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When 2+2 is equal to 1

In common language, all things related to numbers are often used as models of unarguable certainty. After all, numbers are used to counting and by counting you should know that, for instance, 4 is different from 7 and 2+2, which is equal to 4, is different from 7… but it depends!

In everyday life, there are various situations in which it happens to “count” in a different way from what is usually done; however, nobody is less serene. Perhaps somebody would be less serene if he knew it is all about Math, in particular about Modular Arithmetic… but this is a well-kept secret.
Three kids (Annie, Joe and Charlie) are counting out to decide who is going to play first: they are reciting the rhyme “A-u-lì-u-lè-ka-tà-mu-sé-…”. In order, they are pointing one of them for each syllable they are saying until the rhyme ends and the last syllable… chooses the first to play. Assume none of them is cheating, which it may be if syllables overlap or if they swallow their words. We just need to know how many syllables there are in the rhyme so that we could guess which of the three kids is going to play first. This is a rhyme made up of 27 syllables. If Annie starts to recite it and they go on in alphabetical order, the last syllable chooses Charlie.
This is a very strange arithmetic. In fact, a rhyme with 27 syllables has the same effect as that with 3 syllables (or with 6, 9 or 99): it is an arithmetic for which 27 is equal to 3, 6, 9…
If the rhyme had 28 syllables, we would find out that 28 is equal to 25 and 22…, to 13 and 7 and 4 and even to 1. Thus, 2+2 is equal to 4, but also to 1 (or, if you prefer, to 25 or 28).
It is as if all multiples of 3 were equal to zero and we kept track of the remainder when we divide a number by 3. There are the numbers which are multiples of 3 (which correspond to Charlie), those with remainder one when divided by 3 (Annie) and those with remainder 2 when divided by 3 (Joe).
We do not need to go back to our childhood to discover such an arithmetic. Just think of the arithmetic of a clock, where the role of the number 3 in the previous example is taken by the number 12: we thus refer to an arithmetic modulo 12. We can also talk about the arithmetic of the the days of the week (modulo 7) or, more simply, that of the odd and the even numbers (modulo 2). They are different ways of counting in situations when something repeats periodically.
In general, very simple concepts are very deep too. We should not be too much surprised if, since childhood, we are familiar with modular arithmetic (even if we are not aware of it) which, at the same time, is the foundation of current theoretical research in Number Theory. It is also the foundation of some of the most recent applications, which progress in Cryptography has introduced in everyday life, from ATM cards to the e-commerce.

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

When a Doughnut is Equal to a Coffee Mug

If you ask “How many and which are the different surfaces in Topology?”, the answer is given by a beautiful result in 1800. It says that any compact, connected and oriented surface without boundary is “equal” – from the viewpoint of a topologist – to one of the surfaces in a list of “standard” surfaces.

That of the oriented surfaces is recalled by the figures above here and it includes a sphere, a doughnut (a torus in mathematical terms), a sort of double doughnut with two holes, one with three holes, and so on.
They are infinitely many, but in a sense very “neat”.
The number of holes, which characterizes these surfaces, is called the genus of the surface. It represents the maximum number of cuttings (along closed simple curves) without decomposing the resulting surface in two pieces.
To avoid illusions, we must say that the number of “holes”, which is clearly evident for the surfaces in standard form, is not so readable for surfaces not in standard form. For example, the classification theorem says that the tetrahedral surface below is the same as one of the doughnuts of the list. Which one? It is the one with three holes (and not four!), but it is not so evident to recognize it.

We also have an analogous list for non-oriented (compact, connected, without boundary) surfaces. We can also fit in these two lists the case of surfaces with boundary. For instance, a disc is nothing else but a sphere with a hole and a cylinder is a sphere with two holes… a Moebius strip is the projective plane with a hole in it. The projective plane is the first surface in the list of non-oriented surfaces (the Klein bottle is the second one).
Thus, the problem of classifying topological surfaces is completely solved, which does not happen so often in Topology! There is a series of standard forms and given a (compact and connected) surface you know that it is “equal” to one of the standard forms. You also know what must be determined to decide which of these forms it coincides with. In fact, you need to know if it is orientable. Next, you need to count the number of boundary components (if it is just one circumference, or if there are two or seven of them) and, finally, you need to compute the genus.
Now, you may try to recognize the surfaces of this section of the catalog!

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

When a Doughnut is Equal to a Coffee Mug

To agree on the meaning of the word “surface”, you just need to think of the “skins” of the objects (at least, almost any) of everyday’s life. Some of them are smooth like the skin of a ball. Others are sharp-cornered like that of a cube. All of them share the same characteristic. If you imagine to take into account a small piece of it and zoom in, you essentially obtain something which is similar to the plane, where two coordinates can identify the position of a point. We can distinguish scenarios like those of a sphere and a cube from those of a cylinder and a Moebius strip... In these last surfaces, there are points where the surface “ends” and has a boundary. If you zoomed on the points of this boundary, you would find something analogous to a half-plane and not to a plane.
A surface (with or without boundary) is however characterized by a local property: zooming in gives something similar to a plane or, at most, to a half-plane.
What can we say from a global point of view? Surfaces are not all alike not even within Topology, which seems indeed … so compliant that it allows itself to view as equal the surface of the sphere and that of the cube. For a topologist, all information related to measurements are meaningless. You can imagine to stretch an object as much as you want. The object will stay equal to the original one as long as it is made out of a sufficiently deformable material which prevents from breaking it. Cutting into pieces is not an allowed operation for it unless you do not glue the different parts by taking care of glueing the points which were unglued.
Thus, how many and which are the “different” surfaces with these rules of the game?
To begin with, to agree on the surfaces we are referring to, we need to say that they are connected (i.e., made up by one piece), compact (a more technical hypothesis that guarantees that the surface is bounded and that there are no holes like those made by a pin when removing a point), oriented (i.e., they do not contain a Moebius strip) and without boundary.
A sphere and a torus are oriented surfaces without boundary; clearly, they are different.
A disc and a cylinder are oriented surfaces with boundary. The boundary is a circumference in the first case and two circumferences in the second case.

A Moebius strip is a non-orientable surface with boundary (and the boundary is a circumference – of course, with a topologist’s eye!).
An example of a non-orientable surface without boundary is the Klein bottle. Actually, it must be imagined in a different guise than that in which it is usually depicted, or in a three-dimensional model. In fact, with the mind’s eyes you should disregard the self-intersecting curve. You should think of this curve as an “accident” due to the fact that the Klein bottle (which needs wider space) has been “constrained” to the three-dimensional world.

It is like when you draw a knot on a sheet of paper. In a projection, we see some crossings which do not exist in the in the “real” Knot. They are due to the restriction of drawing on a sheet of paper, whereas the knot needs more space. It needs three dimensions: two dimensions are not enough.

Analogously, the Klein bottle (like all other non-orientable surfaces without boundary) needs four dimensions. There is no way of representing it in three-dimensional world without self-intersecting curves.

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

A classical, still current, problem in Algebraic Geometry is the investigation of surfaces of a given degree with the maximum number of singular isolated points of a given type. For example, for quadrics the special singularities are called ordinary double points; their maximum number is one. For cubics, the maximum number is 4.
The model below here on the left, gives an example of a cubic with 4 double points.
For quartics, the maximum number of such singularities is 16. A quartic surface with 16 isolated double points is called Kummer surface and you can see an example of it in the model below here on the left.

Actually, the models below here are called Kummer surfaces too,although you see just 4 or 8 double points. These surfaces have other double points that we can not see in real space. They are double points that you can see in an “extended” space where the coordinates are complex numbers.

The story of the challenge to “rearrange” these surfaces is the fascinating story of a victory, but, most of all, it is a relevant example of what is meant by “making math”.
Guido Castelnuovo described it in this way:
“We had created, in an abstract sense of course, a large number of models of surfaces in our space or in higher space; and we had split these models, so to speak, between two display windows. One contained regular surfaces for which everything proceeded as it would be in the best of all possible worlds; analogy allowed the most salient properties of plane curves to be transferred to these. When, however, we tried to check these properties on the surfaces in the other display, that is on the irregular ones, our troubles began, and exceptions of all kind would crop up. With the afore mentioned procedure, which can be likened to the type used in experimental sciences, we managed to establish some distinctive characters between the two surface families. We tested these properties by constructing new models. If they passed the test, the last step of our investigation was to look for a logic argument.”

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

Where Polygons are Placed

It is not difficult in Mathematics (and in particular in some areas) to come across some apparently “harmless” questions, which are easy to formulate and tell, but with a really complicated solution. Some of these questions (for example, the 4-colour theorem or the Fermat’s theorem) are famous for laypeople too. Their history has shown – also to people who did not know it – that in Mathematics there are still lots of open problems: even the investigation of polygons includes some questions that have not been solved yet in spite of being very old.
What does it mean to classify polygons? It means to determine a “sound” criterion to divide them in “classes” so that it is possible to say that two polygons in the same class are of the same type (or they are equal, if you prefer) and that two polygons in different classes are not of the same type (or they are different). Clearly, there is not only one way of distinguishing different types of polygons. Obviously, the choice of these different ways depends on the problem we are studying.

A first criterion, a bit rough, is to put two polygons in the same class if they have the same number of vertices. Doing so, all triangles are “equal”, all quadrilaterals are “equal”, but a quadrilateral and a triangle are different. In this way, we do not take into account the convexity or non-convexity of a polygon.
Then, we can try to refine the previous criterion to have just one triangle (all triangles are “equal”), only two quadrilaterals (one of them is convex and the other is concave), but four different types of pentagons (the convex, that with only a concave angle, that with two consecutive concave angles and that with two non-consecutive concave angles).
A criterion, which can be used to come up with this classification, requires that two polygons belong to the same class if it possible to “drag” the vertices of the first polygon onto the vertices of the second one (or of its mirror image) without falling into a middle “pathological” situation in which three vertices are aligned. (The existence of such a “catastrophic” situation, which is necessary to send a convex quadrilateral in a concave one, shows exactly that these two types of polygons are different).

If you include the twisted polygons, the description is more interesting and articulate. It is known that there are (in addition to one type of triangle)
• 3 types of quadrilaterals (one of them is convex, one of them is concave and one of them is twisted)
• 11 types of pentagons (one of them is convex, three of them are concave, seven of them are twisted)
• 72 types of hexagons (one of them is convex, seven of them are concave and sixty-four are twisted).

It is not known if there exists a general formula that gives for any integer n the number of different types of n-gons according to this criterion.

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

The devil’s mischief

“Fifty years ago we used to say that if algebraic curves were God’s creation, then algebraic surfaces were the Devil’s mischief”.
With this sentence, Federigo Enriques gave an account in 1949 (in his work Algebraic surfaces) of the difficult problem that he had solved in 1914 together with Guido Castelnuovo: the classification (with respect to birational transformations) of algebraic surfaces, i.e., of surfaces which can be described via polynomial equations, like those depicted in this poster.
The images in this poster are the photographs of some plaster models of algebraic surfaces belonging to the Math Department “F. Enriques”, Università degli Studi di Milano. Clearly, the plaster model is a three-dimensional object. It is the “skin” that gives us the idea of a surface, which sometimes should be imagined unbounded.
Techniques and concepts applied to curves can not be tout court generalized to the investigation of surfaces. On the one hand, this explains Enriques’ sentence; on the other hand, it shows that analogy is not enough although it is an important instrument for a researcher.

The models in the photos above here are examples of surfaces given by degree two polynomials or, briefly, degree two surfaces (quadrics). Those of the photos below here are examples of degree three surfaces (cubics), and those of the leaflet Algebraic surfaces II are examples of degree four surfaces (quartics).

As you can see, two quadrics can already be very different: some of them are bounded (i.e., like the11269, the so called ellipsoid, are contained in a bounded region of three space), while others, like the 11268 or the 11295 are not; some of them are “made up by just one piece”, while others, like the11265 are not; Some of them are ruled, like the 11273 and the 11276 (and models like the 11349 and the 11295 show better this property) while others are not. Some of them are “smooth”, while others have singularities as the vertex of the yellow cone which you see in the 11349 together with the other smooth surface.

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!

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!

The virtual leaflets are the posters of the exhibition "Equal? Different!".

The exhibition explores the concept of equality (and accordingly of classification) in various areas of Mathematics. After an initial presentation we have the Erlangen programme, the projective geometry, the classification of algebraic surfaces (I and II), the classification of topological surfaces (I and II), the classification of mosaics and finally an odd problem of classification about plane poligons.