The Garden of Archimedes
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A Museum for Mathematics:
why
,
how
and
where
.
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 caf?, 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
D?couverte 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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