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.

The borromean rings are the symbol of the famous Borromeo family (and of various others). They can be found around Milan and in Lombardy in various places, on effigies and statues, on doors and fountains. What do knots and links have to do with these rings?

Let us take three strings and weave them into the braid is depicted here. If you join the ends of each string, you will obtain the borromean rings, which provide an example of a link with three components.

These rings have a very particular property, which conceals a symbolic meaning. If you remove one of the three rings, the other two can be separated. All three together, however, cannot be separated.
This does not happen for any "copy" of the Borromean rings, that is, for any link with three components with the same "shadow" as that of the Borromean rings. In fact, the link obtained depends on the under/over choices you make for each crossing. To tell the truth, while you indeed sometimes find "real" Borromean rings around Milan, you can also find some false rings, which appear to be the same, but have some inverted crossings. In this poster from the exhibition matemilano you find all five possible combinations, the Borromean rings included.
One of these combination has the following property: if you remove one of the three rings, the other two remain linked (they form a chain of two rings).
This link belongs to the family of links that can be drawn on the surface of a torus. More precisely, it is a torus link (3,3): see the information site Two Numbers for a knot.

Another "imitator" can be manipulated into a chain with three rings. If you remove the red ring, the other two are free. If you remove the yellow or the green ring, the other two remain linked (they form a chain of two rings). With this link it is possible simply to remove one of the three rings, while the other two remain linked.
Finally, this is the unlink with three components: each component is not linked with the other two.

There are no other "imitators" of the borromean rings. Start from the diagram which represents their shadow (with the three intersecting circles). If you decide which branch overcrosses or undercrosses at each of the six crossings, you come up with one of the six links above. This does not mean that there are no other diagrams besides the ones in this page. There are six crossings. For each crossing you have an under/over choice; so you have 2⁶=64 possibilities. Because of symmetry arguments, it is easy to see that the diagrams are just 10. Among them, some give the same link with three components. The five cases described above give all the possibilities.

The real borromean rings have another property. Their "imitators" are very flexible: one of them (the unlink with three components), is so "flexible" that each ring can be moved as freely as you wish. The borromean rings, in contrast, are very "rigid". In particular, they cannot be constructed with plane circles. This was known to the artisans who reproduced them on bas-reliefs and in cast iron. For example, you can make a rigid model with three plane ellipses that are not circles. If you make a model of them with (flexible) rings, you can form each of them (one at a time) into a circular shape. If you try to do this by laying all three together (by "flattening" them in order to get the diagrams on the poster), you will find that it is impossible: the three rings cannot be laid flat (and they are not circles). It is also interesting to observe that the Borromean rings can be represented in various "symmetric" ways: for instance, they can be made with three plane curves in three mutually perpendicular planes.

In these pages there are several pictures, animations and information on hypercubes. One might think that hypercubes are the only objects in four-dimensional space.....: it is not true! It is only the simplest object and, therefore, it is easier to describe it. In four-dimensional space there is a "zoo" of polytopes which is richer than the "zoo" in three-dimensional space.

To begin with, there are six regular polytopes, which are the analogue of the five regular polyhedra in three-dimensional space: one of them is the hypercube.
There are other possibilities as well. One can define and investigate various families and objects in the 4D world as in the 3D world exist polyhedra , prisms and pyramids, uniform polyhedra and star polyhedra.

Here you find one of the animations on a regular star polytope. Precisely, it is the analogue of the great icosahedron . In the animation this polytope goes through a 3D world. We see a section of it as if an inhabitant of flatland saw all the different polygons, which take shape in his/her flat world, when a star icosahedron goes through it.

It is not surprising that one sees several “detached” polyhedra at the beginning. They look like the great star icosahedron. If we imagine taking a section of the great star icosahedron with a plane which is orthogonal to the direction vertex-vertex, one would see a regular star with five points. If we cut it with a plane which is orthogonal to the direction face-face, one would see three "detached" stars at the beginning.

They would not be regular stars but they would appear a bit deformed.

What else can we find in four-dimensional world? Can we find the analogue of surfaces? Spheres, for instance? Of course, we can. Furthermore, the analogue of surfaces yields a variety of shapes, which is richer than the relative monotony of surfaces.
Let us talk here about the sphere. The hypersphere in four-dimensional world can be defined like the sphere in 3D (and the circle on the plane), that is to say, like the set of points with the same distance from a given point, which is the center of the hypersphere. How can we imagine it? In a sense, our world is a hypersphere as long as we "close it up" with a point at infinity.

To explain this claim, suppose you put a sphere on a plane and next project the sphere on the plane from the North pole (the antipodal point with respect to the tangency point betweeen sphere and plane). We obtain a bijective correspondence - the so-called stereographic projection - between the whole plane and the sphere except the North Pole. Analogously, there exists a correspondence between the 3D space and a hypersphere with a point removed - we still call it stereographic projection.

We can project onto the 3D space (and so "see") some objects contained in a hypersphere. This is the principle on which the following animations are based. In this one, imagine to project the 120-cell on the circumscribed hypersphere (as you can “enflate” a dodecahedron on the circumscribed sphere . Via the stereographic projection, one can pass from a hypersphere to 3D space and imagine to "get into" a 120-cell.

Of course, the faces are not regular dodecahedra, just as the faces of the dodecahedron, which is here drawn, are not exactly pentagons; these are the drawbacks of projections....!

If we want to make a regular polyhedron, we start up with regular polygons that are all alike. We assemble them around a vertex so that there is a gap, in the plane. In other words, the sum of the angles around the vertex is less than 360°.

It is exactly this gap that allows closing up the polygons in three-dimensional space; so we obtain a convex polyhedron. If the sum of the angles were equal to 360°, we would obtain a plane tasselation).

That's the reason why the regular polyhedra can not be more than five. Indeed, the possible combinations are five: 3 by 3 pentagons, 3 by 3 squares, 3 by 3 triangles 4 by 4 triangles or 5 by 5 triangles.
Clearly, in order to be sure that the regular polyhedra are exactly five, we should prove that each combination around a vertex can be carried out with the same pattern so to obtain a polyhedron.
Let us try to mimic this argument in higher dimension, i.e., let us try to make a polytope in four-dimensional space. We can start to choose a regular polyhedron and then try to assemble various copies of this polyhedron, all alike, around an edge. As before, we need a gap (this time in three-dimensional space). In other words, the sum of the dihedral angles at an edge must be less than 360°; As for the polygons that come up in a polyhedron, this gap will be responsible for closing up the polyhedra in four-dimensional space; so we will get a convex polytope. If the sum of the dihedral angles were equal to 360°, we would obtain a space tessellation. For example, think of four cubes around an edge.

Also in this case there are not too many possibilities. If we start with regular tetrahedra, we can assemble them 3 by 3 or 4 by 4, or 5 by 5, and not more. In fact, the dihedral angle of a tetrahedron is greater than 60° and smaller than 72°.

If we start with cubes, we can assemble them 3 by 3 and not more because the dihedral angle of a cube is 90 degrees. If we use four cubes, we will tessellate the space.
If we start with regular octahedra, we can assemble them only 3 by 3 because the dihedral angle of a regular octahedron is greater than 90°(and smaller than 120°).
If we start with regular dodecahedra, we can assemble them only 3 by 3 because the dihedral angle of a regular dodecahedron is greater than 90° (and smaller than 120°).

If we start with regular icosahedra, we can not assemble them even three by three because the dihedral angle of a regular icosahedron is greater than 120 degrees. There are at most six possibilities. In fact, one can prove that each of these combinations can be repeated with the same pattern so to obtain (in four dimensional space) a convex polytope:

• the hypertetrahedron which is made up of 5 tetrahedra (three at each edge);
• the hyperoctahedron which is made up of 16 tetrahedra (four at each edge);
• the 600-cell which is made up of 600 tetrahedra (five at each edge);
• the hypercube which is made up of 8 cubes (three at each edge);
• the 24-cell which is made up of 24 octahedra (three at each edge);
• the 120-cell which is made up of 120 dodecahedra (three at each edge).