**MACHINES
FOR BUILDING SYMMETRY**[1]

* *

** **

** **

** **

**1.****
****Introduction**

** **

What I am
going to describe here is a mathematical interactive exhibition, “*Simmetria, giochi di specchi*”, based on
the theme of symmetry, which has been realized in the last few years by the
Mathematics Department of Milan University.

The main
objects of this exhibition are six “machines for building symmetry”. Three of
them are planar, consisting of three mirrors each, all perpendicular to the
plane of the table where they are posed, and cutting on this plane -
respectively - an equilateral triangle, a right isosceles triangle, a right
triangle with angles p/6 and p/3 (here we shall call them *mirror boxes*).

The other three are 3-dimensional *caleidoscopes*; each one again consists of three mirrors, belonging
to three planes concurrent in a point O and each one cuts on a sphere with
center O a spherical triangle: the first one of these three triangles has
angles p/2, p/3 and p/3; the second
one has angles p/2, p/3 and p/4, while the
third one has angles p/2, p/3 and p/5.

Each one of
the mirror boxes is provided with some tiles, which are the right ones in order
to see in the box a given picture (which will be a tessellation of the plane):
some of these pictures

are shown on the walls of the box,
so that the visitor knows in advance in which one of the three boxes he will be
able to build that picture; other ones are shown mixed up in a poster, so that
the visitor willing to reconstruct that picture has first of all to decide in
which one of the three boxes this can be done. Mathematically, this is a first
not trivial problem: this decision corresponds in fact to detecting the
symmetry group of the given picture.

The
3-dimensional caleidoscopes can be handled in an analogous way: some small
pieces are provided, which correspond to the fundamental domain[1] for the action of the symmetry group on
some polyhedra. By putting the piece in the corresponding caleidoscope, the
polyhedron is reconstructed.

Both in the
2-dimensional mirror boxes and in the 3-dimensional caleidoscopes the same
mathematical concept is underlined: the classification of something (a planar
picture in the first case, a solid object in the second one) with respect to
its symmetry group. So a “machine for building symmetry” builds up different
things, depending on what is put inside it, but all what is built with the same
machine has the same kind of symmetry; while different machines build up
different symmetries[2].

The actual exhibition contains in
fact other objects, which are all functional to this main idea.

This paper is organized as follows:
section 2 is more technical and is directed to the reader who likes to have a
more detailed idea of the mathematical concepts underlying the objects just
described; section 3 contains a discussion about different ways for using these
“machines”, with different sorts of public. Readers mainly interested to this
discussion can skip directly to section 3.

**2.** **The mathematics underlying the
machines for building symmetry.**

**2.1****
****Coxeter groups**

** **

Coxeter, in a series of papers
around 1930, began to study those subgroups of the isometry group in **R**^{n} having the two properties
of being discrete and being generated by reflections in hyperplanes. These two
properties are exactly what is necessary in order to “see” the group in a
system of mirrors. In order to explain what we mean, we shall first illustrate
a couple of examples in the plane.

Let G be the group
generated by the reflections **r** and **s** in two lines r and s, making an angle
of p/4 in point O. G is a finite group,
consisting of four rotations (with center O and angles, respectively, p/2, p, 3p/2, 2p) and four reflections, with respect to the two
lines r and s and other two lines t and u (see figure 1); the set of these four
lines is the *complete set of hyperplanes*
related to the group G (that is, the maximal set of hyperplanes such that the
reflection with respect to that hyperplane belongs to the group); the connected
components of the complement, in the plane, of these four lines are the *chambers* of the group G. All the
chambers are equivalent (in the sense that there exists an element of the group
sending one into the other), each one of them is an angle and the group is
generated by the reflections in the walls of any chamber. There is a bijection
between the whole set of the chambers and the elements of the group: fixing one
of the chambers R_{0}, the bijection is obtained by sending any other
chamber R into the element g in G such that

g(R)= R_{0}.

In order to consider
another example, let H be the group generated by the reflections **a**, **b**,
**c** in the three sides of a right
isosceles triangle. H is an infinite group, and the complete set of hyperplanes
is sketched in figure 2:

It is an infinite grid, and the
complement of this grid in the plane is made up of an infinity of isometric
triangles, which are the chambers of the group.

It is worthwhile
observing that, in both cases, the complete set of hyperplanes (and, as a
consequence, also the set of the chambers) is just what we see if we start from
a set of mirrors (two in the first case, three in the second one) corresponding
to the generators of the group.

In fact, when we say that we are
“seeing the group”, we are not referring only to the fact that the chambers we
see are in one-to-one correspondence with the elements of the group, but, also,
to the possibility of reading out from what we see the presentation of the
group with generators and relations: in the case of figure 1, the group G is
generated by **r** and **s** with relations **r**^{2} = **s**^{2}
= (**rs**)^{4} = 1; in the case
of figure 2, H has three generators **a**,**b**,**c**
and relations **a**^{2} = **b**^{2} = **c**^{2} = (**ab**)^{2}
= (**bc**)^{4} = (**ac**)^{4} = 1.* *In
the photo, we can see another example, corresponding to thegfroup with three
generators **a**, **b**, **c** and relations **a**^{2} = **b**^{2} = **c**^{2}
= (**ab**)^{2} = (**bc**)^{3} = (**ac**)^{6} = 1.

This situation is quite general: any
Coxeter group has a presentation with k generators g_{1},…,g_{k}
and relations of the kind (g_{i}g_{j})^{N}ij=1; if i=j,
N_{ii}=1, as each g_{i} is the reflection in a hyperplane; if i¹j, p/N_{ij} is the angle between the two
hyperplanes associated to the generators g_{i} and g_{j}.

From a geometric point
of view, one can prove that any (irreducible[3]) Coxeter group is generated by the reflections
on the walls of a chamber which can only be:

·
a simplex, if the group
is infinite;

·
the cone on a simplex,
if the group is finite.

There are other limitations on the shape of these
chambers, which lead to the complete enumeration of Coxeter groups in any
dimension; this is reached through an analysis which is quite lengthy in the
general case, but, in low dimensions, is essentially a consequence of the
observation that, due to the discreteness of the group, the dyhedral angles
between the walls of the chamber must be of the form p/n, where n is any positive integral number
(³2). This leaves
very few possibilities, that is:

1. for finite groups in two dimensions:
two mirrors making an angle of p/n (for any n in **N**); the chamber is the cone on a 1-dimensional simplex;

2. for infinite groups in two
dimensions: three mirrors forming a triangle, with angles p/p, p/q, p/r. As 1/p + 1/q + 1/r = 1, the only
possibilities are:

·
p=q=r=3: equilateral
triangle;

·
p=2, q=r=4: right
isosceles triangle;

·
p=2, q=3, r=6: right
triangle with angles p/3 and p/6;[4]

3. for finite groups in three
dimensions: three mirrors forming the cone on a triangle. The angles between
the mirrors must be of the form p/p, p/q, p/r; as these are the angles of the
corresponding spherical triangle, we get the inequality 1/p + 1/q + 1/r > 1,
which leaves the only possibilities:

·
p=2, q=r=3: this
corresponds to the symmetry group of a tetrahedron;

·
p=2, q=3, r=4: this
corresponds to the symmetry group of a cube, or of a octahedron;

·
p=2, q=3, r=5: this
corresponds to the symmetry group of a dodecahedron, or of an icosahedron.

The six cases just obtained in 2. and in 3. are
exactly the six “machines for building symmetry” described in the introduction.

In order to complete the list of Coxeter groups
acting in dimension less or equal to 3, we have to add also:

0.
groups in one
dimension: there is only one finite group, corresponding to just one mirror;
there is also only one infinite group, corresponding to two parallel mirrors.

4. infinite groups in three dimensions:
these are given by four mirrors forming a tetrahedron with all dyhedral angles
of the form p/n; there are three possibilities,
whose corresponding chambers are all shown in figure 3:

5.

·
the tetrahedron of
vertices O,V,O’,V’, which has two dyhedral angles equal to p/2 and four equal to p/3;

·
the tetrahedron of
vertices O,V,O’,M, which has three dyhedral angles equal to p/2, two equal to p/3 and one equal to p/4;

·
the tetrahedron of
vertices O,V,H,M, which has three dyhedral angles equal to p/2, one equal to p/3 and two equal to p/4.

2.2 **Two-dimensional machines and plane
cristallographic groups**

** **

** **In the last
subsection we found three irreducible cases among Coxeter infinite
two-dimensional groups (equilateral triangle, right isosceles triangle, right
triangle with angles p/3 and p/6) and one reducible case (rectangle). The same
four cases appear among crystallographic groups, that is discrete groups of
isometries in **R**^{n}, whose
translation subgroup is generated by n independent translations. For n = 2,
these are the 17 (up to affine conjugacy) *wallpaper
groups*; among them, the ones which are generated by reflections are Coxeter
groups (reducible or irreducible), and there are four of them:** **

·
**p3m1 **corresponding to the
mirror box with the shape of an equilateral** **triangle;** **

** **

·
**p4m **corresponding to the
mirror box with the shape of a right isosceles** **triangle;

** **

·
**p6m **corresponding to the
mirror box with the shape of a right triangle with angles p/3 and p/6;

** **

** **

** **

** **

·
**pmm **corresponding to the
mirror box with the shape of a** **rectangle**.**

** **

It is also interesting to notice that these four groups are not the only ones – among wallpaper
groups – which can be seen in our “machines”: in fact, when we put something in
a mirror box, the planar picture we get has a**
**symmetry group**
**which contains
the group G associated to the box (that is, the group generated by the
reflections in the walls of the box), but not necessarily coincides with it. If
we put in the box something which already has some symmetry, what we get is a
group H which properly contains G as a subgroup and we may also “read” the
index of G in H[5]. So, if we want to
make a list of which ones of the 17 wallpaper groups may be seen in our
machines, we have to add some cases, that is:

·
**p6m **is not contained as a
proper subgroup in any of the other 16 groups; so in the mirror box with the
shape of a right triangle with angles p/3 and p/6 we can only see something with symmetry group **p6m**[6];** **

·
the same happens for the group** p4m **corresponding to the
mirror box with the shape of a right isosceles triangle[7];

·
**p3m1 **is contained as a
proper subgroup of (minimal) index 3 in the** **

** **

crystallographic
group** p31m **so, in the box with
the shape of an equilateral triangle we generally see something with symmetry
group** p3m1 **(by putting in the
box something with no symmetry at all); but we can also see a planar picture** **with symmetry group** p31m**, by putting in the
box something with symmetry group C_{3}, that is a center of rotation
of order 3 in the center of the mirror box.** p3m1** is also
contained as a proper subgroup of (minimal) index 2 in the crystallographic
group** p6m**; this is for us less
interesting, as it does not give a “new” group to see; however, in order to see
the group** p6m** in the equilateral
mirror box it is enough to put inside the box something with a bilateral
symmetry**.**

· **pmm **is
contained as a proper subgroup

· of (minimal) index 2
in the group **cmm**;

·
of (minimal) index 4 in the group** p4g.**

Besides
these two cases, which add two new groups to the list of the ones we can see in
mirror boxes, there are other two possibilities, as **pmm** is also
contained as a proper subgroup

·
of (minimal) index 2 in the group** p4m;**

·
of (minimal) index 6 in the group** p6m.**

Thus in a
rectangular mirror box we generally see something with symmetry group pmm (that
is, this happens when we put inside the box something without any symmetry); we
manage to get a picture with** **symmetry group** cmm**

** **

when we put inside the box something with a
center of symmetry of order 2 in the center of the rectangle. In order to get a
picture with symmetry group** p4g,**we cannot start from
any rectangular mirror box, but we need a square one,** **

** **

** **

and we
have to put in the square box something with symmetry group C_{4}, that
is with a rotational center of order 4 in the center of the square. If we put
in** **the square box something whose symmetry group
is the dyhedral group D_{4} we get a picture with symmetry group** p4m, **but in fact we can get the same group (as a subgroup
of index 2 instead of 8, that is by putting inside the box a picture with just
a symmetry axis along the diagonal of the square. In order to get a picture
with symmetry group **p6m**, we cannot start from any rectangular mirror
box, but we need one such that the ratio between its sides is square root of 3
so that it can be divided in six right triangles with angles p/3 and p/6, each
one obtained from the adjacent one by reflection in the common side: see figure
4.

So, the wallpaper groups which can be seen in a mirror box are seven, the
four ones which are generated by reflections, and other three, which contain a
subgroup generated by reflections.

**2.3****
****Three-dimensional machines and polyhedra**

** **

And what about three-dimensional symmetry machines? The ones we described
in the introduction realise the only possible irreducible finite
subgroups of Iso(R^{3}) generated by reflections, which correspond to
the symmetry groups of the regular polyhedra. As in the two-dimensional case,
the significative mathematical result hidden in these machines is the fact that
they are not just an example, but they are in fact the only possible cases.

As in the two-dimensional
case, by putting something in the “machine”, what one sees is the orbit F of
that “something” with respect to the group associated to the caleidoscope (that
is, the group generated by the reflections in the walls of the caleidoscope);
the symmetry group of F contains the group of the caleidoscope, but does not
necessarily coincides with it. However, finite groups of isometries in the space
are very few, so that for two of the caleidoscopes (the one associated to the
symmetry group of the cube and the one associated to the symmetry group of the
dodecahedron) we may be sure that anything we see inside will always have a
symmetry group which coincides with the group of the caleidoscope: the reason
of this is simply the fact that there does not exist any finite group of space
isometries containing as a subgroup either the symmetry group of the cube or
the symmetry group of the dodecahedron.

Instead, in the caleidoscope associated to the symmetry group G of the
tetrahedron we may see either objects
having G as symmetry group or objects having the same symmetry group H of a
cube (as G is a subgroup of index 2 in H).

Colouring can be used (here as in the planar case) to underline this
phenomenon. For example, an octahedron can be reproduced in the caleidoscope of
the cube, but if we colour it with black and white faces (each black face
touching only white ones, and viceversa) and we do want to reproduce that
colouring also, then we need the caleidoscope of the tetrahedron[8].

Of course, regular polyhedra are not the only objects one can observe in
the three-dimensional caleidoscopes. To give another example, we can consider
the orbit F of a single point x with respect to the group associated to one of
the three caleidoscopes, and the convex envelope of the points in F: in this
way, we naturally get a uniform polyhedron (that is, a polyhedron whose
symmetry group is transitive on the set of vertices). In particular, we get in
this way nearly all the 13 Archimedean polyhedra:

· in the
caleidoscope of the tetrahedron we get
also the archimedean polyhedron (3,6,6)[9];

· in the caleidoscope
of the cube we get the archimedean polyhedra (3,4,3,4), (4,6,6), (3,4,4,4),
(3,8,8), (4,6,8);* *

· in the caleidoscope
of the dodecahedron we get the Archimedean polyhedra (3,5,3,5), (5,6,6),
(3,4,5,4), (3,10,10), (4,6,10).

The only
two archimedean polyhedra which can not
be seen in the caleidoscopes are (3,3,3,3,4) and (3,3,3,3,5), whose symmetry
groups contain only rotations.

The same construction does not only yield the Archimedean polyhedra
(which, besides beeing uniform, have regular faces): in general, for any choice
of x, we always get a polyhedron whose faces are equiangular; for each one of
the described cases, there is just one position for the point x such that the
faces of the corresponding polyhedron are also equilateral.

**2.4****
****Elliptic, euclidean (and hyperbolic) geometry**

** **

Up to now we spoke about two-dimensional and three-dimensional machines
for building symmetry; but there is another way - probably more suitable[10] - to look at this situation. It should be
noticed in fact that the world of polyhedra can be handled in (at least) two
different ways: we can think to a polyhedron as a solid object (homeomorphic to
D^{3}), the analogous in the three-dimensional space of what is a
polygon (homeomorphic to D^{2}) in the plane; alternatively, we can
think to a polyhedron as a two-dimensional object (homeomorphic to S^{2}).
From this point of view a polyedron is not so much the analogous of a plane
polygon, but rather the analogous of a plane tessellation. With the first point
of view one rather sees the symmetries of the object as isometries in the
space; with the second point of view, one thinks more about isometries of the
sphere.

This second point of view is probably the best one to underline the
similarity between the different situations shown with our six “machines for
building symmetry”. In any case we handle surfaces, more precisely triangulated
surfaces; with the first three machines, the mirror boxes, we are in the world
of euclidean geometry, and the restriction on the number of possible machines
comes out essentially from the equality

1/p + 1/q + 1/r = 1,

implied by
the fact that the sum of the angles of a euclidean triangle is equal to p; with the other three machines, the caleidoscopes,
we are in the world of elliptic geometry, and the restriction on the number of
possible machines comes out essentially from the inequality

1/p + 1/q + 1/r > 1,

implied by
the fact that the sum of the angles of a spherical triangle is greater than p.

A very natural extension of this would be to have “machines for building
symmetry” in hyperbolic geometry; and, in this case, we have a much greater
variety, as there exists such a machine for any p, q, r such that

1/p + 1/q + 1/r < 1,

so there
are an infinity of them. Each one corresponds to fixing a hyperbolic triangle
whose angles are p/p, p/q, p/r and
considering the subgroup G of hyperbolic isometries generated by the
reflections in the sides of the triangle. There is no difficulty in simulating
on a computer a virtual hyperbolic machine: it is enough (for example, in the
Poincaré model) to substitute reflections with circular inversions. This is of
course conceptually identical to a physical realization: however, in a plan for
a math exhibition, a real object still makes a great difference, at least in my
opinion, with respect to a virtual one (and it does not seem technically easy
to construct such a machine).

* *

**3.****
****What can be done with the “machines for
building symmetry”.**

** **

**3.1****
****Classification with respect to symmetry, for
different visitors**

** **

The principal aim of an exhibition based on the objects described in the
introduction is to give the visitor an idea of the problem of classifying
something with respect to its type of symmetry – an idea which of course will
be at very different levels, depending on the degree of mathematical knowledge
of the visitor.

The possibility of giving at least a flavour of this idea, even to
someone with no mathematical knowledge at all (like small children) is due to
the fact that, given a group generated by reflections, it is possible to
express the idea of isomorphic or non-isomorphic groups without any technical
algebraic language, but simply by looking at the geometry of the mirrors.

This opens the possibility of making
a lot of non trivial considerations, related to the symmetry group of a planar
picture or of a solid object, without the necessity of introducing the
algebraic language related to groups; for example, planar pictures which can be
reconstructed in the same mirror box (or solid objects which can be
reconstructed in the same caleidoscope) have isomorphic symmetry groups, while
pictures (or solid objects) reconstructed in different mirror boxes (or
caleidoscopes) have non-isomorphic symmetry groups.

Of course, due to what we observed
in the preceding section, the last sentence is in fact not quite correct[11].
However, we do not think this ambiguity makes a serious problem towards
mathematical communication in such a sort of exhibition. In fact, for the
minority of the public who can appreciate this difference, the ambiguity is not
hidden but, on the contrary, some problems are posed on purpose, in order to
provoke questions on the** **subject, and this
gives a new, not-trivial problem to investigate. For the majority of the
public, in order to give a flavour of the different types of symmetry, it is
enough to make them observe how the pictures coming out from certain mirror
boxes are all based on numbers 3 or 6 (and on a triangular grid), while others
are based on number 4 (and on a square grid).

Moreover, the same problems can be used for the more and for the less
sophisticated public with different purposes. To give just one example, the** **reconstruction of the same tessellation with different kinds of colouring
(which can also lead to different symmetry groups) can be used with the more sophisticated
public precisely to provoke the ambiguity we were discussing before, while with
the less sophisticated one the same problem can be used for simpler
observations.

In fact, one of the
reasons why we thought the “machines for building symmetry” were useful for an
exhibition is exactly what is exemplified in the previous comment, that is the
fact that the same objects can be used to communicate different levels of
mathematics to different sorts of public. The fact that they are based on some
non-trivial mathematical concepts from one side allows also an interesting
communication towards a public with more mathematical background and from
another side it is clearly perceivable also by the public with very little
mathematical knowledge: often, it is not necessary to be able to enter deeply
inside a problem in order to understand whether the problem has, or has not,
such a depth.

**3.2****
****The role of interactivity**

** **

Another reason why we think these machines can be a useful example in the
direction of finding ways for the popularization of mathematics is the fact
that they give occasions of “doing mathematics”; and, in saying this, we think
both to the public with less mathematical knowledge and to the public with more
technical instruments. Both will have the possibility of putting their hands on
the objects and meet a problem they will have to solve: for a seven-year old
child the problem can be how to put a given triangle in a mirror box in order
to see an hexagon; for a mathematics university student the problem can be that
of understanding why a crystallographic group cannot have order 5 rotations, …
: in both cases (as well as with other possible categories of public) the
objects can provoke an active reaction by the visitor: which we think is the
only effective way to learn some mathematics.

Another aspect, related
to this one, is the crucial role played by fancy and creativity, which are
human capacities usually (and wrongly) thought to be far away from mathematical
capacities. This crucial role comes out from the fact that the main thing to do
with our machines is just to look what happens when one puts something inside,
and to observe analogies and differences between them.

In order to make these observations, any object would do: if I take the
piece which represents the fundamental
domain of the action of the symmetry group of the cube and I put it, in the
“right” way, in the caleidoscope of the cube, I see a cube.

But if I put it in a “wrong” way,
or I do not put that piece at all, but I prefer to insert a ball, or even a dry
flower I had in my pocket, that will work equally well; in this case also I can
observe that what I see has always the same kind of symmetry of a cube and
always a kind of symmetry different from what we see in the other
caleidoscopes.

So, the “wrong” trials to solve the problems
proposed are equally well useful to observe what happens and thus familiarize
with the concepts involved; this may be very “relaxing”, especially for people
(unfortunately not so rare) who are paralysed by a sort of “fear” towards
maths.

**3.3****
****The role of mathematical proofs**

We spoke of active interaction with
the objects, and of fancy, and of playing. But mathematics is also (or mainly?)
rigorous proofs. What could the role of proof be in this kind of proposal? In
fact, the problem of achieving a rigorous proof is (always, but especially the
context of undergraduate teaching) a matter of subsequent approximations. And
the first stages of these approximations are the understanding of what has to
be proved, and the consciousness that the given fact is not trivial and has
thus to be proved. This seems (and in fact is) an obvious consideration, but it
is unfortunately frequently forgotten: pupils are often forced to prove a
statement in the moment when they have not yet clear ideas about what it does
mean (what it means if it happens to be true, what would happen if it is
false,… ecc.); or they may be asked to prove some facts which are eventually
not so easy to prove, but whose statement is (or appears) evident.

It is of course one of the main
aspects of mathematics the fact that, in a deductive construction, one has to
prove everything, also “self-evident” statements, and we perfectly know how
some self-evident statements are not at all trivial to prove (Jordan’s theorem,
just to give an example), and some are even false. But …; but one needs some
mathematical maturity to appreciate the need to prove self-evident statements;
and it can be useless, or even damaging, to propose their proof in a context
where this maturity is lacking.

This brings to the (apparent) paradox that it may be easier to propose
the proof of “difficult” statements than that of “easy” ones in secondary
school. The role of a mathematical exhibition towards the achieving of proofs
could well be (on some categories of public) that of making people conscious,
and curious, about some facts to be proved.

Let us exemplify what we are trying to say on two statements: the first
one is the fact that a triangle with two equal sides has two equal angles; the
second one is the fact that the frieze groups[12]
are seven. And let us keep in mind two sorts of public, a secondary school
student and an adult with no mathematical background after school. It is very
unlikely that both the student or the grown-up person manage to become
particularly curious about the statement on isosceles triangles; in any case
they both believe it is true, and they would use the statement, without
realising it requires a proof, if in a concrete problem they happen to need it.
The situation is completely different for the statement about friezes: first of
all the statement is strange, and difficult to understand; one does not
understand it at once, but has to think about it, to make a list of the
possibilities, to reason about the fact that whatever drawing he or she is
making, it has to be one of those
seven. When one grasps what this means, usually this is related to a sense of
beauty: the result is beautiful, conceptually beautiful. Moreover, it looks
strange; and it is very natural to ask why
is an eighth case impossible. So, with some time at disposal, it is very easy
that people arrive naturally to the consciousness of the need for a proof.

When we have obtained such a consciousness,
there are still many intermediate stages which can be significant, before
achieving a complete proof: for example, one could begin to observe that, due
to the fact that the group is a frieze group, there must be some restrictions
on the kind of possible isometries in the group (rotations may only be of order
two; the axis of reflections may only be either parallel or perpendicular to
the direction of translations; the axis of glide reflections must be parallel
to the direction of translations; …). This is not yet a proof, but it begins to
give some flavour of it, and the statement, which at first sight appeared
completely mysterious (“*why just seven???*”)
can now be seen as more reasonable (“*I
still do not know why they are exactly seven, but I do understand that the
situation is not completely free, and there are some limitations to be
respected*”).

In fact, these intermediate stages
may be exactly what we would like to be grasped about proofs in mathematics
undergraduate learning, much more than the particular proof of a particular
statement, which is often not relevant in itself (at least at that level).

**3.4****
****The role of beauty**

** **

Beauty has – in many different ways – a crucial role in the exhibition
just described.

The first aspect regards a problem of motivations. We all know very well
that mathematics has on the whole a very bad reputation; it is quite common to
meet persons who have a sort of hate and/or fear towards mathematics; there may** **be also (in the same person)
interest or curiosity about maths, besides fear, but it is very likely that
fear acts as a sort of block towards curiosity. This block is a concrete
problem that any trial of popularizing maths has to consider; one needs a way
to overcome it, to be able to begin to communicate
with the public; and, moreover, this way must be an immediate one, because it
has to win an irrational feeling, not a rational one.

The strong impact of beauty is an enormous help in this sense; and we
found symmetry a very good subject for popularization of maths also for this
reason (besides the ones already discussed).

By saying this, we do not only refer to the trial of involving art, by
proposing posters with the reproductions of some masterpieces where symmetry is
wonderfully used; but we refer also to the “beauty” of the artificial images
that the visitor is invited to reconstruct (and/or to invent) by himself. Of
course these two kinds of “beauty” are not comparable, but it is a fact that
both have a precious role in disposing people to be willing to interact with
maths – a result which is not at all obvious to reach.

A particular role is played by the strong effect of surprise. In our
mirror boxes one “can see infinity” and this effect is very strong, very
beautiful, and also very unexpected, in all sort of public.

Moreover, this effect of surprise does not
come from spectacular, enormous, scenic objects, but from very simple ones.
This has a double positive effect: the first one is that it magnifies the
surprise (if I have to enter into an enormous building, which I see from very
far away, with much light and colour, I do expect I shall see something which
will surprise me; maybe I have no idea of what
I shall see, so I shall still be surprised, but I know in advance that this
will happen; instead, if I put my eye on the border of an object which looks
very simple, I do not expect any particular “special effect”); another positive
consequence is that the objects are easily reconstructible, so for example
teachers realize they can easily build something analogous in their schools[13].

A last aspect about beauty I
would like to remind here is one which has already been mentioned in the
preceding section: mathematics is beautiful not only for the beauty of some of
its images, but, also, for the conceptual beauty of some of its results. Too
often – in my opinion – we do not even try to communicate this kind of beauty:
mathematicians seem to lack any confidence about the fact that this could be communicated to someone,
unless he or she has done the right number of exams in algebra, geometry,
analysis ecc. Sometimes this is true, but it is probably much less true than
what is generally thought, and it is possible to communicate much more than we
think. At least, it is worthwhile trying.

[1] D is a
fundamental domain for the action of G on P if for any point x in P we can find
an element g in G such that g(x) belongs to D and no two points in the interior
of D are related by an element in G.

[2] This is in fact just a first rough approximation, which is not quite
correct: see the following section for a more precise statement.

[3] A Coxeter group is irreducible if it is not isomorphic to the direct product of two Coxeter groups
acting in lower dimensions.

[4] If we try to construct a plane polygon with all
angles of the form p/n, there is another
possibility besides these three triangles, that is a rectangle, with all angles
p/2. This case is less significative among
Coxeter groups, because it is a reducible group: in fact, it can be seen as the
direct product of two copies of the group generated by the reflections in two
parallel lines, each copy acting over **R**.

[5] The index of G in H is equal to the ratio of the areas of the
fundamental domains of G and H: this
can easily be read from what we see, keeping in mind that the fundamental
domain of G is the box itself.

[6] As **p6m** is a subgroup of itself, what can happen is
that by putting something in the box we may get a picture whose symmetry group
H is always isomorphic to **p6m**, BUT is different from
the group G generated by the reflections in the walls of the box.

[7] As **p4m** is a subgroup of itself, the same phenomenon
described in note 6 for **p6m** can happen.

[8] In fact, the symmetry group of the octahedron is isomorphic to H, while
the coloured symmetry group (that is, the subgroup of those isometries which,
besides fixing the object, fix that colouring also) is isomorphic to G.

[9] The notation (a_{1},a_{2},…a_{k}) used for an
archimedean polyhedron expresses the fact that each vertex is adjacent to a
regular a_{1}-gone, a regular a_{2}-gone …, a regular a_{k}-gone
(in this cyclical order).

[10] The distinction between two- and three-dimensional situation can
generate ambiguity in a context where the objects are phisycally shown (and,
therefore, are all necessarily three-dimensional).

[11] For example, we can build in a square mirror box
pictures having four possible (non isomorphic) symmetry groups (**pmm**, **cmm**, **p4m**, **p4g**); while a picture
with symmetry group **p4m** could be built both
in a square mirror box or in a mirror box with the shape of a right isosceles
triangle.

[12] Frieze groups are the symmetry groups of patterns repeating in just one
direction; that is, they are discrete subgroups of the group of plane
isometries, whose translation subgroup is isomorphic to **Z**.

[13] Although this is on the
whole a very positive fact, it also poses some
problems: in fact, it is not always easy tokeep control on this and to
prevent imitations by people who are misunderstanding the conceptual and
didactical role of the objects.

**BIBLIOGRAPHY**

[1]
H.S.M. COXETER, *Introduction to geometry*, Wiley (1961)

[2]
H.S.M. COXETER, J. MOSER, *Generators and relations for discrete groups*,
Springer (1980)

[3] P. CROMWELL, *Polyhedra*, Cambridge
University Press (1997 )

[4] M. DEDÒ, *Forme*, Decibel-Zanichelli (1999)

[5] H. WEYL, *Symmetry*,
Princeton University Press (1952)

Dipartimento di Matematica “F.
Enriques” Università di Milano

[1]** All the photos are taken
from the exhibition “ Simmetria, giochi di specchi” and made by Sabrina Provenzi ; reproduction
authorized by the head of the Maths Department “F. Enriques” of Milan
University.**