiments mentally. Computers have created new possibilities for calculation and representation and so given a boost to experimental and scientific mathematics – which, however, have always existed because mathematicians have always explored their miniature worlds.
Triangle geometry: the Euler line of a triangle
How were the theorems of triangle geometry discovered? I can only conjecture that as with much of mathematics, it emerged from long hours of ‘playing around’.
[Davis 1995: 206]
Euclidean geometry is a model of a physical plane so you can do your experiments on any flat plane surface. Rough experiments may do to begin with though of course you get more convincing results if your experiments are as accurate as possible.
The lines in actual geometrical figures are never infinitely thin; indeed, if drawn with a pencil or felt-tip pen they could even be a millimetre or more thick. Likewise, points can never be drawn to be infinitely small and dots are likely to be even wider than the lines (Figure 11.1).
Figure 11.1 Hand-drawn triangle with mid-point triangle
This causes no confusion however, even to most school pupils who will not be muddled by a sketch like this in which the points are dots and the lines are visibly thick and not even straight. Because it's so easy to draw figures – and because it's fun to experiment! – it has been said that almost all of elementary geometry was originally discovered by experiment, and only later proved. Of course, if you are going to experiment then the diagram should be pretty accurate. Figure 11.2 is an example.
Figure 11.2 Triangle with mid-points and medians
We have drawn lines from each vertex of a triangle to the mid-points of the opposite sides. The lines concur, they pass through the same point which happens to be the centre of gravity of the triangle (usually called G): if this shape were cut out of cardboard then it would balance on a pin at this point.
This is only one of many centres of the triangle. Another centre can be found by drawing the perpendicular bisectors of the sides which concur at the centre of the circumcircle of the triangle, called the circumcentre (Figure 11.3).
Figure 11.3 Triangle with circumcentre
The point where two perpendicular bisectors of two of the sides meet will be equidistant from all three vertices, and so it will lie on the third perpendicular bisector also – and will be the centre of the unique circle through all three vertices.
Figure 11.4 Triangle with altitudes
We could also draw the altitudes, the perpendicular lines from each vertex to the opposite side, which concur at the orthocentre (Figure 11.4). What happens if we draw all three of these centres on the same diagram? We might suspect that we would get three unrelated points because, after all, what connection would you expect between three ‘centres’ constructed in such totally different ways?
Figure 11.5 Triangle with Euler line
The answer is surprising. The orthocentre, H, the circumcentre, O, and the centre of gravity, G, lie on the same straight line (Figure 11.5), called the Euler line of the triangle after Euler who first discovered this fact. As it happens, the figure has an extra property, also noticed by Euler. Point G divides the line OH in the ratio 1 to 2.
Modern geometry of the triangle
The Greeks knew the four basic triangle points, the centre of gravity, the incentre, the circumcentre and the orthocentre. Subsequently, many more points have been discovered, described and analysed, many of them without a doubt by ‘playing around’ which is an aspect of creativity in every human endeavour, not just mathematics.
Euler and Feuerbach between them discovered the nine-point circle (Figure 11.6).
Figure 11.6 Triangle with nine-point circle
The nine points which always lie on a circle are the mid-points of the sides, the feet of the perpendiculars from each vertex to the opposite side, and the points half-way between each vertex and the orthocentre. I can't resist mentioning that when I was aged 12 or 13 we had to learn to prove this theorem, following Mr Straw's explanation on the blackboard. (However, we did not have to prove any of the other properties of the nine point circle, for example that it touches the incircle and the three excircles of the triangle.)
The nine-point circle was a great novelty to traditional Euclidean geometers and the start of a revival. In 1873 Emile Lemoine (1840–1912) read a paper to the French Association for the Advancement of Science on ‘Some properties of a remarkable triangle point.’ It was eventually called the Lemoine point though he was not actually the first to discover it. Henri Brocard (1845–1922) gave his name to the twin Brocard points, found not by experiment but by solving a published challenge, to find a point O such that the angles OAB, OBC and OCA are equal (Figure 11.7).
Figure 11.7 Brocard points
The diagram shows a geometrical construction for one Brocard point: each circle passes through one vertex touching a side there and through a second vertex. If the angles OAC, OCB and OBA are made equal then we get a second Brocard point.
Soon there was a torrent, a stampede, of new points, lines and circles, identified and classified like specimens in a natural history museum and named after the explorers who discovered them. Joseph Neuberg (1840–1926) gave his name to the Neuberg circle, and Christian Nagel (1803–1882), to the Nagel point. Ludwig Kiepert (1846–1934) named the Kiepert hyperbola, and G. de Longchamps (1842–1906) named the De Longchamps point, and so on. However, the rush soon came to an end. As Philip Davis reports, in the comprehensive Encyklopaedie der Mathematischen Wissenschaften of 1914 the Modern Geometry of the Triangle was honoured with a hundred-page entry, yet today the subject is (almost) forgotten. Why?
Mathematical explorers can be naturalists fanning across the plains and through the forests, recording everything that they discover, or they can be geologists and geomorphologists, physicists or chemists with their instruments, studying the broad features of the landscape, digging underneath the earth to discover how the terrain was formed, detecting features that are invisible on the surface, attempting to understand the landscape as deeply as possible. Mathematical creativity can be either broad and shallow or deep and profound.
Pythagoras' theorem is deep because it represents a basic feature of the pla
ne surface of Euclidean geometry. Euclid's geometry itself is deep, not only for the wealth of theorems but for the very idea that you can start with just a few assumptions and infer so much from them. The ancient Greeks, whose work Euclid summarised, were the first people to imagine such a possibility.
Euclid's geometry also presented a powerful technique. However, as that technique was better understood and new techniques added – including the technique of coordinates – the study of Euclidean geometry became more routine, more a matter of merely sound method, and therefore less challenging and less genuinely surprising. The Euler line was a great surprise and so was the nine-point circle, but the Nagel point far less so. There are now literally hundreds of identified ‘special points’ – so they are no longer so special. The great challenges of modern mathematics are now over the hills and far away from triangle geometry.
Figure 11.8 21-point cubic
And yet, there is a chance that it is already reappearing as a serious subject for research, because thanks to the computer it is now extremely easy to search for new special points and to catalogue their properties, which leaves the mathematician free to ask deeper questions about objects such as the cubic curve in Figure 11.8, known as the Neuberg cubic or 21-point cubic, because it was originally proved to pass through 21 special points of the triangle, including the vertices of the triangle, the orthocentre, the circumcentre, and the centres of the inscribed and three escribed circles, and the 6 vertices of the equilateral triangles in Napoleon's theorem.
It is now known to contain many more special points: why so many? It also has many other properties, such as, that the Euler line of the triangle is parallel to the asymptote of the cubic, and so are the tangents at the incentre and excentres. Why?
There are also many other relationships between the special points, many of which are collinear: why? (The answer has to do with group theory.)
There are two distinct ‘Why?’ questions here: the shallow ‘Why?’ can be answered by a straightforward proof which explains how the property follows from more basic facts. The deeper question ‘Why?’ asks why the property exists in the first place. We met this distinction when we looked at Pythagoras' theorem. To prove it is relatively easy and can be done in hundreds of ways: to explain why it exists at all is a much subtler, deeper and more important question.
Treated as individual curiosities, most of the special points are not so special but examined as a whole, new questions beg to be answered and new features are observed and demand to be understood. And that is how mathematics progresses.
The Seven-Circle Theorem, and other New Theorems
As if to illustrate that the variety of geometrical theorems is indeed endless, a curious small book was published in 1974 with this title.
Figure 11.9 Seven-circle theorem
The elegant in Figure 11.9 is almost self-explanatory. A chain of six circles touches the inside of another circle. (They could also touch instead on the outside.) Join the points of tangency of ‘opposite’ circles in the chain and the three lines are concurrent. As always happens with such theorems, we can imagine one or more of the circles ‘blowing up’ until it becomes a straight line. In Figure 11.10, two of the circles have done so, and all six circles are external.
Figure 11.10 Seven-circle theorem for two straight lines [Evelyn, Money-Coutts & Tyrrell 1974: 32, 35, 39]
As the title says, the book was also about ‘other new theorems’ including the heptagon theorem, the three-conics theorem and the nine-circle theorem.
Where did these theorems come from? Did the three authors invent or discover them? Did they reason initially or start from experiment? We don't know because they cunningly remark only that, ‘We derived a lot of enjoyment from evolving them’, adding that they have included rather a large number of diagrams ‘for their beauty of design, which we have found to have an appeal to many of our friends who are not mathematicians’ [Preface].
It is a commonplace that modern science creates vast numbers of beautiful images, and mathematics-as-science is no exception. All the miniature worlds of mathematics are exceptionally beautiful, and elementary geometry is only unusual in making that fact so very obvious.
12 Numbers and sequences
Every teacher knows that pupils are far, far better at spotting number patterns than they are at proving them. Given the Fibonacci sequence: 1
1
2
3
5
8
13
21
…
in which each number is the sum of the previous two numbers, any pupil with enough experience to expect to find extra patterns might look at three consecutive entries, such as 3–5–8 or 5–8–13 and notice that:
and so on. To work out why the differences are first +1 and then −1, and why the pattern (with this slight variation) always works, is far harder. To spot the pattern is rather easy science but to prove it is (relatively) difficult mathematics.
It is no surprise that the most brilliant mathematicians, such as Euler and Gauss, have been great pattern spotters.
The sums of squares
We have seen that,
It is naturally tempting to wonder if you can add up the sequence of squares:
Yes, you can, and the standard result is n(n + 1)(2n+1) which always annoys me because the factor 2n + 1 seems out of place. Anyway, let's try to find the sum of the odd squares instead, which is less well known. A scientific approach is to calculate the first few sums and try to spot a pattern:
Since the sum of the squares is the product of three factors, with an extra factor of 1/6, we take this to be a giant hint and write down the factors of each number in the right-hand column. The last three rows are especially suggestive. The sums to 112 and 132 include the factors 11 and 13 but the sum to 92 does not include the factor 9. Where can it have gone to? Put on your Sherlock Holmes thinking cap or puff on your favourite pipe, and the answer will appear. All we have to do is to pinch the factor 1/6 from the sum of all the squares:
The sum of
is
so the sum,
ought to be,
It is, though we haven't proved that conclusion. Before we leave this little experiment, we will make another observation: the factors 2 in 2n −1, 2n and 2n + 1 suggest that we re-write the sum of the squares which is n(n + 1)(2n + 1), as
Aha! That annoying asymmetry has disappeared!
Easy questions, easy answers
Many puzzles about the polygonal numbers are relatively easy to answer, plausibly because they show very strong patterns and, we might speculate, the stronger the pattern, the easier the proof. After all, proof depends on pattern, and it is by spotting patterns that ideas for proofs appear.
There is a lot to be said for that large assumption, which we will meet again later. Here we are going to test it, by sketching some problems raised by the prime numbers.
The prime numbers
The theory of numbers, more than any other branch of pure mathematics, began by being an empirical science. Its most famous theorems have all been conjectured, sometimes a hundred years or more before they have been proved; and they have been suggested by the evidence of a mass of computation.
[Hardy 1920: 651]
The prime numbers are those with no factors other than themselves and 1. The sequence starts, 2
3
5
7
11
13
17
19
23
29
31
37
41
43
47
53
59
…
It looks extremely irregular and it is. We can see why this is no surprise if we construct the Sieve of Erastosthenes, named after the brilliant astronomer and mathematician who was a friend of Archimedes: (1)
2
3
4
5
6
&nb
sp; 7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
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49
50
