#But with all the tunnel systems it is more likely to be an n-torus with n the number of holes and n being a VERY large number | Explore Tumblr Posts and Blogs

capybaraonabicycle · 1 month

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How many holes does a straw have?

@i-send-you-random-asks

(asking you specifically cause i think you'd have an interesting answer)

Ohhh, yes, this is my question! Thank you, dear!

Short answer:

That depends on your definition of 'hole'. Topology says 1.

Long answer:

Since this depends on your definition of hole, I can think of 5 answers that can be rationalised and make some flavour of sense:

(@marvellouspinecone helped me with some of these a while back and might have additional info, so I am going to credit her here.)

0 holes

You can define a hole as something that makes an object broken, or at least as something you have to put into a finished object AFTER construction. This could be something like a tear in the fabric or a hole you have drilled into the 'wall' of the straw. Ergo, a functioning straw does not have any holes. It looks exactly as it was designed to be.

1 hole

This is the math answer. As said in the infamous post, a straw is 'topologically equivalent' to a torus. To be precise, it is homotopic to a torus.

First question: What is a torus?

Answer: Basically a donut. It looks like this:

[ID: image of a torus. It looks like a donut with a checkered surface. end ID]

Second question: What does 'homotopic' mean?

Answer: This is where it gets math-y technical, but in a way it means that we can continuously transform either of the objects into the other - in a nice way.

Imagine, our straw was made of super-clay: we can't rip it or glue it together at any point, but we can pull and push it together however we like, even changing its density. So we could stretch some parts to become very big and shrink others a lot. We can also bend and twist it a little.

So, we take our straw and we push it together in the direction of its length until the very long straw becomes short like a ring. And then we pull on the 'walls' to make them nice and fat and round. Tada! We have made a donut!

(We can do this in the other direction, too, pull the torus (donut) out long and then make the walls thin - then we get a straw.)

The thing about such homotopies is, they preserve the number of holes an object has. Hence, the straw has exactly as many holes as the torus (donut)!

Third question: How many holes does a torus have?

Answer: In topology, we have something called the Euler characteristic. It is a number that gets assigned to surfaces based on their properties (you can calculate it via triangulation but let's not go there.) A sphere (ball) has Euler characteristic 2. Each hole in a surface lowers the Euler characteristic by 2. The torus (is an orientable surface and) has Euler characteristic 0, so it has one hole.

(If you'd like to have the more exact explanation, it is attaching handles to a surface that reduce the Euler characteristic by 2 and add a hole. And a torus is homotopic to a sphere with one handle attached.)

Thus, a straw has one hole.

2 holes

If we define a hole as an indentation in an object that allows us (or something else) to enter a certain distance into the object, a straw has two holes. One on the top and one on the bottom.

This definition actually makes sense, since we call holes we dig into the Earth 'holes'. In the mathematic sense, they aren't, they're indentations that can (with the super clay idea) be flattened out. But with these holes we don't care about whether it will lead somewhere or just have a floor somewhere at the bottom, you can go in, so it's a hole.

If we forget about the fact that the straw leads 'one hole into the other', so like, if we were very small (or the straw very big) and we would merely walk across the outside and look into the holes, we would find two holes on the straw, one on the bottom, one on the top. If we don't enter, we wouldn't even know they were connected.

With this definition you have to be a little bit careful about when you start calling something a hole. I would reckon there needs to be a certain percentage-relation between depth of hole vs circumference of entrance to hole before you call it such. And maybe also something about size and shape and sharpness of edge - like, you wouldn't call a valley a hole, probably? But like, the straw fulfils the requirements of this hole easily, and twice.

3 holes

Okay, this one is merely for fun and play, don't get mad at me. But, say we define a hole kinda like above, as an entrance to the inside of an object. And we further define hole as any way through an object. Then we end up with something I like to call a 'hole-interval' through the straw.

So, we have one hole (rim at the top) to get into the straw, one hole (the straw, basically) to get through the straw and a third whole (rim at the bottom) to get out of the straw.

This is nonsense, obviously, but I like it, because there is a very nice mathematical feeling to it, resembling a closed interval. A closed interval [a, b] is just one object, but it has three parts that are often regarded independently of the others: the open interval (a, b) in the middle and the edge points {a} and {b}. For example, if you were to test the continuity of a function, you would often regard these three cases separately. So, in a way, there is beauty in regarding the 'three holes' of the straw as separate as well.

Infinitely many holes

This one is kinda nonsense as well, but I like the implications. If we define a hole as any instance of an object that is part of a tunnel through the object - I am using the word 'tunnel' here because actually, that tunnel would be the one hole in this case but for the sake of the definition, it can't be - then a straw is an infinite number of holes, stacked on top of each other. It is important to notice here that a hole cannot possibly have any depth in this case, just like the top and bottom holes in the last case.

This leads to two likely interpretations:

A) We have a hole at any real number (if we consider the straw as an interval along its length again). Then the straw would be made from uncountably infinitely many holes - which I think is an awesome concept.

B) We have a hole at any rational number. This would only give us a countably infinite number of holes in the straw and since Q is dense in R (don't worry about what that means), it would LOOK like the whole straw is made of holes, when in reality most of the straw would actually NOT HAVE ANY holes in it. Now isn't that the best thing you have heard all day?

And the best part : By this definition, not only would any straw be made of infinitely many holes, but any object with a hole in it would have infinitely many holes in it. Remember, for this to make sense, we needed to have holes with 0 depth. But any hole in reality has some depth. Punch a hole into a piece of paper: BAM infinitely many holes stacked on top of each other! :D

What have we learnt?

The most likely answers are 1 hole or 2 holes, depending on whether you take a more mathematical or more language-oriented approach. I think those were the two opinions most vocal in the original post as well.

But if you want to have fun, you can come up with very nice concepts and definitions to count holes by that give you a range of correct answers. Just make sure to think of the implications :)

#Thank you for this question <3 #I think you can tell I had a blast #On a related note: me and my friends were talking about the euler characteristic of the Earth the other day #Bc in meteorology you often consider the Earth as a torus #But with all the tunnel systems it is more likely to be an n-torus with n the number of holes and n being a VERY large number #Instead of - what you would initially guess perhaps - a sphere #math

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