Fantamilano

Look at the map of Milan which dates back to the XVI century. It is more or less a decagon, where the perimeter is represented by the Spanish Walls and the vertices are the Sforzesco Castle and the nine entrances to the city. How can we represent Milan from this map in a "science fiction" way so that the city extends to all the surface of a torus, or a double torus, or a Moebius strip?

To understand how we will proceed (clearly arbitrarily!), it is useful to keep in mind the video games where the cursor moves out of the screen and moves back in on the opposite side of the screen at the same height. For these video games, it is as if the proposed problem on the screen were actually proposed on the surface which is obtained by gluing the two edges where the cursor goes out and goes back in. The glueing pattern is given by the way in which the cursor goes back in.
For example, having a cursor that goes out on the right and goes back in on the left at the same height is equivalent to moving on the surface of a cylinder; Having a cursor that goes out on the right and goes back in on the left not at the same height, but "upside down" is equivalent to moving on the surface of a Moebius strip. The Moebius strip is the surface that can be obtained from a rectangle by glueing two opposite sides after giving to the rectangle a half-twist. Here "upside down" means that when the cursor goes out of the screen up on the left, it goes back in down on the right and vice versa.
If, finally, the cursor can not only go out on the left and go back in on the right (at the same height), but also go out from the top of the screen and go back in from the bottom of it (at the same distance from the vertical left side), it is as if we were on a torus.

For a cylinderand a Moebius strip the operations above can be concretely achieved with a sheet of paper (as long as it is long and narrow enough for the Moebius strip), for a torus paper is not enough. Paper is rigid
and it is possible to glue two sides of a rectangle so to make a cylinder. On the other hand, the cylinder can not be "bent" to glue the two boundary components. To make it, we need to imagine that our material is not paper but a sheet of elastic and deformable material. We can also trust virtual virtuale, animations or our imagination!

Let us go back to the map of Milan , and try to draw it on a Moebius strip, that is to say, describe how we made the model of the Moebius strip as an an exhibit of the exhibition matemilano. It is not a long and narrow rectangle whose opposite sides are to be glued. This is not a problem at all! From a topological point of view, we are allowed to perform any deformation we want. Therefore, we must decide which parts of the Spanish Walls should be identified. These are the parts which will not appear on the boundary of the strip. In the exhibit, the glued sides are those which start from the Castle and which correspond - in a modern map - to via Boccaccio on one side and to via Legnano/ bastioni di Porta Volta on the other side - but this has been completely random!

You can see what we did in the following pictures. In the first one we stretched the map. In the second one (which is somehow redundant) we cut and sewed the first picture following the identification scheme. Thus, we obtained a long and narrow rectangle and, from it, a Moebius strip. Here you see the final outcome.

There are other things to do. For instance, you can obtain a double torus as well. You just (!) need to start from the decagon given by the map of Milan and identify the edges like in this picture. It is not clear at all that we obtain a double torus: the sequence of pictures obtained following the links (starting from this one) is a nice "imagination exercise".