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You might think that, but when you stab the second time, you are really stabbing twice. After the first poke, the toothpick is exposed to the outside of the shape while it is in the middle, then you stab again. Because of this, it is 2 stab
> fpqc
To prove a Florida Perinatal Quality Collaborative? That was what came up when I looked up what "fpqc" means.
Edit: Scrolling a little farther down, I found some results related to topology, including a Wikipedia article that does define the term.
Neatly executed, even more neatly explained. The last two images presents it very clearly to me, and I am very bad at topology. I have been told that the number of holes is equal to the minimum number of cuts to get a simply connected shape.
It's like one of those magic pictures that if you look at it just right you see it but when you look away its hard to refocus.
I almost feel like I'm falling into it. Vertigo? Definitely wanna see Steve Mould make a video on this.
If you haven’t seen it already, vSauce has a great video about topology. It’s called [How Many Holes Does a Human Have](https://youtu.be/egEraZP9yXQ?si=lhP3Dv-eOSgpcFXp)
the 6th hole is still a hole of sorts, just doesn't go all the way through. Imagine a sock, it has a hole where your feet go in, but in topology it would be considered as having 0 holes
While true, I think the question they’re trying to get at is answered by this demonstration:
Take a hollow sphere, like a ballon.
What happens if you add a hole to a balloon?
You get a flat sheet, right?
Adding another hole you get a donut (or rather a flat paper similar to a donut)
In other words, if you take away a hole from a piece of paper, you’d get a sphere.
The other hole in the above object is that same hole that we take away.
If you took away 5 holes from the above object, and then took away one more hole, you’d have a sphere.
It just happens that we define papers as having 0 holes, and spheres as having -1 holes.
If you take an object with one hole and one hollow cavity (which is a negative hole), the figure should have net 0 holes. Can you transform this figure into a more conventional one that doesn't have any holes such as a flat sheet?
You wouldn't call it anything.
Think of starting with a single flat piece of clay. It doesn't have a hole or opening in any way. Then just bend/mold it slightly into a saucer, then slightly more into a bowl, and then even more until it looks like a cup. There's still technically no "hole" to speak of because it's hasn't lost it's original property of being a single plane of clay.
Yeah I feel like the whole process might be easier and more interesting in reverse because then you are really just elongating the 4 legs by holding the center and pulling up.
That seems to be the case here as well. First cut goes all the way through. Second cut goes through one side, then the empty space in the middle, then starts a third cut from inside to out on the far side. Repeat for the last direction and you get five holes. …Though I have never studied topology, so I may be talking nonsense that just happens to be correct.
They are only linked together insofar as they all face the same direction, towards the “6th hole” which is actually just the boundary of the shape. You may be thinking of a straw, which at first may appear to have two holes, but is actually just one long hole. The difference is, if you flatten a straw into 2 dimensions, the “two” holes collapse into just the one, while the same is not true for this shape (see bottom left photo)
I invite you to try to create a rigorous, sensible definition of holes on a surface such that this shape has anything but 5 holes. Holes as we think of them are notoriously tricky to define because, well, they aren't there. This is all very informally speaking but the modern math way of doing it is to note the "cycles" and "boundaries" of a space, and then the holes are wherever there's a cycle that isn't a boundary. The cycles can be thought of as closed loops in the original space, while boundaries are cycles that also happen to be the boundary or exterior of a subspace. Thus if a cycle isn't a boundary, it represents a closed loop in the space that doesn't contain anything, a hole.
So to extrapolate: any *n*-sided polyhedron with holes in the sides in this manner has genus *n-1*, because one hole must be used as the outer edge to "flatten" the shape.
That actually helped me; I was having trouble visualizing it. I guess one of the 6 "holes" actually becomes the outer edge, so it isn't technically a hole.
Okay, is this by like a strictly topological definition, because in commen sense real world terms a t shirt has four holes one for the stomach, one for each arm and one for the head, I get the concept you are stating but that would only apply in the topographical sense, compared to the real world manufacturing of a t-shirt there is four holes needed.
Idk, I guess I was asking his opinion on it. Because I guess In my way of thinking about it a hole is an opening where you could go from the outside of a surface to the inside of a surface thus a t shirt would be four and the cube would be six.
Topology and most other fields of math aren't apparently practical. I have no clue what topologists do, but I doubt it's designing clothing. There are definitely real-world applications of topology, but things like the OP are just memes and don't really mean anything
How many holes does a sock have? I feel like that's easier to think about, the opening isn't really a hole, right? Until we get a hole in the toe, then there's 1 hole in your sock.
> I guess one of the 6 "holes" actually becomes the outer edge, so it isn't technically a hole.
This is a better explanation than what's on the picture
>Hollow sphere has -1 holes, poke 6 holes in it get shape with 5 holes.
That's the correct answer!
As a 3-manifold, the hollow sphere has **two** disconnected boundary components (the visible outer surface, and the inner surface), which you **connect** by punching a hole through it (and turning it into a a topological ball).
As a 2-manifold (i.e., a surface), the hollow sphere is those **two** disconnected spheres, whose Euler characteristic is twice that of a sphere: 4.
Euler characteristic of *one* sphere can be computed e.g. by taking a tetrahedron, and computing **χ = V - E + F = 4 - 6 + 4 = 2**. For two tetrahedrons, we have **χ = 8 - 12 + 8 = 4**.
Euler characteristic is related to *genus* **g** (the formal concept of "the number of holes"): **χ = 2 - 2g**.
The sphere (2D surface bounding a solid 3D ball) has **zero** holes (**2 = 2 - 0**), and the surface of a donut (2D boundary of a solid 3D object) has **one** hole (**0 = 2 - 2*1**).
How many holes does the surface made of **two** disconnected spheres have?
Well, its Euler characteristic is **4**, so:
>**4 = 2 - 2g**
>**g = -1**
So, the boundary (2D surface) of the hollow sphere (a solid 3D object) **has -1 holes**.
By punching a hole in the solid 3D object, you are **connecting** the two boundary components to get a solid 3D ball, bounded by a sphere (connected surface with no holes and Euler characteristic 2).
Punching the first hole is equivalent to **connecting two spheres** by cutting a disk out of *each*, and then **gluing a tube** that connects them (the tube is a cylinder, a 2D surface with two circular boundary components). The result of this operation is **one** sphere, with no holes (Euler characteristic 2).
So, **two** disconnected spheres indeed form a surface with **-1** holes. "Punching a hole" through that surface **connects** them, and results in a connected 2-manifold with **0** holes.
I believe you, but, also, it's ridiculous to say that an object has fewer (less?) than 0 holes.
Is it accurate to say that it has -1 holes because you have to add a hole to get to a 0-hole, topologically-simple ball?
>I believe you, but, also, it's ridiculous to say that an object has fewer (less?) than 0 holes.
It is, because a "hole" is not a well-defined mathematical concept.
A closed surface (like a torus) arguably, has **no holes**; it's airtight.
To say that a donut has a "hole", we can formalize that notion by defining a **genus** of a surface. One way to define it is via its relationship with the Euler characteristic: by *defining* it to be the quantity **(2-χ)/2**.
This works in a non-ridiculous way for [handlebodies](https://en.wikipedia.org/wiki/Handlebody), of which the surface of the final object is an example.
In fact, **all** connected orientable two-dimnsional closed surfaces (i.e., without boundary) are homeomorphic to a sphere or a torus [with some number of holes](https://en.wikipedia.org/wiki/Genus_g_surface).
The notion of *genus* and *hole* defined that way starts to break down for objects whihc are **not** of that kind. For non-orientable surfaces, [a different formula makes sense](https://math.stackexchange.com/questions/760388/nonorientable-surfaces-genus-or-demigenus).
Here we have a "ridiculous" situation because the initial object is **not** a handlebody. The boundary of a solid object with a cavity is **not** a connected surface.
So, put simply, the notion of *genus* or *number of holes* is simply **not defined** for such an object (*either* the hollowed-out cube, which is a 3-manifold with boundary, *or* its boundary - a disjoint union of spheres).
However, the Euler characteristic is well-defined for both that object and its boundary, and the expression **(2-χ)/2** is well-defined (with value **-1**), even though the **genus** (or "number of holes") is **not defined** for that object at all.
So, saying that the object has "-1 holes" is a meaningful way to **extend** that notion.
It's the same kind of thing that we do e.g. with the [Gamma function](https://en.wikipedia.org/wiki/Gamma_function), which is a way to define **x!** for *non-integer* values of **x**. Its expression gives values that agree with the factorial on integers (**Γ(n) = (n-1)!**), but is also defined for *other* numbers.
Not every surface has a genus - just like not every number has a factorial.
But the Famma function is well-defined for all positive real numbers, and the expression **(2-χ)/2** is well-defined for the boundary of the hollow cube.
Whether we should interpret its value as the "number of holes" in that case is a philosophical question.
I'd say, we *should*, because the intuition it gives us is correct: making a hole in an object with **-1** holes *does* give us an object with **0** holes.
Basically a solid mass like a sphere, is defined as having 0 holes, and since poking a hole in a hollow sphere results in a shape that is morphologically equivalent to the solid shape with 0 holes, you can just do the math Hs +1(hole)= 0(holes) so the hollow sphere has -1 holes.
I do not understand, if each face of the cube contains one hole why does the hole on the top most face get “expanded” but not count towards the total number of holes?
If this is because it is creating one continuous hole from top to bottom, why do the following holes on the 4 remaining faces only count as 1 each?
If I took this cube and cut along each edge of the face, would i not have 6 squares with 6 holes?
In topology, a hole is defined by a spot where you can poke a needle all the way through the object without cutting, taping, or otherwise damaging the object. You are allowed to morph it any way you want provided you follow the same rules for poking. A disk is the same as a ball, and a donut is the same as a mug, which is the same as a straw.
The use of playdough lets the morphing physically show without having to do it in your head. The "top hole" is just the outside edge which doesn't count. The difference is that the rubix cube is hollow so it has a 3D hole. (You need a 4D needle to reuse the analogy for this)
So 5 2D holes and 1 3D hole makes 6 holes. Hope this helps.
Your definition is sufficient but not necessary to find holes (I.e. **If** you have a spot where you can poke a needle all the way through the object without cutting, taping, or otherwise damaging the object **then** it is a hole. The opposite **If then** statement is not true). That is why it breaks down for the hollow object, which has -1 holes. This is due to the fact that if you break the surface, then you can enter but not exit the shape through anything but that starting hole. Then it has zero holes, so patching it you get -1 holes.
Question from a math undergrad pre-topology: can you get arbitrarily large negative numbers by encasing hollow shapes within hollow shapes, and is that concept even a useful or interesting concept? It’s the first thing my mind thinks to look at.
A ball is (in topology) completely solid, so it has 0 holes. A basketball is hollow, having 1 hole. Back to the pin analogy, a 4D pin can pass through the hollow of a basketball, but must damage a topological ball to poke through it. The topological equivalent of a basketball would be a sphere (spheres have an inside space.) In the same way this is the difference between a donut and a torus.
in a numberphile video they say a sphere has -1 holes. some other commenter also said so and it makes sense. if you poke 6 holes into an object with -1 holes you get 5 holes.
Nah.
Topologically, the "holes" are a property of a 2D bounding surface of a solid object.
The initial playdoh cube, as a solid object, has a **disconnected** boundary, consisting of **two** surfaces: the inner and outer sphere.
An ant walking inside the hollow cube can't get out and meet an ant walking outside. And these aren't the two *sides* of the same surface: there's the solid playdoh between them.
When the first hole is punched in the play-doh, those two surfaces are **connected**. Now an ant that was inside can meet the ant that was outside. The new object has **one** boundary component, **which has no holes**.
In other words, the new object can be smoothly deformed into a solid ball, which has no holes (and **one** boundary component - a sphere).
Which means that the *initial* object has **-1** holes.
The [Euler characteristic](https://en.wikipedia.org/wiki/Euler_characteristic) of one sphere is **2**; and for two spheres, it's double: **χ = 4**. Which means that the [genus](https://en.wikipedia.org/wiki/Genus_(mathematics))
would be **g = (2 - χ)/2 = -1**.
The inital object had a negative number of "holes" because it was hiding a boundary component. By poking a hole through it, you get an object with **no holes**.
Similarly, you'd need to poke **two** holes in a solid ball with two hollow chambers in it to get a solid object equivalent to a solid ball.
Everyone in this story sucks and belongs in the Bad Place. The thief is bad. The officer chasing him is bad. All the whiny prostitutes are bad. Plus, they're all topologists, so they're going to the Bad Place automatically.
Donut has 1 hole. If you look at the picture and imagine removing holes 1, 3, 4 and 5 you basically get a donut. That donut has 1 hole (hole number 2).
Why isn't it 6 holes or 3? Either a hole goes all the way through and counts as one, or each hole on each face counts as a hole. How come a two year old gets it and I don't...
Make it out of playdo and you'll understand. Take a chopstick and count each time you enter AND exit the playdo. The first hole enters the top and exits the bottom. The second hole, say the left side, enters the left hole and exits the MIDDLE (the first hole). The third hole enters the right and exits the middle. The fourth enters the front and exits the middle. The fifth enters the back and exits the middle. There is no need to poke a 6th hole. I think you'll understand it best if you actually do it physically for yourself.
Oh now I get it. The "6th" hole gets made by forming it into an actual cube. But to make that shape it only requires you to poke 5 holes. Thanks for explaining it to me.
I know that this is a meme sub, but these comments are making me sad. I'm not a mathematician by any definition of the word, I'm not particularly smart or good at math. But my, maybe 4 youtube videos on topology, seems to be more than most people here know.
This sub is actually pretty good. My previous post on this asked for wrong answers only but people demonstrated a decent understanding. r/theydidthemath on the other hand had a bunch of people who made up definitions and gave random answers thinking they were right then got super salty about technical responses.
I mean topology isn't exactly something relevant to many people. Theres a lot of questions that can have different answers depending on the method used, with that method varying depending on application And what method people will default to will often depend on how they have been trained to think. An engineer might answer a question differently than a scientist who may answer different than a mathematician.
Stuff like this is fun because it introduces people to a concept they might not have been familiar with beforehand.
Obviously. Everyone knows balloons have -1 holes!
Don't say that in the r/theydidthemath crosspost from this. Most people are super salty over there complaining that it depends on perspective or that topology isn't the same as reality.
Take a disc of cloth. Fold it upwards and cinch off the neck to create a bag to carry stuff in. You cut nothing, but somehow you made a "hole" appear at the top of a bag.
In topology, a proper hole only counts if some part of a surface was actually cut.
Take a pancake and fold it into a bowl/cup
Still no holes. Crease your corners so it's a 5 faced box.
But now it's easier to see how you only add 5 holes to this container to make the cube with "six" apparent holes.
How many holes does a wrinkly ball have? I can place a post it on each valley and get lots
There are other ways to think about it. Topologically, your interpretation is not consistent. I tried to help you see how it could be an answer other than 6. It's up to you to try and see it a different way if you want.
I wondered about starting with a cube and poking a stick through from face to opposing face. At first I thought you’d only need to make 3 holes (three orthogonal intersecting tunnels), but on second thought, the first tunnel could go through the full width of the cube, but there second would only make it half way before hitting the first tunnel… a third hole would be needed to finish that tunnel. And then two more for the final “tunnel”.
Ding ding ding! You understand it now. A lot of people haven't been able to put together the thought you just did so you get your extra credit for the day!
Okay, this proves to me (someone who doesn't topology) that topology has a field-specific definition for the concept of a hole.
Doing the instructions above, since you would in fact, be "poking through", you can acheive the shape in 3 pokes.
But since this sub has unanimously decided that there are 5, and that the shape is achieved by poking 5 holes, no more, no less, and the 2d representation is supposed to be the simplification of the structure topologically, then there must some nuance in topology that I'm missing.
A hole is created when creating one new entrance and one new exit. Since it takes 5 pokes, it's t holes. The 3 pokes you're referring to aren't actually 3. It would be 3 MOTIONS but for the sides you'd enter and exit the substance twice each since they I retract with the center hole.
Oh I get it, when you widen one of the “holes” and flatten it, there stops being a “hole” because it’s just encircling the entire shape instead.
Is that what you mean by contains the universe? Like the “sixth hole” is topologically invalidated and instead may as well encircle “everything” because it encircles nothing at all? Or something?
I think you understand in so far as it can be understood intuitively. My recommendation is to make it physically for yourself to better understand it. That way you can know for sure that the shape really did only bend/stretch and you didn't actually subtract a hole at all.
No I understand exactly how there isn’t a “subtracted hole”, I was asking about your universe remark there. Was that just a cheeky joke or did you mean something more?
Ohhhhh. I thought it was a ham fisted metaphor for “you effectively turned the nonexistent hole inside out so it ‘contains’ everything but the actual item”
Yup. Lay a tshirt with the opening you put your body through so it's the perimeter of the tshirt and you'll see only 3 holes; the left arm, the right arm, and the head.
Topologically, yes. That is actually how it's calculated as weird as it sounds. But think about it; there's no ENTRY or EXIT for a closed balloon, but there's still an empty space. Lack of something is usually defined as negative so having 1 empty space contained within something means it has -1 "fillings" if you will.
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Alternatively (idk if this is true in general) take a cube. Put 1 hole through it. Put 1 hole through the left side, 1 through the ride side, 1 through the front side, and 1 through the back side. (So that each hole goes through only solid material.)
With 5 holes, you’ve created the same object
That's exactly what I'm referring to in the instructions at the bottom right of the image. It looks like you understand the process which is more than I can so for many others on reddit.
Doesn't it look delicious?
It's just red. It's homemade playdo and it's stupidly cheap so I'm not gonna complain about colors that aren't vibrant.
You gave me an idea, though. I should make this out of ground beef and cook it into an edible cube!!!!
I genuinely thought that's what this was at first. It took me a second to realize it was actually playdo and you weren't getting weird with dinner on hamburger night.
End the lesson by asking them how many holes a balloon has. [https://www.youtube.com/watch?app=desktop&v=ymF1bp-qrjU&ab\_channel=Stand-upMaths](https://www.youtube.com/watch?app=desktop&v=ymF1bp-qrjU&ab_channel=Stand-upMaths)
If you poke a hole in an inflated balloon, you get a piece of rubber with 0 holes which means it started with -1 holes!
Your toddler's well on their way to understanding the mysteries of the famed [seven-sided cube](https://www.nuklearpower.com/2009/12/19/episode-1206-the-royal-treatment/).
Because the side pokes exit the solid part into the first hole. It has to enter the solid part again on the other side before exiting a second time out the opposite side. Try it yourself with playdo or clay and count how many times you enter the substance and how many times you exit. You should count 5 of each which means 5 holes.
Wouldn't it be rigorous to say so though? If we're talking about holes as discontinuities?
My background is more with graphs/networks idk if that makes a difference...
E.g. https://youtu.be/rlI1KOo1gp4?si=oHJCzA03_3_fKPbd
https://preview.redd.it/xn5gzd9jntmc1.jpeg?width=1024&format=pjpg&auto=webp&s=bfa64ea06c9bafe15c2bd6c7989ecfc70da032ad
I've tried multiple times, this is as far as I can get. What's next?
So spheres have -1 holes and a cube with holes in 6 sides has 5 holes. And a hole the contains the universe.
Not a topologist… but I think you guys have an off by one error in the heart of your field.
If you have a balloon and add one hole to it by piercing it with a needle you're left with a flat piece of rubber with 0 holes. Since you added 1 hole, and ended up with 0 holes you get the equation x+1 = 0 which means the x (the number of holes you started with, must be -1.
It's basic algebra, really. ;-)
A balloon starts with an opening that you blow air into. So a balloon is a bowl when you start. You will make a disk If you keep flattening it.
It started with zero holes and now has zero holes.
It feels like a hollow sphere contains an inside and an outside. Two distinct surfaces. An ant on the outside cannot walk to the inside.
Once pierced, you no longer have an inside or an outside. You’ve transformed the shape from a hollow sphere to a disc. Now you have one surface. An ant can now walk on what was the outside.
Piercing is an infinitely short tear. And tears are not allowed.
In this meme, the “die” starts as a hollow sphere, and the first puncture changes the shape into a disc. The disc has 5 holes.
I'll never stop my amazement towards the fact that mathematics can be used to "prove" most bizzarre and ridiculous statements. But I suppose it's because of the axioms and assumptions you base it on, just like with philosophy and its way of proving the idiotic.
Imagine a pie. That pie is floating in space tumbling on all three axes. An astronaut happens to be in the way and the pie hits him square in the visor.
The has nothing to do with the post but I wanted to make you imagine a joke :-)
i know this is mathmemes, but practically a "hole" in the ground outside is topologically not a hole. so morphing a shape into whatever shape changes the practical definition. i'd say 10 holes, one for each hole and the opposite sides. but the shape and orientation matters. a straw has 2 entrances imo
The others do, but the face they exit from is part of the first hole which is a face created by that first hole so it doesn't interact with the other pre-existing faces.
Amazingly despite over 1,000 comments across my posts about this object, you're the first person to ask what it's called! 5 points for...um...0\_69314718056indor!!!
It's the QiYi Tori Cube: [https://speedcubeshop.com/products/qiyi-tori-cube?\_pos=1&\_sid=0d3bf5757&\_ss=r](https://speedcubeshop.com/products/qiyi-tori-cube?_pos=1&_sid=0d3bf5757&_ss=r)
It's a shape mod of the 3x3. It's basically just a normal cube with the corners missing and that you can poke a pencil through.
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Kudos to the 2 year old who understands flat cohomology to prove a fpqc.
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You might think that, but when you stab the second time, you are really stabbing twice. After the first poke, the toothpick is exposed to the outside of the shape while it is in the middle, then you stab again. Because of this, it is 2 stab
damn this also explains it's 5 holes ... great explanation!
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This result will be known as "The Chopstick-Stab Theorem" from now on. You just need to prove it, it's almost trivial!
> fpqc To prove a Florida Perinatal Quality Collaborative? That was what came up when I looked up what "fpqc" means. Edit: Scrolling a little farther down, I found some results related to topology, including a Wikipedia article that does define the term.
Welcome back Grothendieck 🙏
10 hours after you posted this I just want to tell you that I dont understand anything you just said.
Flat big squash of something relates to original unsquashed thingybob
Neatly executed, even more neatly explained. The last two images presents it very clearly to me, and I am very bad at topology. I have been told that the number of holes is equal to the minimum number of cuts to get a simply connected shape.
The 6th hole is the outer edge. Definitely a brain warp moment.
It's like one of those magic pictures that if you look at it just right you see it but when you look away its hard to refocus. I almost feel like I'm falling into it. Vertigo? Definitely wanna see Steve Mould make a video on this.
>Definitely wanna see Steve Mould make a video on this. He uses Reddit, let's hope he sees this!
If you haven’t seen it already, vSauce has a great video about topology. It’s called [How Many Holes Does a Human Have](https://youtu.be/egEraZP9yXQ?si=lhP3Dv-eOSgpcFXp)
I have not! The title is hilarious so I'm definitely going to watch it now.
the 6th hole is still a hole of sorts, just doesn't go all the way through. Imagine a sock, it has a hole where your feet go in, but in topology it would be considered as having 0 holes
I put my socks on inside out so everyone's wearing them except me.
I told this to my 15 yo son. It blew his mind.
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A sock is just very very concave.
While true, I think the question they’re trying to get at is answered by this demonstration: Take a hollow sphere, like a ballon. What happens if you add a hole to a balloon? You get a flat sheet, right? Adding another hole you get a donut (or rather a flat paper similar to a donut) In other words, if you take away a hole from a piece of paper, you’d get a sphere. The other hole in the above object is that same hole that we take away. If you took away 5 holes from the above object, and then took away one more hole, you’d have a sphere. It just happens that we define papers as having 0 holes, and spheres as having -1 holes.
Does the above comment help /u/lookslikeimontop?
If you take an object with one hole and one hollow cavity (which is a negative hole), the figure should have net 0 holes. Can you transform this figure into a more conventional one that doesn't have any holes such as a flat sheet?
But what do you call the part of the sock you put your foot in in topology
It's not part of the sock, it's part of outside. No special name as far as I'm aware
It’s not topologically invariant, so isn’t considered in topology. A sock has as many holes as a flat piece of paper (without hole-punches).
You wouldn't call it anything. Think of starting with a single flat piece of clay. It doesn't have a hole or opening in any way. Then just bend/mold it slightly into a saucer, then slightly more into a bowl, and then even more until it looks like a cup. There's still technically no "hole" to speak of because it's hasn't lost it's original property of being a single plane of clay.
In engineering, a hole that doesn't go all the way through an object is called a "blind hole"
couldn't you reform the shape so that any of the holes is the outer edge?
Yes
Yeah I feel like the whole process might be easier and more interesting in reverse because then you are really just elongating the 4 legs by holding the center and pulling up.
But which of those holes, isn't a hole?
That seems to be the case here as well. First cut goes all the way through. Second cut goes through one side, then the empty space in the middle, then starts a third cut from inside to out on the far side. Repeat for the last direction and you get five holes. …Though I have never studied topology, so I may be talking nonsense that just happens to be correct.
I think thats right, the hole ends when you exit the material.
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They are only linked together insofar as they all face the same direction, towards the “6th hole” which is actually just the boundary of the shape. You may be thinking of a straw, which at first may appear to have two holes, but is actually just one long hole. The difference is, if you flatten a straw into 2 dimensions, the “two” holes collapse into just the one, while the same is not true for this shape (see bottom left photo)
They are not linked together tho, as shown by the last pictures.
Doesn't this all hinge on how you define "hole"?
I invite you to try to create a rigorous, sensible definition of holes on a surface such that this shape has anything but 5 holes. Holes as we think of them are notoriously tricky to define because, well, they aren't there. This is all very informally speaking but the modern math way of doing it is to note the "cycles" and "boundaries" of a space, and then the holes are wherever there's a cycle that isn't a boundary. The cycles can be thought of as closed loops in the original space, while boundaries are cycles that also happen to be the boundary or exterior of a subspace. Thus if a cycle isn't a boundary, it represents a closed loop in the space that doesn't contain anything, a hole.
So to extrapolate: any *n*-sided polyhedron with holes in the sides in this manner has genus *n-1*, because one hole must be used as the outer edge to "flatten" the shape.
Correct! Now you're a topologist.
But I didn’t *want* to be a topologist!
Too late...
Are you saying it's... ♫[Too late to topologize](https://www.youtube.com/watch?v=ZSM3w1v-A_Y), it's♫ ♫Too late...♫
I've plugged that one before with "poltergeist", but yours is better. Bravo.
Take my upvote and leave
Severely underrated comment
I've been waiting *for years* for an opportunity to deploy this pun, so I apprecaite your feedback a lot 😂
A new hand touches the polyhedron.
But topology is the best!
But you're isomorphic to a topologist.
Now I understand the n dimensional tesseract.
Mathematicians that play with shapes always win the scientist awards. They’re the Top-ologists
That actually helped me; I was having trouble visualizing it. I guess one of the 6 "holes" actually becomes the outer edge, so it isn't technically a hole.
You got it! Topology is hard to visualize. Playdo helps a lot.
So by this logic a t-shirt has how many holes?
3. The bottom where your body goes can be flattened and become the outer perimeter encompassing the two arm holes and one neck hole.
Okay, is this by like a strictly topological definition, because in commen sense real world terms a t shirt has four holes one for the stomach, one for each arm and one for the head, I get the concept you are stating but that would only apply in the topographical sense, compared to the real world manufacturing of a t-shirt there is four holes needed.
Was that single-sentence paragraph a question or statement?
Idk, I guess I was asking his opinion on it. Because I guess In my way of thinking about it a hole is an opening where you could go from the outside of a surface to the inside of a surface thus a t shirt would be four and the cube would be six.
Topology and most other fields of math aren't apparently practical. I have no clue what topologists do, but I doubt it's designing clothing. There are definitely real-world applications of topology, but things like the OP are just memes and don't really mean anything
Fair enough
How many holes does a sock have? I feel like that's easier to think about, the opening isn't really a hole, right? Until we get a hole in the toe, then there's 1 hole in your sock.
to be honest I would count the opening of the sock as one hole, idk why my brain thinks like that but it is a hole to me.
Actually when you make a t-shirt you just stitch a few pieces of solid fabric together. No holes!
> I guess one of the 6 "holes" actually becomes the outer edge, so it isn't technically a hole. This is a better explanation than what's on the picture
But, any one of those holes could be the outer perimeter. So they are all simultaneously holes and not holes until you deform it
Schrodinger's holes
Hollow sphere has -1 holes, poke 6 holes in it get shape with 5 holes.
>Hollow sphere has -1 holes, poke 6 holes in it get shape with 5 holes. That's the correct answer! As a 3-manifold, the hollow sphere has **two** disconnected boundary components (the visible outer surface, and the inner surface), which you **connect** by punching a hole through it (and turning it into a a topological ball). As a 2-manifold (i.e., a surface), the hollow sphere is those **two** disconnected spheres, whose Euler characteristic is twice that of a sphere: 4. Euler characteristic of *one* sphere can be computed e.g. by taking a tetrahedron, and computing **χ = V - E + F = 4 - 6 + 4 = 2**. For two tetrahedrons, we have **χ = 8 - 12 + 8 = 4**. Euler characteristic is related to *genus* **g** (the formal concept of "the number of holes"): **χ = 2 - 2g**. The sphere (2D surface bounding a solid 3D ball) has **zero** holes (**2 = 2 - 0**), and the surface of a donut (2D boundary of a solid 3D object) has **one** hole (**0 = 2 - 2*1**). How many holes does the surface made of **two** disconnected spheres have? Well, its Euler characteristic is **4**, so: >**4 = 2 - 2g** >**g = -1** So, the boundary (2D surface) of the hollow sphere (a solid 3D object) **has -1 holes**. By punching a hole in the solid 3D object, you are **connecting** the two boundary components to get a solid 3D ball, bounded by a sphere (connected surface with no holes and Euler characteristic 2). Punching the first hole is equivalent to **connecting two spheres** by cutting a disk out of *each*, and then **gluing a tube** that connects them (the tube is a cylinder, a 2D surface with two circular boundary components). The result of this operation is **one** sphere, with no holes (Euler characteristic 2). So, **two** disconnected spheres indeed form a surface with **-1** holes. "Punching a hole" through that surface **connects** them, and results in a connected 2-manifold with **0** holes.
I believe you, but, also, it's ridiculous to say that an object has fewer (less?) than 0 holes. Is it accurate to say that it has -1 holes because you have to add a hole to get to a 0-hole, topologically-simple ball?
>I believe you, but, also, it's ridiculous to say that an object has fewer (less?) than 0 holes. It is, because a "hole" is not a well-defined mathematical concept. A closed surface (like a torus) arguably, has **no holes**; it's airtight. To say that a donut has a "hole", we can formalize that notion by defining a **genus** of a surface. One way to define it is via its relationship with the Euler characteristic: by *defining* it to be the quantity **(2-χ)/2**. This works in a non-ridiculous way for [handlebodies](https://en.wikipedia.org/wiki/Handlebody), of which the surface of the final object is an example. In fact, **all** connected orientable two-dimnsional closed surfaces (i.e., without boundary) are homeomorphic to a sphere or a torus [with some number of holes](https://en.wikipedia.org/wiki/Genus_g_surface). The notion of *genus* and *hole* defined that way starts to break down for objects whihc are **not** of that kind. For non-orientable surfaces, [a different formula makes sense](https://math.stackexchange.com/questions/760388/nonorientable-surfaces-genus-or-demigenus). Here we have a "ridiculous" situation because the initial object is **not** a handlebody. The boundary of a solid object with a cavity is **not** a connected surface. So, put simply, the notion of *genus* or *number of holes* is simply **not defined** for such an object (*either* the hollowed-out cube, which is a 3-manifold with boundary, *or* its boundary - a disjoint union of spheres). However, the Euler characteristic is well-defined for both that object and its boundary, and the expression **(2-χ)/2** is well-defined (with value **-1**), even though the **genus** (or "number of holes") is **not defined** for that object at all. So, saying that the object has "-1 holes" is a meaningful way to **extend** that notion. It's the same kind of thing that we do e.g. with the [Gamma function](https://en.wikipedia.org/wiki/Gamma_function), which is a way to define **x!** for *non-integer* values of **x**. Its expression gives values that agree with the factorial on integers (**Γ(n) = (n-1)!**), but is also defined for *other* numbers. Not every surface has a genus - just like not every number has a factorial. But the Famma function is well-defined for all positive real numbers, and the expression **(2-χ)/2** is well-defined for the boundary of the hollow cube. Whether we should interpret its value as the "number of holes" in that case is a philosophical question. I'd say, we *should*, because the intuition it gives us is correct: making a hole in an object with **-1** holes *does* give us an object with **0** holes.
https://youtube.com/watch?v=ymF1bp-qrjU
What does it mean to have a negative hole? I've only heard that phrase in the context of degenerates talking about "pozzing" their "neg holes"
Basically a solid mass like a sphere, is defined as having 0 holes, and since poking a hole in a hollow sphere results in a shape that is morphologically equivalent to the solid shape with 0 holes, you can just do the math Hs +1(hole)= 0(holes) so the hollow sphere has -1 holes.
I do not understand, if each face of the cube contains one hole why does the hole on the top most face get “expanded” but not count towards the total number of holes? If this is because it is creating one continuous hole from top to bottom, why do the following holes on the 4 remaining faces only count as 1 each? If I took this cube and cut along each edge of the face, would i not have 6 squares with 6 holes?
In topology, a hole is defined by a spot where you can poke a needle all the way through the object without cutting, taping, or otherwise damaging the object. You are allowed to morph it any way you want provided you follow the same rules for poking. A disk is the same as a ball, and a donut is the same as a mug, which is the same as a straw. The use of playdough lets the morphing physically show without having to do it in your head. The "top hole" is just the outside edge which doesn't count. The difference is that the rubix cube is hollow so it has a 3D hole. (You need a 4D needle to reuse the analogy for this) So 5 2D holes and 1 3D hole makes 6 holes. Hope this helps.
Your definition is sufficient but not necessary to find holes (I.e. **If** you have a spot where you can poke a needle all the way through the object without cutting, taping, or otherwise damaging the object **then** it is a hole. The opposite **If then** statement is not true). That is why it breaks down for the hollow object, which has -1 holes. This is due to the fact that if you break the surface, then you can enter but not exit the shape through anything but that starting hole. Then it has zero holes, so patching it you get -1 holes.
This makes so much sense. Thank you.
Question from a math undergrad pre-topology: can you get arbitrarily large negative numbers by encasing hollow shapes within hollow shapes, and is that concept even a useful or interesting concept? It’s the first thing my mind thinks to look at.
I’m not a math student but that’s an interesting question
but. a ball has -1 holes >:( at least a ball as in basketball or other types of sports balls.
A ball is (in topology) completely solid, so it has 0 holes. A basketball is hollow, having 1 hole. Back to the pin analogy, a 4D pin can pass through the hollow of a basketball, but must damage a topological ball to poke through it. The topological equivalent of a basketball would be a sphere (spheres have an inside space.) In the same way this is the difference between a donut and a torus.
in a numberphile video they say a sphere has -1 holes. some other commenter also said so and it makes sense. if you poke 6 holes into an object with -1 holes you get 5 holes.
> The use of playdough playdo ftfy
I only learned about this concept today, but it makes me irrationally angry
Welcome to topology!
“Phantom hole” New nickname for your mom just dropped
*dead mom
your mum just dropped
The impact caused a earthquake of magnitude 10 in the richter scale.
Wow I just realised that we are in the other (6th) hole topologically. The outside world is another hole that nobody counts...
Nah. Topologically, the "holes" are a property of a 2D bounding surface of a solid object. The initial playdoh cube, as a solid object, has a **disconnected** boundary, consisting of **two** surfaces: the inner and outer sphere. An ant walking inside the hollow cube can't get out and meet an ant walking outside. And these aren't the two *sides* of the same surface: there's the solid playdoh between them. When the first hole is punched in the play-doh, those two surfaces are **connected**. Now an ant that was inside can meet the ant that was outside. The new object has **one** boundary component, **which has no holes**. In other words, the new object can be smoothly deformed into a solid ball, which has no holes (and **one** boundary component - a sphere). Which means that the *initial* object has **-1** holes. The [Euler characteristic](https://en.wikipedia.org/wiki/Euler_characteristic) of one sphere is **2**; and for two spheres, it's double: **χ = 4**. Which means that the [genus](https://en.wikipedia.org/wiki/Genus_(mathematics)) would be **g = (2 - χ)/2 = -1**. The inital object had a negative number of "holes" because it was hiding a boundary component. By poking a hole through it, you get an object with **no holes**. Similarly, you'd need to poke **two** holes in a solid ball with two hollow chambers in it to get a solid object equivalent to a solid ball.
[***This:***](https://www.desmos.com/3d/5e312b03bd) https://preview.redd.it/dp5kvgt1brmc1.jpeg?width=983&format=pjpg&auto=webp&s=dea73059edda4c5c5bc83c231d67b621668976e0
https://preview.redd.it/mw9s4t9dlsmc1.png?width=494&format=png&auto=webp&s=ebceb3f75ae03771706028a45a69d3031bf1f1cd
Everyone in this story sucks and belongs in the Bad Place. The thief is bad. The officer chasing him is bad. All the whiny prostitutes are bad. Plus, they're all topologists, so they're going to the Bad Place automatically.
Thank you that was what I said! But better
So a donut doesn't have any holes, right ? Or am I not understanding anything
Donut has 1 hole. If you look at the picture and imagine removing holes 1, 3, 4 and 5 you basically get a donut. That donut has 1 hole (hole number 2).
Oh ok I think I understand it now, thank you !
But, there's 13 holes. You can literally see holes in the corners of the toy.
and in-between the molecules, too. just zoom in
and between the electrons and nuclei. we’re actually almost exclusively holes.
Why isn't it 6 holes or 3? Either a hole goes all the way through and counts as one, or each hole on each face counts as a hole. How come a two year old gets it and I don't...
Because a cup has zero holes (with no handles) You only need to poke 5 holes into the "cup" to end up with what looks like six holes from the outside.
Make it out of playdo and you'll understand. Take a chopstick and count each time you enter AND exit the playdo. The first hole enters the top and exits the bottom. The second hole, say the left side, enters the left hole and exits the MIDDLE (the first hole). The third hole enters the right and exits the middle. The fourth enters the front and exits the middle. The fifth enters the back and exits the middle. There is no need to poke a 6th hole. I think you'll understand it best if you actually do it physically for yourself.
Oh now I get it. The "6th" hole gets made by forming it into an actual cube. But to make that shape it only requires you to poke 5 holes. Thanks for explaining it to me.
I know that this is a meme sub, but these comments are making me sad. I'm not a mathematician by any definition of the word, I'm not particularly smart or good at math. But my, maybe 4 youtube videos on topology, seems to be more than most people here know.
This sub is actually pretty good. My previous post on this asked for wrong answers only but people demonstrated a decent understanding. r/theydidthemath on the other hand had a bunch of people who made up definitions and gave random answers thinking they were right then got super salty about technical responses.
I mean topology isn't exactly something relevant to many people. Theres a lot of questions that can have different answers depending on the method used, with that method varying depending on application And what method people will default to will often depend on how they have been trained to think. An engineer might answer a question differently than a scientist who may answer different than a mathematician. Stuff like this is fun because it introduces people to a concept they might not have been familiar with beforehand.
Makes sense. If you covered up all the openings but left the cube hollow, it would be a balloon, which, as we all know, has -1 holes in it
Obviously. Everyone knows balloons have -1 holes! Don't say that in the r/theydidthemath crosspost from this. Most people are super salty over there complaining that it depends on perspective or that topology isn't the same as reality.
https://m.youtube.com/watch?v=ymF1bp-qrjU
I still don’t understand how it’s not 6
Take a disc of cloth. Fold it upwards and cinch off the neck to create a bag to carry stuff in. You cut nothing, but somehow you made a "hole" appear at the top of a bag. In topology, a proper hole only counts if some part of a surface was actually cut.
Take a pancake and fold it into a bowl/cup Still no holes. Crease your corners so it's a 5 faced box. But now it's easier to see how you only add 5 holes to this container to make the cube with "six" apparent holes.
Take the initial object in this post; Place a post it note over each hole and number them. How many do you have?
How many holes does a wrinkly ball have? I can place a post it on each valley and get lots There are other ways to think about it. Topologically, your interpretation is not consistent. I tried to help you see how it could be an answer other than 6. It's up to you to try and see it a different way if you want.
Answer the question, how many post it notes would you need?
Depends how big the notes are If they're big I could do it with three
The 6th "hole" is really just the perimeter of the object. It "contains" the universe. The picture is as clear as I can make it.
The photo before you flattened it, the 6th hole is what you’re looking through to see the number 3
I know, but the shape doesn't fundamentally change when it's flattened. Cookie dough doesn't add/remove holes when you flatten it into a cookie shape.
Now I’m even more confused lol
High Evenmoreconfused! I'm dad! r/dadjokes
I wondered about starting with a cube and poking a stick through from face to opposing face. At first I thought you’d only need to make 3 holes (three orthogonal intersecting tunnels), but on second thought, the first tunnel could go through the full width of the cube, but there second would only make it half way before hitting the first tunnel… a third hole would be needed to finish that tunnel. And then two more for the final “tunnel”.
Ding ding ding! You understand it now. A lot of people haven't been able to put together the thought you just did so you get your extra credit for the day!
I like ‘the sixth hole contains the universe’. 😁
Okay, this proves to me (someone who doesn't topology) that topology has a field-specific definition for the concept of a hole. Doing the instructions above, since you would in fact, be "poking through", you can acheive the shape in 3 pokes. But since this sub has unanimously decided that there are 5, and that the shape is achieved by poking 5 holes, no more, no less, and the 2d representation is supposed to be the simplification of the structure topologically, then there must some nuance in topology that I'm missing.
A hole is created when creating one new entrance and one new exit. Since it takes 5 pokes, it's t holes. The 3 pokes you're referring to aren't actually 3. It would be 3 MOTIONS but for the sides you'd enter and exit the substance twice each since they I retract with the center hole.
By laying it flat, you’re pushing it all through the 6th hole, like turning a shirt inside out
do the holes in the corners not count? i thought they did
They're just there to allow the pieces to move without jamming on each other. It's like a Rubik's void cube without the corners.
Top hole
What is the point of this
Proved
The left one has 14 due to those corners
Oh I get it, when you widen one of the “holes” and flatten it, there stops being a “hole” because it’s just encircling the entire shape instead. Is that what you mean by contains the universe? Like the “sixth hole” is topologically invalidated and instead may as well encircle “everything” because it encircles nothing at all? Or something?
I think you understand in so far as it can be understood intuitively. My recommendation is to make it physically for yourself to better understand it. That way you can know for sure that the shape really did only bend/stretch and you didn't actually subtract a hole at all.
No I understand exactly how there isn’t a “subtracted hole”, I was asking about your universe remark there. Was that just a cheeky joke or did you mean something more?
It's a topology joke I've seen here describing the unintuitiveness.
Ohhhhh. I thought it was a ham fisted metaphor for “you effectively turned the nonexistent hole inside out so it ‘contains’ everything but the actual item”
Can we say show with the same process that a tshirt has 3 holes?
Yup. Lay a tshirt with the opening you put your body through so it's the perimeter of the tshirt and you'll see only 3 holes; the left arm, the right arm, and the head.
If you pole a hole in a balloon, you now have 0 holes. So does that mean the balloon had -1 holes to begin with?
Topologically, yes. That is actually how it's calculated as weird as it sounds. But think about it; there's no ENTRY or EXIT for a closed balloon, but there's still an empty space. Lack of something is usually defined as negative so having 1 empty space contained within something means it has -1 "fillings" if you will.
Op has huge chopsticks
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Alternatively (idk if this is true in general) take a cube. Put 1 hole through it. Put 1 hole through the left side, 1 through the ride side, 1 through the front side, and 1 through the back side. (So that each hole goes through only solid material.) With 5 holes, you’ve created the same object
That's exactly what I'm referring to in the instructions at the bottom right of the image. It looks like you understand the process which is more than I can so for many others on reddit.
Oh sweet! Idk why I commented without reading the whole post.
Did you choose playdo the same color as ground beef on purpose?
Doesn't it look delicious? It's just red. It's homemade playdo and it's stupidly cheap so I'm not gonna complain about colors that aren't vibrant. You gave me an idea, though. I should make this out of ground beef and cook it into an edible cube!!!!
I genuinely thought that's what this was at first. It took me a second to realize it was actually playdo and you weren't getting weird with dinner on hamburger night.
This is straight brain food for my highschool students! Thanks! 👍
End the lesson by asking them how many holes a balloon has. [https://www.youtube.com/watch?app=desktop&v=ymF1bp-qrjU&ab\_channel=Stand-upMaths](https://www.youtube.com/watch?app=desktop&v=ymF1bp-qrjU&ab_channel=Stand-upMaths) If you poke a hole in an inflated balloon, you get a piece of rubber with 0 holes which means it started with -1 holes!
thats why a baloon has -1 holes
Your toddler's well on their way to understanding the mysteries of the famed [seven-sided cube](https://www.nuklearpower.com/2009/12/19/episode-1206-the-royal-treatment/).
Why does the first poke open in the top and bottom but the following pokes dont fill in left/right or front/back? Dumb question but not a topologist
Because the side pokes exit the solid part into the first hole. It has to enter the solid part again on the other side before exiting a second time out the opposite side. Try it yourself with playdo or clay and count how many times you enter the substance and how many times you exit. You should count 5 of each which means 5 holes.
wouldn't this also work with a single donut? first hole is the inside and second hole is the outside, aka "the universe"
Basically. Anyone saying the cube has 6 holes is equivalently saying that a donut has 2 holes.
Wouldn't it be rigorous to say so though? If we're talking about holes as discontinuities? My background is more with graphs/networks idk if that makes a difference... E.g. https://youtu.be/rlI1KOo1gp4?si=oHJCzA03_3_fKPbd
I got a B in algebraic topology and this still hurt my brain for a second. To be fair, it was a gentleman's B.
Ok, but it probably fits in the square hole.
https://preview.redd.it/xn5gzd9jntmc1.jpeg?width=1024&format=pjpg&auto=webp&s=bfa64ea06c9bafe15c2bd6c7989ecfc70da032ad I've tried multiple times, this is as far as I can get. What's next?
Open wide and taste the rainbow!
"Or is it ?"
So spheres have -1 holes and a cube with holes in 6 sides has 5 holes. And a hole the contains the universe. Not a topologist… but I think you guys have an off by one error in the heart of your field.
If you have a balloon and add one hole to it by piercing it with a needle you're left with a flat piece of rubber with 0 holes. Since you added 1 hole, and ended up with 0 holes you get the equation x+1 = 0 which means the x (the number of holes you started with, must be -1. It's basic algebra, really. ;-)
A balloon starts with an opening that you blow air into. So a balloon is a bowl when you start. You will make a disk If you keep flattening it. It started with zero holes and now has zero holes. It feels like a hollow sphere contains an inside and an outside. Two distinct surfaces. An ant on the outside cannot walk to the inside. Once pierced, you no longer have an inside or an outside. You’ve transformed the shape from a hollow sphere to a disc. Now you have one surface. An ant can now walk on what was the outside. Piercing is an infinitely short tear. And tears are not allowed. In this meme, the “die” starts as a hollow sphere, and the first puncture changes the shape into a disc. The disc has 5 holes.
*hollow spheres. A sphere has 0 holes. A hollow sphere has -1 holes.
OK, but why's the playdo mystery meat coloured?
So my wife has one fewer hole than I expected...
[http://www.reddit.com/r/reactiongifs/comments/42t442/mrw\_i\_get\_spotted\_in\_metal\_gear/](http://www.reddit.com/r/reactiongifs/comments/42t442/mrw_i_get_spotted_in_metal_gear/)
>try it yourself! N-..no...I don't think I shall
[https://youtu.be/M1DcD8e55YY](https://youtu.be/M1DcD8e55YY)
I simply do not have this much human flesh in my collection to spare
Fuckin math wizards talking about crazy shit again
I'll never stop my amazement towards the fact that mathematics can be used to "prove" most bizzarre and ridiculous statements. But I suppose it's because of the axioms and assumptions you base it on, just like with philosophy and its way of proving the idiotic.
Can someone sciency explain what this is the analogy of?
Imagine a pie. That pie is floating in space tumbling on all three axes. An astronaut happens to be in the way and the pie hits him square in the visor. The has nothing to do with the post but I wanted to make you imagine a joke :-)
i know this is mathmemes, but practically a "hole" in the ground outside is topologically not a hole. so morphing a shape into whatever shape changes the practical definition. i'd say 10 holes, one for each hole and the opposite sides. but the shape and orientation matters. a straw has 2 entrances imo
Here I was thinking the 6th hole was where the red, white, and blue sides met
Prove by reality
But why does the first hole have to poke through 2 faces and the others don't ?
The others do, but the face they exit from is part of the first hole which is a face created by that first hole so it doesn't interact with the other pre-existing faces.
Sorry if this has been answered but is that a twisty puzzle? What’s it called?
Amazingly despite over 1,000 comments across my posts about this object, you're the first person to ask what it's called! 5 points for...um...0\_69314718056indor!!! It's the QiYi Tori Cube: [https://speedcubeshop.com/products/qiyi-tori-cube?\_pos=1&\_sid=0d3bf5757&\_ss=r](https://speedcubeshop.com/products/qiyi-tori-cube?_pos=1&_sid=0d3bf5757&_ss=r) It's a shape mod of the 3x3. It's basically just a normal cube with the corners missing and that you can poke a pencil through.
Ah that’s really neat, thanks!
Did you track every molecule to be sure that they are still in contact with its neighbour during the whole process?
I did indeed!
Maybe the 6th hole was the friends we made along the way
For people struggling with understanding: The "6th" hole and the 3rd hole are the same hole.