This is the text of the poster you can see here.

From Shadow to Shadow

What is left invariant in pictures (taken from different angles) of the facciata of the same house, in pictures of the same discor of something else? What changes in them?

What is left invariant in the three different shadows that the same cardboard contour projects on a plane when it is enlightened from different angles? What changes in them?
If we disregard chromatic issues and we just observe the outline of the images, in both cases differences are more numerous than common characteristics.
In a close-up shot, the shape of the doors will be similar to a rectangle. In a foreshortened picture, the shape of the doors will be that of an irregular quadrangle. If one of the picture is taken from above and the other from below, the proportion among the different parts will be totally distorted and so on.

The shadow of a disc can be a circle as well as a more or less stretched ellipsis or something else; it depends on the position of the plane with respect to the light.
Yet, there is something in common between the two pictures or the two shadows: for example, the straight lines in a picture stay straight in the other picture too. The shadows of the disc have a “soft” outline without tips nor bumps.
If you ask if there exists a transformation of the plane that sends a picture of the façade to the other or if there exists one that sends a shadow of the contour to the other, the answer is positive in both cases. The transformation in both cases is a projectivity of the plane.
The two pictures of the façade – or the two shadows of the contour – are equal from the viewpoint of projective geometry, which investigates them as though they were the same thing.

Projectivities alter the length and the ratio of length of segments. Two identical windows in real life are rather different in a foreshortened picture. None the less, everyday experience says that there is something that allows one to recognize regularity in foreshortened images. Look at the images below…

Evidently, the pins that appear in picture 1 are not positioned in a regular way. On the other hand, the distance between those in picture 2 is the same although the actual distance between each image $is different and decreases.
What makes us recognize regularity in the first picture? Or, more specifically, what makes us exclude regularity in the second one?
It is the cross-ratio, a numerical invariant of projective geometry. Although lengths and ratios of lengths of segments are not preserved under projectivities, the ratio of the ratio of aligned segments is. Given four points A,B, C and D on a line, the ratio between the two ratios AC/BC and AD/BD – or their cross-ratio (ABCD) – coincides with the cross-ratio (A’B’C’D’) of their images under any projectivity.
If you take four points on a line and at the same distance, their cross-ratio is 4/3. Thus, all their projective images have the same cross-ratio 4/3. In picture 1, the cross-ratio of the “bases” of the pins is not 4/3, whereas the cross-ratio of those in picture 2 is 4/3. Although we are not aware of it, our mind is trained to detect it: it can “measure” some cross-ratios before we realize it!

Imagine to glue the opposite edges of a rectangle so that the points at the same height on the vertical and horizontal edges are identified. You will get a doughnut, which is called a torus.

Now, imagine to glue two opposite edges as before and the remaining edges with a half twist. You will get something odd. At the first step you will get a cylinder as for the doughnut. Second, you can’t simply put together the two circles to glue them. Indeed, the orientation on the first one is opposite to that of the second one. To glue them we need to "go into" the cylinder so that the surface has self-intersections; so you will get the Klein bottle.

The self-intersection curve is not an instance of how we obtained the surface. One can prove that it is not possible to embed the Klein bottle in three-dimensional space without self-intersections. If we want to have a mental picture of the Klein bottle, we should "not see" the self-intersection curve with the eyes of our mind. We should regard it as an "accident" due to its embedding in three-dimensional space.
We can compare it with the diagram which represents the projection of a trefoil knot on the plane. Under this projection there are some crossings. If we know that the diagram represents the projection of the knot, we know that a crossing on the projection corresponds to two points on the knot. These points belong to two different branches whose projections on the plane intersect. The same happens for the Klein bottle. The self-intersection curve corresponds to two curves on the surface, one on each sheet, which are forced to coincide on the "projection" of the bottle in three-dimensional space.

The Klein bottle as well as the Moebius strip is a non-orientable surface. If it were labeled, and the label could move on the bottle, the label would be flipped after a full round on the "handle".

The Klein bottle and the Moebius strip are "relatives". More specifically, it is possible to cut a Klein bottle in order to have two Moebius strips (or glue two Moebius strips so to get a Klein bottle).
The Moebius strip and the Klein bottle are different because the former has a boundary and the latter does not. This difference makes it possible to embed the former in three-dimensional space without self-intersections - and not the latter. There is a family of surfaces like the Klein bottle that are non-orientable and withour boundary. They can’t be embedded in three-dimensional space without self-intersections, but it is possible to do it if one makes a hole on them so to get a boundary curve.

The first example of this family is the projective plane. It can be visualized by "seaming" a disc and a Moebius strip along their common boundary, which is a circle. It is difficult to imagine seaming a disc along the boundary of the Moebius strip when the strip is "manipulated into" this position. To help our imagination, we can deform the strip until the boundary takes the shape of a standard circle. Then we can imagine to seam a disc along the circle. Doing so, the Moebius strip takes some crossings; thus, even in this case, there are some self-intersections. In fact, this may not happen. This deformation of the Moebius strip, which turns the boundary into a "standard" circle, does not yield any self-intersections. Note, however, that if you tried to glue a disc to the circle, you would not be able to do it without intersecting the shell.
Therefore, also in this case it is not possible to embed the projective plane in three-dimensional space without self-intersections.

From "Matemilano, percorsi matematici in città", Ed. Springer

Let us suppose to connect the three main railway stations in Milan (Centrale, Nord and Garibaldi) with the Cathedral, the stadium and the Linate airport via a system of fast trains. It is therefore necessary that the connecting paths do not cross. Is it possible?

This is a revised version in the Milan area of a classical topological problem, which can be worded in this way. Fix three points (in the figure three coloured dots; in our example the cathedral, the stadium and the airport). You must connect each black dot with each of the coloured dots so that the paths do not cross.

This is a topological problem because it is easy to realize that the distance between the dots does not matter as well as where they lie. Actually, our original problem is not realistic because we should have trains for which the manufacturing cost does not depend on the length of the paths and the travelling times are not length-dependent!
You may start drawing the possible paths from a black dot. You may also get to draw all the paths except one. After that, you are done. You can not connect the Nord Station with Linate without crossing the other paths. This does not depend on a possible mistake in drawing the first paths: it is actually impossible!
To realize that the problem is in fact impossible, you may think of the circuit (sketched in the picture) with six paths: Central Station - Cathedral - Nord Station - Stadium - Garibaldi Station - Linate - Central Station.
However you draw the paths, this circuit is a closed curve without crossings (for the paths do not cross). Such a curve separates the plane in two regions: an internal part and an external one. Thus, any one of the three paths we have to add is completely in the internal region or completely in the external one.

It is easy to see it is not possible to solve the problem. If, for instance, one of the three paths to be added is in the internal region of the circuit, the second path should be in the external region; so the third path is blocked both inside and outside.

The crucial point that makes the problem impossible to solve is that any closed curve without self-intersections divides the plane into two regions, an "inside" and an "outside". This statement is not surprising. Yet, it is the typical example of a theorem (the Jordan curve theorem) which has a very intuitive, and apparently trivial, statement, but it is highly non-trivial to prove. Indeed, it is necessary to apply very sophisticated techniques or, if you do not want to use them, you need very involved arguments. To realize that this theorem is non-trivial, you may observe that the same statement is not true on a torus, i.e., a doughnut, nor on a Moebius strip. This is the surface which is obtained from a rectangle if you glue the two short opposite edges after you give a half twist to the rectangle. Notice that the rectangle should be quite long and narrow if you really want to make a Moebius strip.
On these surfaces it is possible to trace closed non-intersecting curves that do not divide the surface into two regions! We expect to solve our original problem, and in fact this is the case, as you can see from our models (a torus and a Moebius strip)from the exhibition matemilano.

Let us go back to our problem. An alternative - here you see another map - ) makes you connect three black dots with three coloured dots via nine paths, which do not cross, on a map where you can go out from one edge of the rectangle and come back from the opposite edge following some rules.
This situation corresponds to ask the question on another surface, that is to say, on the surface which is obtained by gluing the edges following the rules which settle how to go back from an edge after you go out from the opposite edge - see the virtual leaflet fantamilano. The problem can be solved in the two cases above. In fact, they correspond to the torus and to the Moebius strip respectively.

This is the text of the poster you can see here.

One Geometry, lots of Geometries

When are two figures (for instance, two plane figures) the “same”? It seems an “innocent” question but you just need to give an answer to realize that it is not so easy. It is not difficult to imagine scenarios in which a different meaning to the word equal must be given – it seems a pun!

Anyone ordering an interlocking puzzle to a carpenter must know if the piece of wood (from which the parts will be cut out) is painted on one of the two faces or is the same on both sides. In the former case, the two figures above must be considered different; in the latter case they are equal.
There are scenarios in which what matters most is just the shape of the figure (the squares of a professor on the blackboard are never equal to those on the exercise books of his students, but nobody seems to realize it and everybody refers to the diagonal of a square….). There are scenarios in which two figures must be positioned in the same way with respect to a coordinate system (writing is a good example: a letter can not be upside down!)
None the less, the procedure is not completely random: the criterion of comparison can change, but not the “rules” underneath it.

In a sense, Felix Klein was the first in 1872 who pointed out a way of giving a good definition of “equality of figures”.
He did it in his dissertation as a new professor at the University of Erlangen; he had to present the new scientific program he would be inspired by in the following years. He managed to combine “the various research areas in Geometry, thus creating a system which made emerge an overview of numerous and new problems, which gave way to further developments”. A fundamental step in the history of modern Mathematics!

The key to arrange the results which crowded geometry books was to interpret Geometry as the investigation of the properties of the figures, which do not change under the action of a “transformation group”. Therefore, there is not just one geometry, but there are lots of them. To choose one of these geometries means to select the “allowed” transformations. Through them, we change a figure into another that is “equal” to the first one. In other words, we choose the “glasses” through which we look at the objects.

For the carpenter mentioned before, the “allowed” transformations are all the isometries if the two faces of the piece of wood are equal. They are the so-called direct isometries (rotations and translations or, in other words, those that do not switch right and left) if just one of the two faces is painted. As another example, in the case of writing, the glasses are those given only by translations.
It is not enough. Some pc programs allow one to take an image and to "stretch" it via two numbers: one is for horizontal "stretching" and the other one is for vertical "stretching". If the two numbers are equal, the shape of the two figures does not change. Even in common language, we say that the two figures are equal by similarity. In general, if the two numbers can also be different, the two figures are said to be equal by an affinity.

It is really difficult to say what is “equal” between two affine figures: even two different shadows (or two different photographs) of the same object (for instance, a circle and an ellipsis) can be equal in this sense.
Yet, our brain can do better. There are even more general methods of recognizing when two objects are equal. In a sense, they are very “natural” for our brain. They are connected to projectivities, which govern the geometry of vision: two different shadows of the same object (for instance a circle and an ellipse, but also a circle and a parabola) can be linked by a projectivity and our brain is well trained to “give substance to shadows” even without projective geometry.

And further… there are the glasses of topology: for them, a doughnut is “equal” to a coffee mug.

This is the text of the poster you can see here.

Different? Equal!

In our country, it has been long stated in a clearly way that the white, black, red or yellow kids in first grade – as well as the members of a university senate or of a board or directors – are all alike in their splendid (or sad) difference. People like us, who still believe in it, know well that understanding in which way we are different one from the other helps us to understand how and why we are all alike.

From Darwin’s theory to a shopping list, any process of abstraction is based on a classification activity. In early childhood, first examples of this activity are related to the build-up of language: when a child learns that… the name “glass” stands for his own glass with a purple cat on it, as well as for the less colored glass of any grown-up close to him.

Thus, it is not by chance that one of the first operations carried out by a mathematician (when he/she analyses a given scenario) is to classify the objects of his/her investigation in a way that is functional to the prefixed goal he/she has chosen. Then, the difference between two objects is not an intrinsic property, but a characteristic which depends on the parameters chosen to study them. Reality has different facets which heavily depend on the methods of investigation.
When it is successful, i.e., it yields a complete classification of the objects under investigation, this operation of “rearranging” gives an important simplification because it allows one to consider only the containers and not the objects therein contained. When it fails the aim and it does not give a complete classification, the work done to set suitable criteria for equality/difference is however important to make considerable progress in understanding the question under investigation. The history of mathematics is rich of successful tasks of classification as well as unsuccessful, albeit fruitful, classification trials!
The topological classification of varieties can show well this scenario: the case of surfaces (i.e., two-dimensional varieties) was completely solved relatively early (at the end of 1800). On the other hand, the investigation of the general case has presented enormous difficulty. At the beginning of this millennium, a thorny question in this setting has been recently solved when Grigori Perelman has proved the famous “Poincaré Conjecture”, which had been open since the early years of the previous century.
Henri Poincaré tried to classify three dimensional varieties via homology theory. He came up with the existence of a variety, which is different from the sphere from the viewpoint of Topology; however it had the same homology of the three-dimensional sphere. He tried to find out other methods of distinguishing the sphere from other varieties and in particular, he focused his attention on what is called the “fundamental group” in Mathematics. He did not find a proof for his choice being adequate, but he set an unsolved problem until Perelman published a series of results which, in particular, confirm Poincaré Conjecture.

In this exhibition, we describe other and diverse scenarios in which you must decide (arbitrarily but consistently!) the criteria of equality and difference in order to classify some “objects” (sometimes concrete, sometimes abstract). In these scenarios, you can imagine to “put on” a special type of “glasses” to decide in which container you can put a given object. Through these glasses, two apparently different objects may look alike.

May it be arithmetic for which 2+2 is not always 4 or geometries for which circumferences and ellipsis are equal. Or may it be surfaces that transform one into the other and stay astonishingly different as the ones below, or mosaics that, in spite of any appearance, come from the same machine. The exhibits of this exhibition show that the classification of objects requires an open mind and imagination.

Once again, Mathematics gives way to experiences of great freedom and creativity!