Worner Research Lecture 1996

Join the Dots and See the World

Professor Terry Mills

Joining the dots is a game for children. But it is also a mathematical activity with a history dating back hundreds of years and with many modern applications. In mathematics, this activity is called "interpolation" which is the subject of this lecture. The lecture has several purposes.

  • To present an overview of the study of interpolation, its history and its place in the modern world by establishing connections with science, engineering, information technology, business, social sciences, humanities, the fine arts, and health sciences.
  • To illustrate how the mathematical traditions of different countries contribute to this field.
  • To explore the relationship between teaching and research.
  • To share with members of the audience some modern mathematical concepts.

Table of Contents

  1. Interpolation
  2. Interpolation and approximation
  3. Smoothing
  4. Smoothing and approximation
  5. Complex examples
  6. International traditions
  7. Teaching and research
  8. Conclusions
  9. Acknowledgements
  10. Bibliography
  11. Biographical sketch
Interpolation
In this section we will define the concept of interpolation and mention some aspects of its historical development.

The concept of interpolation
Joining the dots is a game for children, but, as we will see in this lecture, there are many variations of this game which are important in various fields of knowledge. This well-known puzzle is one variation:

Find the next number in the sequence:   2, 4, 8, 16, 32, .....

Such puzzles may be encountered in primary school mathematics lessons and also in some aptitude tests used for screening job applicants. Now we can turn this problem into a curve fitting problem.

We will use x to denote the position of the number and y to denote the value of the number itself. The third number is 8: so when x = 3, we have y = 8. Thus the sequence can be represented as shown in the following table.

x 1 2 3 4 5
y 2 4 8 16 32

Table 1: Sequence of numbers

Then we plot the data as shown in Figure 1.

y                                                  y

-                                         *(5,32)  -
30+                                                  +30
-                                                  -
-                                                  -
-                                                  -
-                                                  -
20+                                                  +20
-                                                  -
-                                 *(4,16)          -
-                                                  -
-                                                  -
10+                                                  +10
-                         *(3,8)                   -
-                                                  -
-                 *(2,4)                           -
-         *(1,2)                                   -
0+                                                  +0
       x+-------+-------+-------+-------+-------+-------+x
0.0     1.0     2.0     3.0     4.0     5.0     6.0
Figure 1: Plot of data in sequence

If we find a curve through these points then we extrapolate that curve and find the value of y when x = 6. So we would have found the next point in the series. Thus the puzzle is, essentially, one of joining the dots.

Interpolation is defined to be the construction of a curve through given points. Thus, the above argument shows that the puzzle can be regarded as a problem in interpolation. We now present some historical notes and a couple of simple modern applications of interpolation.

Historical notes
Interpolation methods have a long history. Just over 300 years ago, Isaac Newton (Fraser (1927)) stated that the aim of interpolation is

To describe a geometrical curve which shall pass through any given points ..... Although the problem may seem to be intractable at first sight, it is nevertheless quite the contrary. Perhaps it is one of the prettiest problems that I can ever hope to solve.
These are strong words from a man who seemed to dominate intellectual life of the time in a way that would seem impossible these days. (For an excellent biography on Newton, see Westfall, R.S. (1980).) About 200 years ago, the French mathematician J.L. Lagrange (1792/3) wrote
La méthode d'interpolation est, après les logarithmes, la découverte la plus utile qu'on ait faite dans le calcul. [The method of interpolation is, after logarithms, the most useful discovery in calculus.]
The status of Lagrange is measured by the fact that he is the only mathematician and one of the very few scientists buried in the Pantheon in Paris.

Newton, Lagrange and other scientists were interested in interpolation methods in order to assist them to use mathematical tables which were so important at the time. Remember that Newton and Lagrange did not have the computers that we have today. In Newton's time a "computer" was a person who performed calculations or computations. This terminology appears also in publications which are not so old: on p.6 of Interpolation and Allied Tables (H.M. Nautical Almanac Office (1956)) we find the statement

Every computer makes mistakes....
- presumably a "computer" is a person.

To illustrate the applications that Newton and Lagrange had in mind, consider the following example. If we could find a formula which generated the 4 known square roots in Table 2, then we could use this formula to estimate the square root of 6, and, this way, the table of square roots could be extended to include the square root of 6.

x 1 4 6 9 16
y = square root of x 1 2 ? 3 4

Table 2: Square root table

In the closing years of the twentieth century, interpolation methods are still used to drive our tables further. The Australian Government Actuary produces life tables which describe the mortality rates that apply to the Australian population. These tables are extended using modern interpolation methods. (See Office of the Australian Government Actuary (1991), especially p.9.) Here we see a contemporary application of interpolation which is relevant to the life insurance industry and the study of demography in Australia and also builds on the work of Newton and Lagrange.

In 1996, VCE students who were studying Mathematical Methods Units 3 and 4 were asked to undertake a research project which dealt with interpolation (Board of Studies (1996)). Students had a choice of one of three projects all based around a common theme of "fitting functions". One of the projects deals with designing a bicycle track along the Yarra River in Kew. The design was to be developed by choosing various points along the river bank and then fitting a smooth curve through these points. The smooth curve thus found was the design for the track.

So, a problem which challenged the great Isaac Newton is now expected to be solved by high school students in Victoria. This illustrates quite dramatically how knowledge percolates though our society.

A problem
There is a fundamental problem in the study of interpolation methods. Although many clever people have considered interpolation methods for a long time, we cannot ignore the inescapable fact that, in general, there is more than one solution to the interpolation problem. We can see this in our puzzle.

Consider the relation y1 = 2x and the data in Table 1. As shown in Table 3, this certainly fits the data. With this formula we arrive at the prediction that when x = 6, y1 = 64.

Thus, we have found a mathematical justification for saying that the sequence is {2, 4, 8, 16, 32, 64}.

However, I can find a formula which would make the next number anything that we choose. Let us suppose that we want 42 to be the the next number in the sequence (Adams, D. (1979)). To achieve this, let

equation

A little checking shows that this too fits the data exactly as shown in Table 3. Ultimately, it leads to the prediction that when x = 6, y2 = 42.

x 1 2 3 4 6 - 6
y1 2 4 8 16 32 - 64
y2 2 4 8 16 32 - 42

Table 3: Two interpolating rules for the one data set

Thus, we have found a mathematical justification for saying that the sequence is {2, 4, 8, 16, 32, 42}. Hence our puzzle does not have a unique answer.

This lack of uniqueness is part of any interpolation problem. Like our puzzle, curve fitting problems tend to have infinitely many answers. This may seem strange to many who believe that each mathematical question has exactly one answer and all other answers are wrong. But it is quite common in mathematics to be confronted with a question which has many answers - possibly infinitely many answers. Some mathematical problems have no answer.

Return to the Table of Contents.

Interpolation and approximation
In this section we introduce the idea of approximation. Approximations are estimates, rough answers, numbers which are "close enough". Let me illustrate with the simple hand-held calculator. My CASIO fx-82LB calculator tells me that the square root of 2 is 1.4142136, but we know that this cannot be correct because

1.4142136 × 1.4142136 = 2.00000010642496.

Thus, the answer from my calculator is approximately correct, or close enough for most practical purposes. It is an "approximation" to the real value of the square root of 2.

As I like to tell my students about so many different problems in mathematics, you can spend your whole life studying approximations and getting paid to do it. (These days with modern computers, there is probably better money in working out approximate answers than there is working out answers which are exactly correct!) There are international journals and many text books devoted to the study of approximations.

In fact if you can work out a way to calculate square roots faster and more accurately than anyone else, then you can probably sell the idea to a large computer company. Furthermore, this sort of question never goes away: it is like setting a world record. Your algorithm, which is better than everyone else's, is only the best until someone who is smarter than you comes along with a better method. In their computing text, Cody and Waite (1980) devote almost 30 pages to modern aspects of calculating square roots approximately. In a more recent mathematics text, Petrushev and Popov (1987, pp.78-86) discuss the history of this problem.

Approximation problems crop up in the study of interpolation problems. The data in the graph in Figure 1 was generated by some function

y = f(x)

- but what is f(x)? Our little puzzle showed that this problem is impossible to answer since, from the data, there appears to be infinitely many answers. So can we find a function which is close to f(x) or is approximately f(x)? If so, then we will be able to estimate the value of y when x has some other value - such as x = 6. It turns out that, if we are prepared to make a few assumptions that are not very demanding, then we can often find good approximate answers to the underlying function in an interpolation problem even though we cannot find the underlying function.

This may sound a little odd. If we can find answers which are approximately correct, why cannot we write down the exact answer? Is this merely a reflection of the times when "close enough is good enough"? The answer is more subtle that this. Let me illustrate with the square root of 2 again.

Here are some numbers which get closer and closer to the square root of 2:

1, 1.4, 1.41, 1.414, 1.4142, 1.41421, 1.414213, 1.4142135, 1.41421359.

Each number in this sequence is closer to the square root of 2 than the previous one - but it is impossible to write down the square root of 2 as a decimal number since it goes on and on ad infinitum without repeating. This is a simple example of a situation in mathematics where we can write down approximate answers and these answers can be as close as we like to the exact answer, but we cannot write down the exact answer (at least as a decimal).

Return to the Table of Contents.

Smoothing
Smoothing problems are related to interpolation problems. Let me illustrate with a typical smoothing problem which arises in a problem in regional economics.

For several years, the ANZ Banking Group has collated time series data of the number of job advertisements in metropolitan daily newspapers in Australia. These series are reported widely in the Australian media and have become accepted as important economic indicators in national debates on the assessment of economic policy and planning. The ANZ series for Australia is regarded by many economists as a leading indicator of the number of persons in full employment in the Australian workforce.

Some years ago my colleague Graeme Byrne and I decided to consider a similar series based on data in regional Victoria, and, with the help of our students Janine Lloyd and Paul Gregg, we collected data over several years. (A report of this project is in Byrne, Lloyd and Mills (1991).)

The data collected were the numbers of job advertisements placed each Saturday in the Bendigo Advertiser, Geelong Advertiser, The Courier (from Ballarat) over the decade from mid-1983 to mid-1993. Each advertisement was counted as one even if it advertised several jobs. This method agrees with the method used for the ANZ series. While these series do not measure the total number of jobs available in the respective regions, they are indicators of same.

The Bendigo series is presented in Figure 2. This figure shows the average number of job advertisements per week for each month in Bendigo over about a decade. The smooth curve represents the underlying trend.

jobsdata
Figure 2: Smoothing Job Vacancy Data with ANUSPLIN

Thus, in smoothing data, it is not necessary that our curve pass through the data points: it is sufficient that the curve be close to the points.

Return to the Table of Contents.

Smoothing and approximation
Just as interpolation methods give rise to approximation problems, so too do smoothing methods. Let me illustrate this with an example which is based on some research work in chemistry which is currently being carried out at La Trobe University, Bendigo.

Associate Professor Bruce Johnson and Dr John Wells are research chemists at La Trobe University, Bendigo, who have a particular interest in environmental chemistry. Together with their PhD student, Michael Angove, they have been successful recently in improving our understanding of the chemistry behind the accumulation of the toxic element cadmium in soils. (See Angove, Wells, and Johnson (1996).) A key part of their research involves looking at data like that in Figure 3 and trying to find a smooth curve which passes close to these data points. They do not particularly need the curve to go through the points because inevitably the points contain some of that ubiquitous "experimental error". So they have a smoothing problem.

cadmium
Figure 3: Experimental results of M.J. Angove, J.D. Wells and B.B. Johnson showing Cd(II) adsorption on Comalco kaolinite at two different pH values

Angove, Wells and Johnson have in mind that there is a particular mathematical relationship between the two variables depicted in Figure 3 (known as a Langmuir model). The interesting mathematical question is "How well do the data and the theoretical model match?". We need to compare "observation with hypothesis" or "reality with theory". The difference between reality and theory is a measure of the effectiveness of an approximation:

ERROR = REALITY - THEORY

This example illustrates how smoothing gives rise to approximation problems.

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Complex examples
This section is devoted to describing, in general terms, some more complex applications of interpolation and smoothing methods.

Computer aided geometric design
In recent years we have witnessed the development of a new field which is known as Computer Aided Geometric Design, or sometimes as Computer Aided Design (CAD). The basic idea is to develop computer models and graphics to depict 2-dimensional or 3-dimensional objects. These computing techniques grew out of work in manufacturing, especially the manufacturing of cars.

For example, the pre-CAD methods for designing a car began with preliminary drawings from which a patternmaker constructed a clay model. The clay model was then measured at many points and these 3-dimensional measurements were interpolated by hand (that is, smoothed out) to form the stamp-die. Nowadays, these measurements are processed by a computer using interpolation and smoothing methods.

A key part of this process is the development of mathematical interpolation and smoothing methods to calculate the surface.

These methods played an important role in the development of aircraft (e.g. the Concorde) as well as in car manufacturing. As shown by Davis, P. (1996), these methods are now used in many other fields such as surgery, animation in movies, flight simulators.

This illustrates also how mathematical ideas developed for one industry can find applications in many other industries as well. Such cross-fertilization of ideas are commonplace in mathematics.

Finite element method
For the last 300 years, mathematicians have been using calculus to describe the ways in which aspects of the physical world change or move. Calculus is used to describe:

  • the motion of the planets
  • vibration of a string on a guitar
  • bending of a beam
  • cooling of a cup of coffee
  • the growth of a population.
Scientists and engineers have found that certain sorts of equations - called differential equations in the language of calculus - are powerful tools for coming to grips with change in the world.

These differential equations reflect the physical world in the mathematical world. If the mathematician can solve these equations then the scientists are in a strong position for predicting how the particular phenomenon under investigation will behave.

The difficulty is that, quite often, these equations appear to be impossible to solve. So, we try to solve these equations approximately and our approximate answers yield predictions which are accurate enough for our purposes even though they are only approximations.

The finite element method is a powerful method for solving differential equations approximately which has been popular with structural engineers. These days, it is important for engineering students to develop some understanding of this method.

The relevance of the method to this lecture is that the finite element method calls on fundamental ideas in both interpolation and approximation.

Two of my colleagues, Associate Professor Joe Petrolito (an engineer) and Dr Katherine Legge (a physicist), have combined their skills to develop a new application of the finite element method. Joe Petrolito is an expert in the finite element method and Katherine Legge is an expert in acoustics and the physics of musical instruments.

Together they have developed an approach to designing a marimba by using the finite element method to find the optimal undercut which is necessary to tune the instrument. This is a nice illustration of an application of the finite element method which is based on important ideas in interpolation and approximation. (See Petrolito and Legge (1995).)

We also see in this example how the methods used by engineers to study building structures can be applied to an instrument such as the marimba once we regard the marimba as a structure rather than an instrument. Throughout the history of mathematics, we see instances of mathematics which was developed for some particular purpose subsequently being used in a vastly different setting.

Computerized tomography
X-rays were discovered a century ago by Wilhelm Röntgen. Many of us have had our limbs X-rayed as a result of sporting injuries or had chest X-rays to enable us to travel overseas.

Some of us may have had our feet X-rayed when our parents took us shopping for school shoes. Fortunately X-ray technology has come a long way since then!

These days we read about new approaches to medical imaging such as computerized tomography (CT) or magnetic resonance imaging (MRI). These modern techniques have replaced invasive radiological examinations such as carotid arteriography and pneumo-encephalography which were extremely uncomfortable for the patients. The significance of these advances was recognised in 1979 when the Nobel Prize for Medicine was awarded to A. McLeod Cormack and G. Newbold Hounsfield for their work in computerized axial tomography.

It is not so well-known that the fundamental mathematical ideas on which computerized tomography is based were developed by the German mathematician Johann Radon in 1917. Now when Radon wrote his paper, X-rays were the furthest thing from his mind. His prime motivation was to solve a problem in pure mathematics. However, his results were exactly the tools required when medical scientists started to develop CT machines fifty years later in the 1960s. But since Radon's work was an exercise in pure mathematics, some applied mathematics was needed to link Radon's ideas to the technology.

Two fields of applied mathematics which have played key roles in the development of CT are interpolation and approximation theory. A good survey of these ideas can be found in Rowland (1979) whereas the paper by Herman et al. (1992) shows how tomography is still providing challenging problems in interpolation for the mathematician. A short readable article is Swindell and Barrett (1977).

To see how interpolation plays a role, suppose that we wanted to produce a scan of a skull, perhaps in looking for a tumour. Imagine a thin slice taken through the skull. A large number of parallel beams are projected through the skull and the attenuation of the X-rays along each beam is measured. This process is repeated for many slices and all these measurements are used in the calculations which lead to the final scan or picture of the skull.

I have deliberately left out a lot of detail in this description because I want to draw attention to the connection between the data (i.e. measurements from a large - but finite - number of beams) and the final outcome (i.e. a continuous picture or image of the skull).

This is an interpolation problem: we have a finite amount of data and we want to produce an infinite amount of information in the form of a scan. It turns out that algorithms used in producing CT images call on ideas from interpolation, smoothing and approximation.

Radon's results from 1917 have found applications in other fields such as radioastronomy, physiology and scientific instrumentation - and interpolation and approximation theory play their roles in making these applications feasible too.

Note that this is another example of some mathematics being developed for one particular purpose (viz. to solve a problem in pure mathematics) yet finding applications is vastly different areas many years later. In fact, these days, Radon's name is associated more with tomography than with mathematics.

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International traditions
Different countries have different traditions in many aspects of their cultures: food, wine, the arts provide obvious examples. In this section, I will describe mathematical traditions in France, Hungary and Bulgaria with special reference to interpolation and approximation.

French school
The twentieth century has been the Century of the Abstract. We have seen the rise of abstract painting, abstract dance, and abstract sculpture. My colleague John Robinson (1995) of the Department of Art and Design has recently discussed some of these issues from an artist's perspective.

Mathematics is no different in this respect. Perhaps mathematics has always been abstract. But, in this century the power of mathematicians to generalise, or to think in abstract ways, has had a dramatic impact on mathematics and its applications. There is no better demonstration of this than the ideas associated with dimension.

Most of us think that we live in a three dimensional world - a few of us think about the fourth dimension (mathematicians, physicists and sci-fi writers are most notable). But mathematicians have been busy developing all sorts of ideas about 5 dimensional space, or 10 dimensional space or any size dimension that you please.

In fact very often mathematicians talk about infinite dimensional space - but mainly we talk about this only among ourselves. (We also talk about fractional dimensions, but that is for another lecture.)

French schools of mathematics have contributed as much to the development of abstract ideas in mathematics as anyone. The French mathematician N. Bourbaki wrote a series of books called The Elements of Mathematics. To quote Bourbaki (1968):

The Elements of Mathematics Series takes up mathematics at the beginning and gives complete proofs. In principle, it requires no particular knowledge of mathematics on the readers' part, but only a certain familiarity with mathematical reasoning and a certain capacity for abstract thought. Nevertheless, it is directed especially to those who have a good knowledge of at least the content of the first year or two of a university mathematics course.

Bourbaki was a prolific writer and produced a dozen or so major books starting in the 1930s. This would make a remarkable academic record except that Bourbaki was not one person. After some time it was revealed that Bourbaki was the pseudonym of a group of French mathematicians (and one non-French mathematician). One can find a description of the development of the career of Bourbaki in the autobiography of one of the founding members of the Bourbaki group - André Weil, brother of Simone Weil. (See Weil (1992, Chapter V).)

Anyway, we see in the above quote the reference to the importance of "abstract thought". Bourbaki's style has had a great impact on the language and methods of modern mathematics generally, as well as on French mathematics in particular. For example, infinite dimensional spaces are now part of modern engineering research in the study of structures and mechanics. They are also part of modern approaches to the study of interpolation and smoothing. (See Champion, Lenard and Mills (1996).)

Hungarian school
In spite of very difficult circumstances, especially during this century, Hungary has produced some remarkable mathematicians. In the field of interpolation and approximation, major contributions have been made by L. Fejér (1880-1959), G. Szegö (1895-1985), G. Grünwald (1910-1942), P. Turán (1910-1976), G. Freud (1922-1979), P. Erdös, J. Szabados, P. Vértesi and many others. The Heyward Library at La Trobe University contains a fine collection of important works from which one can get an overview of the work of this school. (See Erdös (1990), Fejér (1970), Freud (1971), Szegö (1939), Szabados and Vértesi (1990).)

After studying these works, one senses a great capacity among these mathematicians for tackling difficult, technical problems using methods from classical mathematical analysis. Problem solving is a hallmark of Hungarian mathematics. This Hungarian tradition is embodied in the two volumes of problems written by G. Pólya and G. Szegö which are now available in English (Pólya and Szegö (1972, 1976)). In the preface of Volume 1, Pólya and Szegö discuss the relation between problem solving and constructing broad theories in mathematics:

There is an analogy between the task of constructing a well-integrated body of knowledge from acquaintances with isolated truths and the building of a wall out of unhewn stones. One must turn each new insight and each new stone over and over, view it from all sides, attempt to join it to the edifice at all possible points, until the new finds its suitable place in the already established, in such a way that the areas of contact will be as large as possible and the gaps as small as possible, until the whole forms one firm structure.
The outstanding Hungarian problem solver is Paul Erdös, one of the most prolific mathematicians who has ever lived. He has published about 1400 papers across many different fields of mathematics. Recently SBS TV screened a program on Erdös which described his remarkable achievements and his rather unusual lifestyle. Many of his papers on interpolation were written with P. Turán and can be found in Turán's collected works (Erdös (1990)).

Paul Turán too was a great problem solver. In an article on "Personal Reminiscences of the Work of Paul Turán" (Erdös (1990, v.1)) Erdös writes of Turán:

Even in high school he showed a considerable ability in mathematics and was one of the best problem solvers in the mathematical periodical published for high school students.

This sort of problem solving usually requires pencil and paper only. At times, Turán had very few tools of trade. G. Halász wrote the following in a letter to Turán (Erdös (1990, v.1)):

While in a Nazi labour camp, mounting poles and transporting bricks, you sought problems not requiring much computation, and it was there that you discovered your classical graph theorem that became the starting point of a flourishing theory on extremal graphs.
The flame of mathematical thinking continued to burn even in these dark days of Europe's history.

Just before he died, Turán wrote an important paper listing 89 open problems in interpolation and approximation (see Erdös (1990, v.3, pp. 2432-2494)). This collection of problems has kept many mathematicians busy for the last 20 years or so.

J. Szabados and P. Vértesi (1990) have written an important treatise on interpolation and approximation which gives a good insight into the contribution by Hungarian mathematicians. Later in 1996, Dr Vértesi will be a La Trobe/CRA Distinguished Visiting Fellow at Bendigo and mathematicians at La Trobe University will have the opportunity to share ideas with a member of this important Hungarian school.

Bulgarian school
Bulgarian mathematicians have in the last 30 years or so made significant contributions to the study of interpolation and approximation methods. Two aspects of the Bulgarian school deserve mention.

Firstly, the school has developed under the leadership of one mathematician - Professor Blagovest Sendov. Professor Sendov's leadership has been evident in Bulgarian politics as well as Bulgarian mathematics.

Secondly, there is a distinctive theme which flows through papers produced by Sendov and his colleagues. I spoke earlier of the idea of an approximation. An approximation is involved when one tries to approximate some complicated mathematical object (e.g. the square root of 2) by some simpler object (e.g. 1.414). The error in the approximation is measured by the "distance" between the two objects (e.g. subtract 1.414 from the square root of 2). Now there are many ways to measure "distance". This point confronted VCE students this year. (See Board of Studies (1996).) The distinctive feature of the work of Bl. Sendov is that his work has concentrated on a measure of distance called "the Hausdorff distance". (See Sendov (1990, 1996), Mills and Sendov (1980).)

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Teaching and research
In universities, we often discuss the relationship between teaching and research. In this section I will illustrate how I see the connections between the two activites.

Teaching through research
One of the difficulties that we have encountered in mathematics is that, having developed these abstract notions and methods, we have trouble communicating them even to other mathematicians. These days it is very difficult for the average person who has a PhD in mathematics to understand the average article in a mathematics research journal. Macquarie University's Alf van der Poorten, who is one of Australia's leading expositors on number theory, described the situation as follows in van der Poorten (1996, p.viii).

One of the difficulties in reading, or listening to, mature mathematics is its immense vocabulary and the volume of notions that seem to be required. Nor can one readily discover the meaning of the more popular ideas because all too often they are described in terms of more obscure words.

We have a communications crisis in mathematics. (This is also true in other fields such as computing, economics, medicine, and science generally.) Hence, there is a need for mathematicians to explain their work to others who may not necessarily be experts in the field.

Thus I see a need for Teaching through Research: by this I mean that when we publicise research results, we can regard this as a form of teaching. There is a need for propagating the truth as well as discovering the truth.

I regard academics such as Paul Davies (physics), Umberto Eco (semiotics), Germaine Greer (feminism), Gustav Nossal (medical research), Peter Singer (philosophy), David Suzuki (environmental science) as academics who are teaching through research. (Needless to say, these persons may not see themselves this way, nor may they like to be on the same list as some of the others. However, I am merely stating my perception of their works which I regard as inspiring.)

This suggests a strategic approach to begin the process of developing research projects is the following. Pick an important, difficult area which, although it is well understood by experts, others would like to know about without having to become experts themselves. Then set about understanding the field and reporting on it to a wider audience through seminars and publications. To do this sort of research you need to be able to understand the work done by the experts and you need superior communication skills: thus you need a combination of expertise in research and teaching. After a while you may begin to make important contributions yourself to the field.

Research inspired by teaching
Even though the subjects that I teach seem to me to be a long way from the areas in which I conduct my research, there are still many surprising connections. Let me give an example.

A few years ago, two students (Rachel Mattingley and Michelle Seppings) were conducting a research project with me in the area of quality control. They were investigating a method called "acceptance sampling by variables" which is used in controlling the quality in certain manufacturing processes. This method is set out in the Australian Standard AS2490-1981. In their project, they constructed a mathematical model of this quality control process (Mattingley and Seppings (1994)). If we skip all the details, it turns out that it was essential that Rachel and Michelle calculate a certain curve called the operating characteristic of the quality control method. After a great deal of computation they developed a picture which looked like Figure 4.

Pa                                                     Pa
1.00* *                                                   +1.00
-     *                                               -
-      *                                              -
-       *                                             -
0.70+        *                                            +0.70
-         *                                           -
-          *                                          -
-           *                                         -
-             *                                       -
0.35+               *                                     +0.35
-                  *                                  -
-                      *                              -
-                          *                          -
-                                   *                 -
0.00+                                                 *   +0.00
 +---------+---------+---------+---------+---------+
0.00      0.10      0.20      0.30      0.40      0.50
p
Figure 4: Points on an Operating Characteristic

The task now is to fit a curve through these points - but more importantly the curve must go down as we move along the p-axis. Thus, not only do we want a curve to go through the given points as did Newton and Lagrange, but we want the curve to have the additonal geometric property that it goes down as well.

Indeed we want the curve to satisfy several constraints. The curve should

  • go through the points
  • be decreasing (i.e. going down as p increases), and,
  • initially its shape should be convex down and then be convex up.

We call this field of research constrained interpolation and approximation. The study of constrained approximation and interpolation is a very large field, but for many years I have paid it almost no attention. It was only through the work of these students that I found some nice applications of this field and hence I was led to learn even more about results in this field.

I am sure that all of us who have been teaching for some time have had similar experiences which illustrate how Teaching Inspires Research.

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Conclusions
In this lecture I have presented a general discussion on interpolation methods and their applications and made some general points about modern mathematical research including the following.

  • There are many mathematical research problems still to be solved. Some of the recent publications in the bibliography are evidence of this. From the lecture, a few research problems have emerged.

    1. Study the size of errors in any approximation. Suppose that we know that we have approximately $1000 in our bank account. It is one thing to know that we have in our bank account $1000 ± $50, but it is quite a different thing to know that we have in the bank $1000 ± $500. To be more blunt, we cannot say that we know all about any approximation until we have a very good idea of the error. In Bendigo, we have put a great deal of effort into studying the size of errors which arise in interpolation. (See, for example, Goodenough and Mills (1981), Mills and Smith (1992), Byrne, Mills and Smith (1995).)

    2. Investigate the history of interpolation. Interpolation has attracted the attention of many famous mathematicians. Thus, the history of interpolation is a fertile field for research. A starting point would be Goldstine, H. (1977) and recent work on the history of interpolation and smoothing can be found in Brezinski, C. (1994), Chabert et al. (1993), Champion, Lenard, Mills and Petrolito (1996). Recently, my colleague Dr Robert Champion has presented some seminars in which he has shown the potential in studying the history of smoothing.

    3. There is plenty of biographical information about Newton, but much less about Lagrange. As far as I know there is no monograph devoted to the life and works of Lagrange. There are some papers by Luigi Pepe, an Italian expert in the history of mathematics and a work by Judith Grabiner (1990) devoted to various aspects of the works of Lagrange. Now Lagrange was a senator as well as a distinguished mathematician - his collected works are published in many volumes. One would guess that there ought to be lots of material about his life as well as his work. So, it would be a very interesting project to write a book on the life and works of J-L. Lagrange. If one could write a book to rival Westfall's biography of Newton then one would have made a major contribution to the history of mathematics.

    4. Investigate the approximations to your favourite functions. In Section 2, I mentioned the problem of calculating square roots approximately. The square root button on the calculator is only one of several buttons. One can ask exactly the same questions about finding better ways of calculating logarithms or trigonometric functions. Each function button on the calculator generates new problems in approximation and it would appear that these problems demand almost individual answers. Thus the humble calculator provides inspiration for many problems in modern mathematical research in the study of approximations.

    5. The study of interpolation and smoothing in many dimensions is now receiving a fair amount of attention in light of the increased capability of modern computers.

  • Mathematics is not merely the study of numbers. We have encountered some important concepts such as dimension, distance, infinity, interpolation, approximation and smoothing in this lecture alone. One of the points that I am trying to get across is that these abstract concepts are very useful in a great variety of different branches of knowledge as well as being practical.

  • We have seen several examples of mathematics which is developed for one purpose being used later in completely different applications. Thus, mathematicians have become quite comfortable with the idea of developing mathematical ideas for their own sake or developing them for applications which may not seem to be vital to society. In this respect, mathematicians tend to be ahead of their time because they often create the mathematics which is necessary for technological breakthoughs a century before they are needed.

  • There is an obvious international aspect of modern mathematical research as well. My own study of Latin and French at school has helped me a great deal over the years to learn a little of other languages so that I can read mathematical papers and books in foreign languages. Not all great mathematicians wrote in English! Students who are studying languages at school should be aware that their language skills are valuable research skills which can give insight into different schools of thought around the world.

  • In my own approach to mathematics, teaching and research are intimately connected: each feeds the other. There has always been a need in our society for well trained, enthusiastic mathematics teachers at all levels. However the roles of teachers in stimulating research interests are not acknowledged enough. Many academics to whom I have spoken attribute their initial enthusiasm for their subject to a teacher.

Hill Worner remembers one of his mathematics teachers at Bendigo 60 years ago, W. Spencer Lake, as a teacher "who made mathematics into a lively and living subject" (UCNView, v.2, no.2, Dec. 1991, p.19). I hope that this lecture has helped to maintain Lake's philosophy in Bendigo.

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Acknowledgements
I wish to thank La Trobe University for granting me OSP Leave during the period December 1995 - February 1996 to enable me to visit Université Paul Sabatier (Toulouse), since part of this leave was devoted to preparing this lecture.

At La Trobe University, Bendigo, I thank the Research Committee and Professor L. Kilmartin for financial assistance in producing this paper. I am grateful to many colleagues who have assisted me with this paper: these colleagues include Mr G.J. Byrne, Dr R. Champion, Ms A. Cooper, Ms A. Forden, Mr I. Glanville, Associate Professor B.B. Johnson, Dr C.T. Lenard, Mr J.L. Penwill, Associate Professor J. Petrolito, Mr P.T. Scott, Dr J.W. Schutz, Dr S.J. Smith, Dr J.D. Wells, Ms M. Whitby.

I am also grateful to Dr P. Vértesi (Mathematical Institute of Hungarian Academy of Sciences) for details concerning the Hungarian school, and to Professor G. Herman (University of Pennsylvania), Ms G. Lee (Charles Sturt University) Mr L. Adorni-Braccesi, Ms P. Moran and Ms F.L. Mills (Bendigo Health Care Group) for helpful ideas on computerized tomography.

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Bibliography

  • Adams, Douglas (1979) The Hitch Hiker's Guide to the Galaxy. London: Pan Books.

  • Angove, M.J., Wells, J.D. and Johnson, B.B. (1996) 'Cadmium Adsorption on Kaolinite: a Two-Stage Adsorption Process', Australian and New Zealand National Soils Conference 1996, University of Melbourne, 1-4 July, v. 2, pp. 5-6.

  • AS2490-1981: Sampling Procedures and Charts for Inspection by Variables for Percent Defective. Homebush: Standards Association of Australia.

  • Board of Studies (1996) Common Assessment Task 1: Investigative Project. Mathematical Methods Units 3 & 4. Victorian Certificate of Education. Melbourne: Board of Studies.

  • Bourbaki, N. (1968) Theory of Sets. Paris: Hermann.

  • Brezinski, C. (1994) 'Historical Perspective on Interpolation, Approximation and Quadrature' in Handbook of Numerical Analysis, Vol. III. ed. P.G. Ciarlet and J.L. Lions. Amsterdam: North-Holland, pp. 3-46.

  • Byrne, G.J., Lloyd, J.C. and Mills, T.M. (1991) 'Charting the Future', UCNView, v.2, no.1, September, pp. 16-17.

  • Byrne, G.J., Mills, T.M. and Smith, Simon J. (1995) 'The Lebesgue Constant for Higher Order Hermite-Fejér Interpolation on the Chebyshev Nodes.' Journal of Approximation Theory, v.81, no. 3, pp. 347-367.

  • Chabert, Jean-Luc et al. (1993) Histoire d'Algorithmes. Paris: Belin.

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  • Champion, R., Lenard, C.T., Mills, T.M. and Petrolito, J. (1996) 'Smoothing Methods and Engineering', Proceedings of Australian Engineering Mathematics Conference '96 (AEMC96). eds Yuen, W.Y.D. et al. Canberra: Institution of Engineers Australia. pp. 401-407.

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Biographical sketch

Terry Mills was born in Sydney in 1948, educated at St Patrick's College, Strathfield (NSW), and has gained qualifications from the University of Sydney, University of Melbourne, University of Florida, and La Trobe University.

He has held positions at the University of Melbourne, University of Florida, Eastern Montana College and La Trobe University, Bendigo (and some of its predecessors). He was appointed as Lecturer in the Department of Mathematics at Bendigo in 1975. Later he held positions as Head of Department of Mathematics (11 years), Warden of the Halls of Residence (12 years) and has been a Professor since 1993.

In addition he has spent periods of time visiting the Institute of Mathematics of the Bulgarian Academy of Science, Mathematical Institute of the Hungarian Academy of Science, Technion - Israel Institute of Technology, and Université Paul Sabatier (Toulouse).

He is Associate Dean (Teaching) at Bendigo, Editor of the Australian Mathematical Society Gazette, member of the Council of the Australian Mathematical Society, and a member of the Statistical Society of Australia and American Mathematical Society.

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© Copyright 1996 by T.M. Mills. All rights reserved.
Terry Mills can be contacted at t.mills@latrobe.edu.au