## Q: What Is a Hole? A: We’re Not Sure!

How many holes does a donut have? That’s pretty easy: one. What about a straw? Two (one at each end) or just one? (Isn’t a straw just an elongated donut?) Does a coffee mug have one hole or two? Does a bowl have a hole? If no, then what about a hole in the ground or a hole in a wall that doesn’t pass all the way through? Does a basketball have a hole? A Reddit user asked 1600 people how many holes were in various objects and the results are fantastically all over the place.

This is a trivial question, but it reveals something interesting about people’s perceptions. The dictionary definition of “hole” includes two main meanings for the purposes of this question: “an opening through something” and “a hollowed-out place”. Mathematics offers another possible meaning:

A hole in a mathematical object is a topological structure which prevents the object from being continuously shrunk to a point. When dealing with topological spaces, a disconnectivity is interpreted as a hole in the space. Examples of holes are things like the “donut hole” in the center of the torus, a domain removed from a plane, and the portion missing from Euclidean space after cutting a knot out from it.

But a hole isn’t clearly defined in math or topology. From What We Talk about When We Talk about Holes in Scientific American:

Here’s my short answer that is also the reason I’m not an algebraic topologist. If you can put it on a necklace, it has a one-dimensional hole. If you can fill it with toothpaste, it has a two-dimensional hole. For holes of higher dimensions, you’re on your own.

That answer isn’t very satisfying. Is there a better way to describe holes? I talked with some of my topologist friends and discovered two things: topologists don’t all agree on what a hole is, and it’s fun and interesting to think about different interpretations of a word whose mathematical definition isn’t completely settled. I think my larger conclusion, in the spirit of the season, is that holes are like Santa Claus: the true meaning is in your heart.

No wonder those poll results are all over the place. But at the same time, it’s interesting that many more people say that donuts have a hole than washers or rubber bands. I guess donut holes have better marketing? As for straws โ reason tells me they only have one hole but I know in my heart they have two. (via the whippet)

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