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.”

Some virtual leaflets ("The nets of a hypercube" and "How to split a hypercube in two doughnuts") go over the structure of a hypercube.

In others you may find some different shapes of four dimensional world. In "Why only six?" there are the analogue of the regular polyhedra and in "Not only hypercubes" you may find the analogue of the sphere and of the star polyhedra. Finally, in "Beyond four dimensions" you may see other constructions which can be repeated in higher dimensional space.

As we can imagine a hypercube or other objects in four-dimensional space, could we think of 5, 6, or more, dimensions? The answer is positive and it is not even more difficult conceptually: at the most it is our imagination to be challenged. For instance, we can think of the n-dimensional space as the set of n-tuples of real numbers. As for the plane, we can introduce a coordinate system and identify a point on the plane with a pair of real numbers; in three-dimensional space any point is identified with a triple of numbers. Therefore, an n-dimensional space has the right to exist for any n. Of course, this does not help much to imagine objects in this space.

Let us go back to a hypercube. We obtain a hypercube from a cube with the same abstract procedure as the one we use in order to obtain a cube from a square. This will be called the prism construction. According to it, we can start from a point (dim. 0) and imagine to obtain a segment (dim. 1); next a square (dim. 2), next a cube (dim. 3) and finally a hypercube (dim. 4).
There is no reason why we should stop: the prism over a hypercube of dimension four will be a hypercube of dimension 5 and so on.
Also using coordinates, we can center our objects in the origin and arrange them so that the edges are parallel to the coordinate axes. Then a square (with edge two) has four vertices (±1,±1); The vertices of a cube are the eight points (±1,±1,±1); The vertices of a hypercube are the 16 vertices (±1,±1,±1,±1). So forth, we will obtain that the vertices of a hypercube in n-dimensional space are the 2n points (±1,±1,...,±1,±1).

There are other constructions which can be repeated infinitely many times, for instance, the pyramid construction. It can be applied to obtain a tetrahedron from a triangle. It suffices to join a point not on the plane where the triangle lies with all the vertices of the triangle. If you start with an equilateral triangle you will get a regular tetrahedron if the following conditions are satisfied: i) the point is on the line perpendicular to the plane where the triangle lies and passing through the barycenter, and ii) the height is taken in such a way that all faces are equilateral triangles.
Coordinatewise, the easiest way to visualize this construction is to start from a triangle on the plane x+y+z=1 in three-dimensional space. The vertices are the points (1,0,0), (0,1,0), (0,0,1).
In (n+1)-dimensional space, the n+1 points (1,0,...,0),...,(0,...0,1) are vertices of a hypertetrahedron in dimension n.

Another construction, that we will call the bipyramid, allows one to get a regular octahedron from a square and, at the next step, a hyperoctahedron in four dimensional space from an octahedron.

Coordinatewise, the four points (±1,0) e (0,±1) are vertices of a square. The six points (±1,0,0), (0,±1,0), (0,0,±1) are vertices of a regular octahedron. In general, the 2n points (±1,0,...,0),..., (0,...,0,±1) are vertices of a hyperoctahedron in n-dimensional space.

It is interesting to note that in dimension greater than or equal to 5 one can prove that these are the only regular polytopes. The 120-cell, the 600-cell and the 24-cell are specific “accidents” of dimension four. Analogously the icosahedron and the dodecahedron are specific “accidents” of three-dimensional space.