## Mathematics and Knots

### by Ronald Brown and Tim Porter

School of Informatics
University of Wales
Bangor, Gwynedd LL57 1UT
United Kingdom

Abstract: We explain how we came to be involved in making a mathematical exhibition; some mistakes we made; and the philosophy we evolved on the principles which underlay our presentation. The main conclusion is that the exhibition should tell a story and aim to give an impression of the character of mathematics.
Over the three years 1986-89, a team in the School of Mathematics at the University of Wales, Bangor, designed the exhibition Mathematics and Knots[EG1] , which was exhibited at the ICMI89 meeting at Leeds University on the Popularisation of Mathematics, and at the accompanying PopMaths RoadShow, which then toured the UK.

In this paper we would like to explain what we were attempting to achieve, and the problems we had in getting to this stage.

We do not claim to have achieved all our aims, or to have reached a final version. The exhibition will be useful if it is enjoyed by the public and by mathematicians. We hope it will also stimulate others to think about the problems of exhibition design in mathematics and to encourage them prepare for themselves presentations of mathematics in a variety of topics and media.

## 1. How it began

Our involvement in making a mathematical exhibition came about in the following way.

One of us (R.B.) was invited to give a London Mathematical Society Popular Lecture on knots, one of two lectures in an evening, for June, 1984. It seemed a good idea to have material to display in the foyer for people to view when they arrived and in the coffee interval. So coloured enlargements were made of slides of knots in art and in history, and also a few of the overhead transparencies used in the lecture were enlarged to A3 size. All this material, and some models of knots made of copper tubing, was rather randomly distributed over some display boards.

During the next year we accumulated more material, which was used successfully for an Open Day for the Centenary of the University College of North Wales, in 1984, and in Royal Institution Mathematics Masterclasses for Young People in Gwynedd. It was also shown as an accompaniment to a Mermaid Molecule Discussion (by R.B.) in London in November 1985, again on "How mathematics gets into knots".

In the Summer of 1986, it turned out that there was some money left over from the grant by Anglesey Aluminium to our Masterclasses. Permission was given to use this money to develop the exhibition in a more professional way, by bringing in a designer. We also set up a design team of R.B. (Chairman), T.P. and Nick Gilbert (now at Herriot Watt University), and asked for help from a designer, Robert Williams, and writer, Phiop Steele. It was the continual development of ideas from this team, with a full and frank range of suggestions and criticisms, both from within the team, from our designers, and from others who saw earlier versions, which has led to the exhibition you now see.

Versions and parts of the exhibition were shown in 1986 at the Royal Institution in January, and at the British Mathematical Colloquium in April; in 1987 at the Eisteddfod in Pwllheli (a Welsh version of a part) and at the British Association for the Advancement of Science at Belfast in August; on the occasion of talks by me and others at various universities and schools; in 1988 at the Royal Institution to accompany a Discussion by Sir Michael Atiyah on "The geometry of knots". On the last occasion there was also an exhibition of some John Robinson sculptures, of Celtic Interlacing by Lady Wilson from the British Museum, and of knots by Geoffrey Budworth of the International Guild of Knot Tyers.

## 2. Major problems

The design team and the designer had not realised what we were taking on. The design work extended over almost three years, and involved many drafts. The reasons for this were of several kinds.

### 2.1. Novel aims

We set ourselves some novel aims for the exhibition (see section 3). In so doing, we were setting out into uncharted territory, and had to learn as we went along.

### 2.2. Lack of design experience

We knew not even the elements of the mechanics of producing an exhibition. Yet our aims required a close marriage of content and medium. So we had to learn some of the problems and techniques as we went along. We were not able to employ the designer for the continued help we needed because of the lack of funds.

### 2.3. Insufficient funds

In order to finance the exhibition further,we wrote to a number of firms and organisations, and obtained a lot of help. A list is given at the end of this article. Also, when the Committee for the Public Understanding of Science started in 1987, we made an application to them and were fortunate to obtain a grant of £2,000. Without this total sum of about £3,300 the exhibition would not have been able to be produced.

To produce anything to a professional standard is expensive. An estimate we had at the time from a member of a Government department was that a 30 square metre exhibit for Olympia had cost £20,000, of which half was for design and half for construction.

In fact our designer gave us a lot of help for very little financial reward. He suggested the use of polystyrene boards with aluminium surround for durability, lightness, and transportability. However it gradually became apparent that his proposed layout with two columns per board was not appropriate for the exhibition, where the graphics had to predominate. The layout was considerably revised with the help of another designer. Finally, on the advice of the head of a local Arts course, we paid for the layout to be redesigned yet again using a grid approach, in which all modules of text and graphics fit into a centimetre square grid, and for all the knots to be redrawn to have a three dimensional effect and so as to fit into the modular layout.

A severe problem is printing costs. To produce an A2 board one first produces an A4 board with the text and graphics. This is photographed to an A2 negative from which the print is made. The enlargement from A4 to A2 (four times the area) means that the typesetting has to be with 1200 dots per square inch: the usual output of a standard office laser printer is quite unsuitable for this job. Commercial computer typesetting is expensive. The only way of proceeding within the budget then available was to use facilities in the University system.

At that time, the only system available was the Oxford Lasercomp system. This was then an excellent system for producing a book in one or another standard text format. It also has a good variety of fonts, though not so many as commercial printers. It worked on what is called a "markup" system, where all control is by inserted commands in the text. This has the effect that the printed version is received a week or so later. Further, a combination of graphics and text can only be produced by photographing large drawings to a small size for pasting on the A4 page, so that the enlargement produces a good quality product. Thus the design has got to be completely decided, and there is little room for experiment.

In the end the current production would not have been feasible without the use of a Desktop Publishing system, in this case Pagemaker, with Postscript output. The graphics were drawn, scanned, and then positioned within the text, allowing tight control of spacing and layout. However there is at present a smaller range of fonts available than in the Lasercomp system. For example, the standard range of fonts does not include an extra bold. A further problem was that in each case the techniques had to be learnt from scratch. The current boards were typeset at the University College London Computer Service.

### 2.4. Stand-alone exhibition

The key difference from the design viewpoint between the foyer exhibition and our new aims was that we intended a travelling exhibition independent of any lecture. This meant that the boards had to be self-explanatory, and not just titillation for an explanation to be obtained later in the lecture. So we had to decide what story we wanted to tell. It was at this stage that we began to formulate and clarify our aims, while we were designing the text and graphics for each board.

### 2.5. Visual impact

The visual impact in an exhibition has to dominate. Each board has to tell a story, but the story has to be told mainly through the eye rather than through the text. So the exhibition format is one of the hardest to get right for the conveying of ideas, rather than simply the presentation of images.

### 2.6 Use of language

It is very easy for a mathematician to use words and phrases which mean something to him but which convey nothing to the general public. As an example, we found ourselves using the phrase "uniquely up to order" in discussing the factorization of a number into primes. In fact, this is a sophisticated idea, which we eventually conveyed by specific examples with numbers.

The aim was to use a simple and clear language. The stripping of inessentials and unclear language was a part of the design process. On the other hand, we also found that some ideas needed a more leisurely exposition than we gave them at first.

## 3. Aims

The decision as to the aims of the exhibition was crucial. The issues involved are part of "exhibitology". This term has been coined by Len Brown (R.B.'s brother). He learnt the basics of this study in his period as Head of the Engineering Section of the Science Centre at Toronto, and suggested the following simple example as illustration of some basic principles.

Suppose that you wish to produce an exhibition on "Bridges". There is certainly a lot of material available. However a decision has first to be made on the point of view to be taken. Is the exhibition to be about transport? structure? geography? history? rivers? trade? Each of these themes could lead to an acceptable exhibition, the various exhibitions would have much common material, but in the end each would be telling a different story, and give the visitor a different impression of the subject of Bridges.

In our case, we had to decide what impression of mathematics we were intending to convey, and then seek to find the means to do so.

The aims we set for our exhibition fell into two kinds: structure, and content.

### 3.1. Structure

We agreed that the exhibition should be:

1. self contained
2. easily transportable
3. reasonably cheap to produce
4. reproducible in several copies
5. able to be set up and managed without continual supervision

### 3.2. Content

We agreed that the exhibition should:

1. suggest that the making of mathematics is a natural human activity, part and parcel of the usual methods by which man has explored, discovered, and understood the world
2. present each item with a purpose and context, and not just because it was something that could be shown or demonstrated
3. convey an impression of some of the key methods by which mathematics works
4. show mathematics in the context of history, art, technology and other applications

## 4. Consequences of the agreed aims

### 4.1. Structural

Our requirements tended to rule out hands-on material, at least for the moment. Such material is expensive to produce and maintain; if it does not work it is worse than useless; it can be stolen, and indeed this is more likely the more attractive the material.

In any case hands-on material can also suffer from being a gizmo, designed because it is hands-on rather than to make a point which illuminates the themes of the exhibition; nice to play with, but superficial. The participant is expected to exclaim "Wow!", but there is still a question as to what he or she has learnt. Of course this tension between the requirements of entertainment and arousing interest, and the requirements of instruction and information, is basic to the whole activity of exhibition design.

Our requirements also meant that we were initially intending a static exhibition: something to be looked at, and enjoyed, but not involving an activity. Once the structure and the content had been decided, it would still be possible to design hands-on or animated material which would advance our overall mathematical aims and which could be used as occasion demanded and allowed.

### 4.2. Content

It was in terms of content that we felt we were taking the more radical line, and the various features of these aims deserve separate paragraphs and discussions.

### 1. Mathematics and discovery

The novelty and excitement of mathematics is conveyed by some of the major mathematical exhibitions (Horizons Math?matiques in Paris and as a travelling exhibition, and the Mathematika at the Boston Science Centre). However we felt that it is helpful to analyse the basis for mathematical novelty and excitement, in order to clarify what we were intending to present.

We felt that this excitement and interest comes from the vision of new relations, and new kinds of order or patterns. It is not the whiz! bang! excitement of the amusement arcade. For example, it is extraordinary that the number \pi is involved not just with the ratio of the circumference of a circle to its diameter, but also with the description of population distribution. It is extraordinary that whereas we think a negative number cannot have a square root, such does exist if we allow a new kind of number, and even more extraordinary that these new numbers should have applications to the study of prime numbers, to the design of electronic circuits, to cosmology, and to the study of elementary particles.

An exhibition should convey some flavour of the real achievements of mathematics. If instead it simply presents an assortment of, for example, strange polyhedra, and states that these are the wonderful things mathematicians study, then it will be very easy for the public to be convinced that mathematics is hard or weird or both. Each exhibit should have a mathematical point and should explain its relations with other parts of mathematics and with other disciplines.

In the case of this exhibition, we felt the most surprising idea that could be conveyed in a way related to everyday experience was the analogy between knots and numbers revealed by the notion of a prime knot. It is for this reason, as well as for the needs of exposition, that we devote more than one board to this topic.

Surprising applications are also important for conveying some of the excitement of mathematics. In this case, we stress recent applications, such as to knotted orbits in weather systems, and to knotted DNA.

Key aims of mathematics are to show new perspectives, views, and order in what seems initially a tangle of unanalysable phenomena. This is one impression of mathematics that we wish to convey to the viewer.

### 2. Normality of mathematical methods

Here is where we feel we are really breaking new ground. Mathematics lacks an adequate discussion of methodology. A majority of students of mathematics do not know what they are doing or why they are doing it, they know only that they have to learn how to do certain things. Very few university courses attempt to explain the reasons for the development of a particular piece of mathematics, in some cases because the teachers are unaware of these reasons. However, experience shows that an analysis of the particular methods used in mathematics, and a relating of them to standard methods by which we explore and manage the world, is welcomed with relief by school teachers and students, who often seem starved of a global viewpoint.

It has been said that the difference between a professional and an amateur is that an amateur can do things, in many cases as well as a professional, but a professional also knows how he or she does things. It is this knowledge, based on tradition, experience, perception, judgment and analysis, which gives the professional the confidence to produce work on demand and to certain standards.

Of course, in this exhibition we cannot hope to convey the whole gamut of mathematical ways of working. We are not interested in conveying technique. What we want to express is the mathematical equivalent of musicality - perhaps we should call it mathematicality? This is a horrible word, but its derivation should at least convey what is intended.

Often, mathematics is presented as a completed body of knowledge, whose development has been unrelated to the activities of human beings. The questions which motivated the whole theory in the first place are in teaching often simply omitted, and students and pupils are asked to appreciate the methods and the theory without context, without relevance to other mathematical or scientific activity, one might even say, without meaning. For example, how many books on group theory are there which mention the range of applications of group theory, from crystallography to modern physics, and which show how the exposition given fits into the wide mathematical and scientific context? The dehumanising of the presentation of mathematics has gone very far.

Our aim was to use the theory of knots to illustrate some of these basic methods of mathematics. Our listing and analysis of these methods carries no claim to finality. However such a listing is useful as a systematisation, and, more crucially, as a way of relating these mathematical methods to standard methods of exploration and analysis. Thus we illustrate the claim that the peculiarity of mathematics lies not so much in its methods, but in the material and the objects with which it deals.

### 3. The context of history and art

Our aim here was to remind the viewer of how knots have enormous rich- ness and importance in the history of man. It has even been suggested that the Stone Age should be called "The Age of String". We were fortunate to have been told by Joan Birman of the oldest known knot: the Antrea net, dated 7,200BC, from the Helsinski National Museum. The net, found in a peat bog, was 30 m. by 1.5 m. with a 6 cm. mesh. It had stone sinkers and bark floats, and was made of willow bark. The knot used then is still used today.

One can only speculate on the social organisation and lives of the people who constructed this net, and on the length of time such a technological achievement took to evolve. One can also sense that the early understanding of the form of knots, and its link with survival, is an expression of an early but by no means primitive geometrical feeling, an understanding that the form not only can be so but has to be so, by virtue of its logic.

We had planned a series of boards on knots in history, practice and art. However we found that the amount of work for the mathematical boards took up all the time, and we were not able to prepare boards on the wider aspects. This motivated asking John Robinson if he would like to do an exhibition. The result was an exhibition of thirteen major sculptures, and four tapestries, with a catalogue [JR]. This exhibition was also shown in the Anglican Cathedral at Liverpool, in parallel with the PopMaths Roadshow at the Catholic Cathedral, 1990, and toured Bangor, Oxford, Cambridge, London, Barcelona, Zaragoza.

## 5. The methodology of mathematics

We considered that the aim of the exhibition was also to convey in spirit some of the basic methodology of mathematics. This meant that we had to analyse what we considered the basic method which each of the boards was supposed to convey. Thus there was a continual analysis and comparison of what we thought we could show, with what should be considered the intention of the board, how it fitted with the other boards, and what was the visual impact.

But first we had to decide what were the basic methods we were trying to convey. We believe the following list of basic methods to be useful for our purpose:

• Representation
• Classification
• Invariants
• Analogy
• Decomposition into simple elements
• Applications

We now analyse these in detail.

### 5.1 Representation

In the case of knots, we have to show them.

### 5.2 Classification

Making lists is a basic human activity. However, in view of the complexities of the world, you cannot make a list of everything, at least not if you are to lead a sensible life.

You may recall the Memory Man described by Luria in [L] who was unable to forget anything, and consequently was not able to lead a normal life. One of the diagnostic features for autistic children is that they remember nonsense patterns as easily as other patterns [W].

So in making lists we impose or find order: we classify. For example, a zoologist does not list all the animals in a game reserve, he lists antelopes, elephants, lions and so on. In order to do so, he needs criteria for saying that two animals are the same. In mathematics the notion of equality, or, in more precise mathematical terms, equivalence, is basic.

Knot theory presents interesting mathematical points in this area. Firstly, "When are two knots the same?" is a non trivial question. Secondly, to make an initial list of the first elements of an infinite family involves some classification into the "simplest" elements. Thirdly, the presentation of even such a simple list is likely to suggest the need for some further order and classification.

We address the first question by showing in our boards how diagrams of knots can be transformed, without changing the knot.

This is one area where computer animated graphics could have greatly improved the presentation. However, we were limited by the cost of producing the graphics, and the cost of presenting the graphics at an exhibition display. The latter is not so hard to overcome, since a video could be made and easily displayed, for example at most schools. It is more important for us to show the context in which any computer graphics would run, as this would dictate what graphics should be produced in order to support the overall themes.

One of the goals that we set ourselves was to explain the meaning of a list of knots. Although such a list is apparently simple, the explanation involves the following ideas:

• when are two knots the same;
• crossing number;
• mirror images;
• the arithmetic of knots;
• prime knots.

These themes give interconnections between the different boards.

### 5.3 Invariants

Classification of knots involves two aspects:

• when are two knots the same?
• and when are they not the same?

The first usually involves transforming one diagram of the knot into another. The second involves the more subtle point of deciding when such a transformation is not possible. Such a decision involves the notion of invariants.

We deal with four invariants in our presentation:

• crossing number;
• unknotting or Gordian number;
• colouring number;
• bridge number.

We also mention briefly the new knot polynomials which enable one to distinguish easily between a trefoil and its mirror image. The advantage of the four invariants we deal with in detail is that they can be easily presented at this level, and that they suggest many detailed exercises and examples which people can try for themselves.

The further point made by the discussion of invariants is that we do not claim to give a complete set of invariants, that is, we do not have some method of distinguishing all possible knots. Thus many problems remain in the theory, and this again is a point which is easily conveyed. We do want the reader to see that mathematics is, and will continue to be, an open-ended activity.

### 5.4 Analogy

In order to explain our list of knots, we have to describe the notion of prime knot. This is done by explaining a basic composition of knots.

There was a decision to be made here: whether to call the composition of knots addition or multiplication. The literature uses both terms.

We chose the term addition for two reasons. One was that the notation 0 for the unknot is more descriptive than the notation 1. The more important reason was to emphasise that the analogy is not between things but between the way things behave, between their relationships. For this it is helpful to have a different notation for the two operations which are being taken as analogous. Instead of making an analogy between two multiplications, we make an analogy between an addition and a multiplication. This, we hope, is more striking and also illustrates a general point, that such analogies may be available in other situations.

Although we use the term "The arithmetic of knots", we also use algebraic notation and emphasise laws. This makes the point that it as the study of analogies that algebra obtains its generality and power. By using the commonplace word "analogy", we aim to demystify, and to show that one aspect of the method of algebra is as a standard process by which we understand and try to make order of the world. The other point to be made is of course the excitement of an unexpected analogy, of "That reminds me of ...!".

We are also able to state the deep fact that knots have a decomposition into a sum of prime knots, and this decomposition is unique up to order. The appreciation of this led one small boy after the Mermaid Molecule Discussion by R.B. to ask: "Are there infinitely many prime knots?". It so happened that the lecturer had not previously formalised the question for himself, so had really to think in order to be clear that all torus knots are prime, and that there are an infinite number of them. However the proof that torus knots are prime is not so easy. It is good to have something to state which is comprehensible and believable, but which it is not at all clear how it might be proved. This is one of the great advantages of knot theory for expositions at this level.

### 5.5 Decomposition into simple elements

Decomposition into simple elements is a basic process in mathematics, or indeed wherever one deals with complicated matters. In knot theory the process crops up in a variety of guises.

1. We have already mentioned the prime decomposition of knots. Here the prime knots are the simple elements and the fact that any knot can be expressed uniquely (up to order) as a sum of prime knots is clearly an important fact about knots.
2. The process of transforming one diagram of a knot into another may be quite complicated. It is therefore of interest that such a complex process can be resolved into a sequence of simple moves, the Reidemeister moves.

We illustrate this in the process of changing the Bowline, and also in illustrating why the colourability of a knot is an invariant.

### 5.6 Applications

The exhibition starts with a picture of the sculpture Rhythm of Life, by John Robinson. It ends with an indication of some applications, including how knotting is involved in DNA, one of the building blocks of life itself.

This section on applications could be expanded, and made more vivid in a variety of ways, given funding, space, and so on. But all we hope to do in this exhibition is to catch the imagination of some people. The exhibition is intended to be small scale, and unambitious in its use of technique. Indeed, this is imposed on us by the criteria which we outlined at the beginning. Such limitations are still compatible with high expository aims.

## 6. Conclusion

We believe that if the exhibition is successful on the terms initially laid out, then it should be possible to build out from it as wider funding and staffing, and more ideas, become available. The honing of ideas and presentations, the discarding and developing of innumerable drafts, the criticism and comments from many, all have been valuable in clarifying our aims and our methods. In particular, we give our thanks to the long suffering designers, Robert Williams, Jill Evans, and John Round, who have been involved at various stages.

The existence of various drafts of the exhibition has enabled Heather McLeay to start designing a set of worksheets for young pupils [M]. Drafts of these worksheets have been used successfully at Royal Institution Mathematics Masterclasses in 1988 and 1989 at Bangor and Cambridge.

## References

[B] Brown,R., "Conversations with John Robinson", in Symbolism: sculptures and tapestries, by John Robinson, catalogue for the exhibition at the International Congress of Mathematical Instruction, Leeds, 1989, published by Mathematics and Knots, University of Wales, Bangor (1989).

[EG1] Bangor ExhibitionGoup, "Mathematics and knots", Exhibition of 16 A2 Boards, Mathematics and Knots, 1989.

[EG2] Bangor ExhibitionGoup, "Mathematics and knots", A4 Brochure of the Exhibition, Mathematics and Knots, 1989.

[JR] John Robinson, Symbolism: sculptures and tapestries, catalogue for the exhibition at the International Congress of Mathematical Instruction, Leeds, 1989, published by Mathematics and Knots, University of Wales, Bangor (1989).

[L] Luria,A.R., The mind of a mnemonist, trans. by Lynn Solotaroff, Basic Books, New York (1968).

[M] McLeay, Heather, "Worksheets on knots", (in preparation), UCNW (1988).

[W] Wing, Lorna, Early childhood autism: clinical, educational, and social aspects, Pergamon Press, Oxford (1976).

[E] Evans, Nancy

Click on for information on obtaining copies of [EG1 (A3 version), EG2,JR].

## Design team

Ronnie Brown, Nick Gilbert, Tim Porter

Anglesey Aluminium; University of Wales, Bangor; British Gas; London Mathematical Society; Committee for the Public Understanding of Science; Midland Bank Plc.

## Designers

Robert Williams, Jill Evans, John Round

John Round

## Assistance

John Derrick, Gwen Gardner, Tony Jones, Philip Steele.

## Showings

1989-90 With the PopMaths RoadShow at Leeds, Liverpool, Glasgow, London, Bristol, Cardiff.

Royal Institution, 1990.
British Association for the Advancement of Science: Swansea, 1991; Southampton, 1992
Royal Institution Friday Evening Discourse by R. Brown on Out of Line, May, 1992.
Edinburgh Science Fair, 1993.
Coventry Teachers Centre, 1994.

## Loan

The exhibition may be borrowed for educational purposes, provided the costs of transport and insurance are covered. A Welsh version is also available.
This is a slightly revised version of a paper published as  Making a mathematical exhibition'', in The popularization of mathematics, edited A.G.Howson and J.-P. Kahane, ICMI Study Series, Cambridge University Press, (1990) 51-64.