The Garden of Archimedes
A museum for Mathematics |

One of the characteristics of mathematics in the last century was undoubtedly its growing influence on scientific and technological development. Sectors that were traditionally open to the application of mathematics, such as physics and at least in part, engineering, have been totally mathematicised, so much so that often it is difficult to say where mathematics ends and where theoretical physics begins: others, who were more resistent to the use of mathematical methods, like biology, medicine or economics, have now opened themselves to mathematical language and formalisation. This is thanks not only to the widespread use of computers and to the new possibilities opened by their use, but also to the important progress of the art of mathematical modelisation of complex phenomena. It is easy to predict that the contribution of mathematics to the scientific disciplines will continue and increase in the near future, touching sectors that have been up to now outside its sphere of influence, and that mathematics will be increasingly one of the main factors of scientific and technological progress.

Before this undeniable centrality of mathematics in science, lies, however, the fact - also undeniable - that it is becoming less and less understandable to the non-specialist, and that it is becoming a language for the initiates, thus losing its cultural relevance. A 17th century traveler, maybe an English or German man having come to Italy to admire its artistic treasures or enjoy the clemency of its weather, could have receive an invitation for an evening in the palace of prince Cesi in Rome, to assist to a session of the recently founded, but already famous, Academy of the Lynceans. There he could have heard Luca Valerio exposing the results of his research on gravity centres, or if he was lucky he could have met Galileo, who with his telescope would show him the just-discovered satellites of Jupiter, and would discuss with his academic colleagues the pros and cons of the Copernican system and of the Ptolemaic one. One century later, in Milan, he could have asked Maria Gaetana Agnesi to explain the new calculus of fluxions, or differential calculus as it was called on the continent, and receive understandable answers. In the twentieth century, a descendant of our traveler, now a tourist, who for a strange reason might ask and be allowed to attend a meeting of the same Academy of the Lynceans, would hardly be able to understand a word of what is said. Modern mathematics is not for the man on the street.

But let's follow our tourist, who, after having sat on for some time, mostly out of politeness, listening without understanding, takes advantage of a break to leave the Academy; to pass the hours remaining until lunchtime, which he had thought of occupying with mathematics, he enters a café, after buying some newspapers. Luckily, there is a newsstand just outside the exit, and our friend buys the three dailies that he's been advised are the most authoritative. To calculate the total, the newsstand man takes out his pocket calculator, sums three times 1500 lire, and asks him for 4700 lire. At the customer's protest, the seller adds up again and this time gets the right sum, 4500 lire, which he pockets apologising for the error. He justifies it by saying that the calculator's keys are too near and he often presses the wrong ones.

Although not continuous, scenes like these are today very common, and they are the sign of a new type of mathematical illiteracy, in which pocket calculators are substituting even the simplest mental calculations. With their systematic use they bring about a conceptual regression from multiplication to repeated sums, retracing the historical path in reverse. In cases like these, the usefulness of mathematics becomes evident: if the newsstand man errs in his own favour, the mathematical customer can challenge the result and require the operation to be repeated, otherwise he can pretend not to notice and happily pocket the difference.

We all have been part, at one time or another of situations like our imaginary tourist's. In fact, notwithstanding the growing relevance of science in our everyday life, the understanding of even the simplest scientific facts is quite rare, even among the most learned sections of the non-specialist public, as we have occasion to notice all too often. Luckily, in contrast to this tendency, there is a growing demand for scientific information on the part of an attentive audience. This demand is the reason for the success of a number of scientific magazines, the foundation of some science museums, the fortune of science-oriented TV programmes, and the publication of many books dedicated to the diffusion and popularisation of science. Many of these books deal with mathematics, and with some exceptions they are well written and well liked by their readers. Of course one can do better and more, but when it comes to popularised books mathematics is even with other sciences.

The situation is radically different if one looks at scientific magazines and science museums, not to speak of TV programmes. Of course I cannot say that I read every magazine or that I have visited every science museum or that I have seen every scientific TV programme, but I am sure I am not too wrong if I claim that there mathematics finds a very restricted space, if any. And often, when one finds some mention of mathematics, this is in a substantially marginal position. One cans seldom get more than little crumbs of mathematical knowledge, or acquire an idea of the relevance of mathematics in modern society.

The reason for this difference between mathematics and other sciences is not - or at least is not only - the fact that mathematicians are too lazy or too presumptuous to lower themselves and explain their discipline to the general public. Nor is it that mathematics is too difficult, if not impossible, to popularise. Of course not every part of mathematics can be explained in simple words, but this is true more or less for all the other sciences, and in general of all human activities. On the other hand there are vast portions of mathematics, not only the most basic but some of the relatively more complex ones, that can be described in a non-technical way, and that in fact have been covered in books, some of which were quite successful. If this does not happen in museums, it will be for some other reason, for example in the different way mathematics is communicated than other sciences; a characteristic which is not apparent, or not relevant, in popular books.

One of the main differences between a book and a museum is clearly their narrative structure. By their nature, books are slow and analytical: the topic is covered from different and complementary points of view, and it is possible to emphasise the most important or difficult points, until they are explained in the most exhaustive and complete way possible. On the contrary, the language of an exhibition is concise, almost elliptic, and in most cases it is reduced to essentials - a museum is not a book glued to the walls. While the book is based on words and language, a museum is based on objects and phenomena, and explanations must be limited to the strictly necessary. The audience sees an object on display, such as, for example e.g. the reproduction of a spaceship, or even better the real one, and reads a short text explaining its construction, its purpose, its history. In another section, they see a dinosaur egg, and listen to an explanation of the environment in which dinosaurs lived and of the hypotheses on their disappearance.

The same happens when you show a phenomenon. In the mechanics sector, for example, the visitor is invited to sit on a rotating chair holding two weights. When she opens her arms, her angular velocity decreases, while it increases again if she brings her arms near the body, that is near the rotation axis. The corresponding panel will explain the law of conservation of angular momentum. In the next experiment, she is invited to hold the axis of a fast-rotating wheel, and to try turning it to one side, thus verifying the existence of a force which opposes the change of the rotation axis. The panel relates this experiment with the top we played with when we were children, explaining why the top gyrates on its tip without falling, and also with the modern gyroscopic compasses, one of which might even be on display. The visit then continues with other objects and experiments, that illustrate other important scientific phenomena.

None of this can be done directly with mathematics, which does not have objects to exhibit or phenomena to display. Or rather, has its own phenomena and objects, which however cannot be immediately visible as such, but must be extracted from other objects and experiments. You cannot show objects like a group, a complex number or a Riemann variety, nor can one directly experience the phenomenon of the distribution of prime numbers, or Euler's formula

f+v=s+2

linking the vertices (*v*), the faces (*f*) and
the edges (*s*) of a convex polyhedron. What one can
instead do in a museum, is to build a polyhedron and invite the
audience to count its faces, vertices and edges, verifying that
their numbers confirm the formula. But this is not sufficient, to confirm its validity (in fact no verification,
even on an with many specific examples, can guarantee
the validity of a theorem); it is even insufficient to convince
the visitor of its plausibility, in any case not in the same way
in which she was instructed in the conservation of angular
movement. The verification on a single polyhedron can be true by
accident, and it is not even necessary that a formula of this
kind exists, that is, that the number of faces, of vertices and
of edges of a polyhedron are linked in some way and that they are
not totally arbitrary. What is necessary is to repeat the
experiment several times with different polyhedrons. Only after a
considerable number of cases the formula becomes obvious and the
visitor is convinced of its validity. This could be the time to
exhibit a doughnut, or, as we call it in mathematics, a torus,
for which Euler's formula is not valid anymore, and start
experiments with another series of polyhedrons and surfaces,
until we introduce the concept of topological genre. The
difference with the rotating stool or the top experiment is
obvious: if you look at museum language, mathematics is the true,
maybe the only, empirical science.

This fact is not without its consequences. The first and foremost is that mathematics in a museum needs a lot of space. A mathematical object (or concept, if you prefer) is not something you can see and touch, and it cannot be illustrated with a single exhibit: it can emerge and become real only as the ideal substrate which links a succession of physical objects. In other words, mathematical objects, even the most elementary such as numbers or plain figures, are complex constructions whose description must contain at least their main properties, each of which needs at least one experiment to be described and explained.

It is very difficult to find the required space in a generic
science museum, where a balance is needed with other sciences,
also from a quantitative point of view. Here, mathematics is
treated like other disciplines: it is shown in single objects,
some very beautiful, some quite arid (like for example the number
*pi* with 10,000 or even 1,000,000 decimals), but which are nearly
always inadequate to correctly communicate the mathematical ideas
at play within them. What is shown are physical objects, like
some magnificent soap bubbles, and in the best of cases one says
that they admit a mathematical description, or that they have
stimulated important mathematical discoveries. In any case,
little or nothing is said about this mathematics, which always
remains behind the scenes. The trees are hiding the forest.

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What we have written so far brings naturally to the idea of a museum completely devoted to mathematics in its widest sense, including, that is, not only that which goes under the name of pure mathematics, but also its application to other sciences, to technology, and most of all, what is maybe the most important thing of all, its role in everyday life. The objectives of such a museum are manifold. Firstly, the audience can come into contact with the central core of mathematical ideas that reside inside the exhibits and determine their connections. Like a skeleton, which cannot be seen directly but requires the appropriate instruments and can be deducted from the posture of the animal that owns it, mathematics can only emerge from the comparison of different objects and physical phenomena, at first sight very diverse, but which depend on a single mathematical concept or result, which links and unifies them.

Secondly, the visitor will be led to recognise the importance of mathematics, and its determining role in her everyday life. The museum will insist on the fact that, although it is not immediately visible, mathematics permeates many objects of common use, and lays behind many of man's normal activities. In other words, mathematics is not a subject for specialists, but in many ways an important factor in the life of everyone of us.

Finally, mathematics is fun. This is an important message to communicate. Mathematics is not a boring sequence of exercises lacking any common sense, it is a stimulating universe of ideas and methods studied to solve important problems: ideas and methods that can be approached without formal or pedantic procedures, in a simple and interesting way.

Of course, the museum does not want, nor could it, teach mathematics, just like a concert does not teach how to play the piano. Just as the study of an instrument requires exertion and sacrifice, the same applies with the study of mathematics - one does not learn it effortlessly. Hence the distinction in roles - but also their complementarity - between the museum and school. For the latter, the museum for mathematics can perform, although with the obvious differences, a role similar to the one that concerts have for the study of music. Just like one does not go to a concert to learn music, one does not go to the museum to learn mathematics: for this, in both cases, one goes to school. On the other hand, however, just as not everyone is a musician, not everyone is a mathematician. The museum's purpose will, therefore, be that of bridging the gap between mathematics and the people, a place where you can approach mathematics and its most important ideas without difficulty.

Not a museum of mathematics, then, a museum where a dusty,
fossilised mathematics is exhibited inside closed cases; but a
museum *for* mathematics, a place to meet with the most
brilliant and growth-provoking ideas of common culture.

As we have said, this project needs space. The minimum museum unit is not the single object, but rather a path made of a sequence of objects, which it might even be possible to reduce to one, linked and unified by a single mathematical concept. Each of them illustrates a specific property of the latter, so that at the end of the path the visitor can have an idea of how the same mathematical object can be at the basis of all the phenomena she has experienced. Thus, for example, an ellipse can be traced through conic compasses (the perfect compasses of Arab geometers), shortly after its definition as one of the sections of the cone; its properties will, instead, stem from exhibits such as the gardener's ellipse, which the visitor will be invited to draw, or by the examination of the workings of elliptical instruments, illustrating the fact that the sum of the distances from a point on the ellipse to the ellipse's foci is constant. Later she will see that the waves generated at one focus of an (elliptical) baking pan, after being reflected by the sides, will concentrate on the other focus. Finally (although the experiments could go even further), she will verify how the shadow of a tennis ball has an elliptical shape, or if one wants to use a language nearer to mathematics, that the projection of a circle (the tennis ball) is an ellipse. And if the rays of light are parallel (and those coming from a light bulb can be made such by putting the bulb in the focus of a parabolic mirror) the same experiment with the tennis ball also shows that an ellipse can be obtained as a section of a cylinder.

This great number of objects to discuss a single shape is not only needed to describe the many properties of the ellipse, but it also allows to reduce written explanations to a minimum: as we have written, and as we repeat here, a museum is not a book hung on the walls.

Naturally, not all visitors will approach the exhibited objects with the same spirit and the same knowledge. The exhibitions that make up the museum must then be constructed so that they can be read at various levels. This is all the more necessary, since the museum is directed to visitors of all ages and cultural levels, and it is not possible to choose the audience on the basis of their mathematical knowledge or ability. Everyone must be able to appreciate the structure of the museum according to their level of scientific culture, or simply according to theirs willingness to follow the proposed paths with some attention.

This multi-level articulation begins at ground zero, of pure play, with very little mathematical contents. At this level the visitor simply has fun with the exhibited objects and instruments, trying to make them work as intended. This level is particularly indicated for children under 10 (generally speaking, elementary school pupils),outside of the sections that have been created specifically for them.

Next comes level one, in which the visitor reads the posters that give a general idea of the mathematical contents of the exhibition, and if possible of the historical context in which the mathematics in question has developed. If one wants to go a step further, one can buy a guide to the exhibition, from which one can learn something more about the mathematical ideas that determine the sequence of objects, without entering yet in the details of the demonstrations. For groups of visitors, normally groups of students belonging to one class, the written guide can be substituted with a guided visit to be booked in advance.

Finally, the highest level consists of the collection of a series of cards, each placed near the object or group of objects it refers to, and which illustrates with greater precision their mathematical details, including sometimes a simple demonstration. In any case, rather than to formal rigour, these will tend towards the illustration of particularly important concepts.

A series of paths like the one on the ellipse above, linked by a broader common theme, constitute an exhibition, which can be shown on its own or be a part of an even bigger structure, the museum. One of these exhibitions, titled "Beyond compasses: the geometry of curves", was built several years ago as a prototype for the museum, and since then it has been shown in numerous Italian cities and in some cases abroad, with a total of over 350,000 visitors. At the time of writing, it can be seen at the museum and at the Palais de la Découverte in Paris.

In its definitive configuration, the museum will contain several of these exhibitions. Others, if possible different from the ones seen at the museum, will be transportable and temporarily shown elsewhere, in order to create a rotation between the exhibitions visible at the museum and the itinerant ones, and at the same time to promote the cooperation between the museum and other hosting institutions. This arrangement suggests that the size of exhibitions should be kept compact, so that they can be hosted in mid-sized structures such as schools, and therefore be visible also in areas devoid of large exhibition facilities.

There is another type of issue that suggests not to go over 500 square metres of exhibition surface. Because of the peculiarity of mathematical exhibition language, organised as we have seen around articulate paths, the visit requires from the audience a higher level of attention than a traditional museum, where every single object is shown in itself. This level of attention cannot be kept for long, especially when paths are very complex and coordinated, and even considering pauses, the visit to a single exhibition cannot last longer than one hour. This time can be stretched somewhat alternating descriptive parts, such as documentary or historical exhibits, to the more difficult ones. In any case, the optimal size of the exhibition area of the museum can be estimated at around 1,000 square metres.

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The experience of the exhibition "Beyond Compasses" has shown that about three-fourths of the public is made up of students, from the elementary to high school, and the rest of individual visitors. Students usually come in classes led by their teacher, and they take advantage of guided tours. They mostly come from the city in which the exhibition is held, or from nearby towns from which it is easy to reach the exhibition.

The same thing is happening with the museum, which has its own hinterland of users, made of the towns and cities that lie no more than two, at most three hours away by bus. Visitors coming from further away are very scarce, especially when they are from schools - unless the museum is in a city that offers other attractions to school tourism, justifying a stay of several days.

In any case, a number of small museums, organised so that their user regions do not substantially overlap, complement each other. A diffused structure also allows to keep the size of each museum relatively small, and thanks to a rotation of exhibitions among the various locations, it encourages the repetition of the visit after some time.

The museum's project encompasses a network of local entities, situated in such a way that they will cover the biggest possible part of the territory, linked by a central scientific directorate and by common services allowing management savings, but otherwise managed locally. Of course, the creation of a museum section depends on the availability of an appropriate building and on enough financial resources to ensure an efficient local management.

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