It's easy to say hole

A concept familiar to all becomes rather elusive when it needs to be defined: a straw, for example, how many holes does it have.

Asking a friend if the straw he's drinking from has a hole or two can be a great way to ruin his drink. Finding an answer that everyone agrees is not easy, and it can further complicate the debate. Does the glass that contains the Spritz technically have a hole? And how many holes do the taralli to accompany it have? And olives, do they have the same problem as straws? Ultimately, what is a hole really?

Our general knowledge of holes is rather incomplete and their definition has long kept philosophers, linguists and mathematicians busy. The word "hole" is used to mean quite different things, which usually have an opening of some kind in common: the keyhole, for example. In the philosophical field, on the other hand, there are some more complications, which derive from the difficulty in defining holes from the point of view of their existence.

Let's take one of the "taralli" from the aperitif: if we eat it entirely in one bite, have we also eaten its hole? The most logical answer seems to be yes, but what if we ate it gradually instead? In that case we would have broken the "tarallo", which would have lost its hole, and we would not have eaten it. This tells us that holes derive their existence and the very possibility of existing from their surroundings.

In a sense, and playing a little with words, holes can be called parasites: their existence depends entirely on the existence of something else. There cannot be a hole if there is not something enclosing it.

In everyday practice, things are simpler and everyone knows what a hole is when they hear about it. Engineers, who are quite practical types, distinguish between "blind hole" and "through hole": the former identifies an opening that only partially penetrates an object, while the latter a hole that passes completely through it. For these distinctions they usually prefer the term " hole ", basically a synonym for hole, but used above all to define something with regular margins and width: a hole in the wall made with a drill, for example.

Blind holes
A glass jar has a blind hole: it is the opening through which biscuits can be inserted and removed. Imagine being able to reshape it , as if it were made of plasticine, and change its shape - without removing material, adding or combining it - until it assumes that of a glass. We changed some of the characteristics of the object, but the blind hole remained: technically the glass has a hole, thanks to which we can fill it, empty it and drink.

Now let's imagine being able to shape the glass, widening it and reducing its height, to obtain a bowl. We are less inclined to think that a bowl has a hole, but if it was true for the glass, we can apply this definition also in this case. The bowl can then be molded into a deep plate, which would still have a blind hole as we understand it, and finally into a flat plate, which would have lost its opening instead. 

In this hypothetical experiment, in the transition from jar to glass to bowl to deep plate and finally to flat plate we never subtracted or added material, nor broke something or joined anything together (for example the edges of the opening). The material has always remained the same and has simply been remodeled: the blind holes can be removed without the need to close the opening that originates them, nor to weld the edges or to add other material.

Holes and topology Through
holes, on the other hand, are more complicated. The hole in a tarallo ready to be baked cannot be eliminated by reshaping it in the way we changed the shape of the glass jar, if not by crushing and welding together the dough that makes up the tarallo, or by adding more.

We can consider a tarallo as a close relative of the donut, which in turn is geometrically definable as a " toroidal " (empty inside). To obtain one, simply take a circumference and make it make a revolution around an axis external to it.

Defining a toroidal hole, and ultimately any through hole, requires some mental gymnastics, and among the most gymnastic in this area are mathematicians. Their training ground is the "topology", the part of geometry that deals with the study of the properties of mathematical objects, which do not change when they are deformed (as long as they do not create tears, overlaps and glues, as we have seen with the examples above ). This seamless modeling in topology is called “homeomorphism”.

In topology, a sphere and a cube are homeomorphic (i.e. equivalent) objects, because one can be deformed into the other and vice versa, without having to add material, glue or overlap it. On the other hand, a torus and a sphere are not homeomorphic, precisely because the torus has a hole that cannot be eliminated in any way with a simple deformation (no, closing the hole by bringing the parts together would not be a simple deformation).

These conditions explain the saying, to be honest, widespread almost exclusively among those who deal with these things, according to which "topologists do not distinguish a cup from a donut". The two objects are in fact homeomorphic: a donut can be obtained starting from a cup, simply by deforming the original object without gluing, creating tears or overlaps. The two objects are homeomorphic because they both have only one through hole (the blind hole of the cup, as we have seen before, can be eliminated).

For topologists, blind holes are not particularly interesting, since they can be eliminated, while through holes attract great interest, because they have unique characteristics that affect the way we can use geometric shapes.

How many holes
Returning to the aperitif, how many holes does a straw have? The question went viral on the Internet a few years ago, following a BuzzFeed article on the subject, which received a lot of attention in the United States. At the time, most people had replied that there were two holes, colloquially referring to the two openings in the straw.

In reality, a straw and a bull have only one hole. To realize this, just imagine modeling a bull by lengthening its shape, until you get that of a straw. The same holds true in reverse, imagining to reduce the height of the straw more and more, until you get a torus that will have a hole in its center.

In topological terms , a straw can be described as the product between a circumference S 1 and an interval I , which in turn can be defined as [0, L] (hence L defines the length of the straw). On the geometric plane, the circumference isolates a space that we can consider as a hole, because the only way to fill it would be by adding material or by welding / gluing some of its parts together. I , on the other hand, has no hole, and consequently the straw has only one hole.

Starting from these basic elements, which we have simplified a little while trying not to pierce the main concepts, not only can shapes and their transformations be mathematically described, but other important information on the properties of objects can also be derived. Homology, for example, allows the algebraic objects to be traced back to sequences of groups, which encode the quantity and type of holes present in each object. Taralli included.