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 (Ronnie Brown.'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.

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

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

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.

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 its material, in the objects with which it deals.

3. The context of history and art

Our initial aim here was to remind the viewer of how knots have enormous richness and importance in the prehistory and history of man. Their usage is very old.

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 [Robinson, J., (1989), Symbolism: sculptures and tapestries, (catalogue for the exhibition at the Pop Maths Road Show and the International Congress of Mathematical Instruction, Leeds, 1989), Mathematics and Knots, University of Wales, Bangor]. 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.