I think the problem is there are two ways 'hole' is commonly used. A hole in an object (like a piece of paper) goes all the way through. Holes in the ground often only have one opening and aren't topological holes. That is, holes dug by people. I think burrows are a lot more complex.
Name Brand Cereal this is a great point and I think it’s probably what’s underlying the debate. In fact, there’s even a third, more practical or normative definition of “hole” in common use: any opening that’s there by mistake or causing a malfunction. And indeed when people say “that straw has a hole in it” in everyday life, they typically mean it will leak out the side if you try to use it, so by that definition a normal straw has 0 holes
"through" and "blind" holes, respectively. If we see the human body and all its orifices and pores and consider them all to be holes, we're a manifold of uncountably many on them. To make things even more difficult, there's plenty of room to debate whether those holes are through or blind. You could say that pores are blind holes, but they secrete sweat which came from water we drank. Surface area becomes a problem, too, since those pores are breaks in the skin that blur the line between inside and out.
But the idea of a 'hole' in the ground that has been dug, is called such due to the changes in language over the years to simplify the explanation by shortening one's vocabulary. This becomes evident when you use the word 'pit' to describe the same thing.
I believe that a hole dug in the ground and a hole cut in paper are one and the same, it's just that in the paper, the hole's depth is deeper than the paper's depth, but with the one in the ground, the earth is deeper than the hole.
@@gurth-quake1627 but they’re physically different : one is a cavity water can pool into, the other is a passage with an entry and an exit, like a pipe. They’d be the same if you dug that mythical hole all the way to china. Otherwise, to a topologist, a pool or a ditch are no holes at all
This comment and it's comment comments made me realize that we've being calling "black holes" "holes" even though, as far as we know, it doesn't have a hole. You could say we found a loophole in the name.
If it was mentioned, I didn't catch why the ability to loop a string through the same hole multiple times matters, since the thickness of that string relative to the size of the hole will determine how many loops are possible, unless using a mathematical line with 0 width and 0 height but infinite depth, to which it becomes infinite, but seems the same test should suffice. If you can floss the object and at no place can you pull the sting through the object when looped around, you've got a hole. All other things are like the cup and merely cavities. From there, holes can be calculated based on possible two part combinations of the easily defined mouth, nostrils, and anus and the less obvious however many tear ducts the average person has. Every other "hole" has some barrier like the eardrum. The only complication then is if we include all the microscopic channels throughout the body that allow various chemicals to selectively move through various membranes like in the lungs, kidneys, and brain. At that point, it's incalculable.
The problem with looping the string through multiple times is that that's a loop which can't be deformed into the "go through once" loop. That ruins our plan of counting the different loops and using that as our hole count. Upshot being we need to be a bit more careful than just calculating the different possible loops, we need to count the generators (which can be combined to make the multiply looped loops)
A hole: • has at least two openings that connect (3d) • is an empty space within/through an object (2d) A straw is inherently 3d. A straw has one hole. A bowl is inherently 3d. It lacks a second opening. It has no holes. A handle is inherently 3d. It has two openings. It has one hole. A bottle is inherently 3d. It has one opening. It has no holes. A shirt has 4 openings - it has, at least, 2 holes. Depending on how one counts holes (I'll ascribe by the rope concept), the shirt has up to 6 holes (Neck-to-L.arm, Neck-to-R.arm, Neck-to-Hips, L.arm-to-R.arm, L.arm-to-Hips, R.arm-to-Hips). The tricky question isn't whether there is a hole. The tricky question is determining how many holes if there are more than 2 openings.
If holes are not something that can appear and disappear by stretching or bending without tearing or joining, that tricky question is where deformation comes in handy. Shirt: shorten the sleeves (reversed stretching) and widen the torso from the bottom (regular stretching) so the top eventually lies flat. A disc with 3 holes.
Interesting thing about shirt one... Is that if you take a knife, in example, and stab a shirt... Then if you count that shirt already has 6 holes, then you add 4 more with just one stab.
@@pretentiosaurus It's a hole in the sense of the relatively flat surface of the ground, as an abstract. Like if you closed your eyes and walked over it you would fall through the ground you _expect_ to be there. But in terms of the _actual_ surface of the ground it's not really a hole. Or, it's colloquially a hole, but not technically, topologically, a hole. Different definitions for the word based on context/intent.
I watched action lab video too, and I thought he was too philosophical and didn't explain the topological hole well at all. Thanks for doing justice to topological definition. An idea: I think it might be interesting to geometrically define a hole instead of topologically such that it captures the intuition that "a cup has a hole and straw has 2 holes". The rough idea is to do with notion of concave geometric shape, or what kind of smallest missing volume needed to be added so the original shape becomes concave.
In my mind, as a 3D designer, through holes and blind holes are one and the same, it's just that with through holes, the hole's just deeper than the material, and with the blind holes, the material is thicker than the hole.
@@gurth-quake1627 Sure, but the fact that the material isn't as thick affects the resulting object in a definable way. I'd agree that they're related (potentially depending on how they were made), but there's an explicit difference between the two in that a hole "reaches the other side".
If a straw has 2 holes, then nothing that exists in this dimension has only 1 hole. This means that if you cut a hole into paper, then you're actually cutting 2 holes because the paper has thickness therefore there can be a hole on the top and the bottom of a paper laying on the table.
You're arguing semantics, I wanted to believe straws have one hole but it doesn't work they have to have two because thinking about clothes a shirt has 4 holes even though there are 2 pair lined up together like a straw is there are novelty straws for couples with 1 intake and 2 outputs that clearly have 3 holes but only 1 hole was added therefore a straw must have 2 holes and we all need to accept the only way too have a 1 hole item is with one end of the object being closed
You could argue that a straw and a shirt have 0 holes because they don't have any holes poked in them and function normally. They are made with some empty space but still if you tear your shirt you don't say "aw man my shirt got a fifth hole in it" its just A hole. The only hole.
@@nickd1218 let me ask you something. How deep does the hole have to be to be considered 2 holes because you're arguing that I'm wrong which means that if you cut a hole in the paper, it only has 1 hole, but straws have 2 holes. It's the same exact thing except one is deeper than the other, so tell me, how deep does the hole need to be to be considered 2 holes?
"How many holes does this object have?" "Well. It depends on what you mean by a "hole", and it depends on what you mean by "this object"" Tonight I discovered that topology is wild
here's a good example of why it could be two holes: let's say you dug a hole in the ground. doesn't have to be very deep, just has to be a hole. then, you dig a hole 20 feet away from it. its two holes, right? now, dig deeper and connect the two holes, making a tunnel, almost like a straw. are the two holes now suddenly one hole? or are they two sides of the same hole? who knows
Before getting too into the video, I'd like to comment my definition. A hole is a volume of space that passes through a plane/object. A cup cannot have a hole in it as the plane is deformed, but at no point can you pass through it without contacting that plane.
You've said previously that a torus is genus one because it has one hole, so my immediate thought was, "Why should a straw be considered any different than a (stretched) torus?" Does elongating an object increase the number of holes? I wouldn't think so. But I guess it's always a matter of perspective and definition (I'm certainly no expert on this).
@@amandalin4102 All the way through what ? I say it goes all the way though the skin. But does a hole actually have to go all the way trough something ? When I dig a hole in the dirt, it doesn't go all the way trough Earth, but I still call it a hole.
What about a tennis ball, does it have -1 hole? If you drill a "hole" in a tennis ball, then you have the same as a cup or a bowl, which have zero holes. So isn't that negative one hole if you then patch it up again?
A hole is the thickness which is removed and is completely separate from the opening. A hole in the ground, which is then covered over, is still a hole in the ground. That area is still removed. It just has no openings. It does get tricky and just falls into the category of “all words are made up” when you distinguish a whole in a glass vs. a plate, which is mostly flat, but still concave. Interesting thought process for sure on this. But, I really believe the term hole is very loosely defined and can change very much on the situation it is used in. There are far more precise terms available to describe what a specific hole is. A straw can certainly be described much differently than a cylindrical object with a hole running though the middle of it. But, it depends on how exact someone chooses to be.
Before I watch this video, my answer is a straw has one hole. Because a hole is a break in the material that can contain a substance or let a substance pass through. So a straw has one hole, one thick long hole.
I mean a straw is like paper rolled up. If you roll up paper you got a straw. Does the paper suddenly because it's in straw form have 2 or 1 hole, and stop having holes as soon as you unroll it? Oh god. Help.
I'll always contend that a straw has no holes being that a properly structured and functional straw is itself a tube. A hole would be a puncture in the side of the structure
How many holes does a Y pipe have? Three possible inputs, two possible outputs for each input (assuming you can't output through the same input hole). Water going in , could leave through 1 or 2 possible outputs, but generally will only use 1 output. ? What do you think?
A hole is defined as "a hollow place in a solid body or surface." A straw has one hole that goes all the way through what would otherwise be a solid cylinder. If you take a cylinder and drill a hole 40% into the object from one end, then the other, it has two holes. If you then drill through the remaining 20% in the center you merge the holes into one hole larger hole that runs completely through the object. This is the obvious linguistic definition and what people mean. Anything else is sophistry that primarily only serves the purpose of corrupting language. Fundamentally language is a tool for communication, and corrupting it in this way ultimately leads to further difficulty using that tool. The end result of breaking definitions that are perfectly serviceable without a *serviceable* replacement is the erosion of our capacity for communication. . . Communication breakdowns often result in disaster.
at least when people bring unneccesary sophistication into conversations they typically preface it with "technically..." which is at least a good disclaimer for when people want to get annoying about details. But yes, technicalities are almsot entirely useless in day to day lives and dont serve to help the average person. Good comment.
Me ordeing a coffe The cashier: gives me a donut Ask for a staw The cashier: gives me a donut I ask why they gave me donuts? The cashier: this is not a dunken donut, we are in a math class. What are you doing here sir?
That last part made me smile :D imagine if every person was born with a hole from mouth to backside... we would have very little time to digest anything XD
Question: you say a pair of scissors has two ways of passing a rope through their loops, but isn't there at least one more? Rather than passing through one single hole or the other, couldn't you deform it so a rope passes through both holes at once? Are there ways to pass it through both holes that are non-homologous to any other sequence?
Getting a little bit more technical: the homology group is Z x Z, or the product of two copies of the integers. Another way to think about it is vectors in the plane with integer coordinates. Going right one step corresponds to going through one of the holes in the pair of scissors, and going up one step corresponds to going through the other hold. Any other way of passing rope through the loops is homologous to some number (possibly negative) of copies of each of "right" and "up".
A basketball has a 2 dimensional hole, but a solid metal ball does not. A donut shape has two 1 dimensional holes and only one 2 dimensional hole unless it is solid, in which case it has only one 1 dimensional hole. If a straw is considered only in the limits of 3d space, it could have only one 1 dimensional hole or it could have two 1 dimensional holes plus a 2d hole that aligns with the two 1d holes. This would change if there was continuous shearing. Then there is the possibility, if you are considering higher dimensions, that a straw is a 2d plane that approaches itself on an single axis for all points in a line, in which case it has no holes, because you can make it expand and contract infinitely without changing what its shape will be at this point in 4d. A black hole is a hole in at least the 4th dimension, and a 3d sliver is visible in the 3d dimension, so a black hole is at least one hole of an object that exists at least in the fourth dimension. But it is entirely possible that groups of black holes are part of a single 5+ dimensional object, like a single 2d hole of an empty donut (the ring) will look like two separate circles in a 2d cross sectional universe.
0:52 that's a divit or dent. If you take a rod it is solid. Now take that rod and drill out the center. Now you have a metal tube. 1 hole, an entrance and exit.
I overlooked the thumbnail and immediately neglected it like "what would he say in the video if it's obviously two holes", then after scrolling down a bit more realized, "wait.. It's... One long hole..." and came back up here. Learned about hole identification, and learned that non perforated humans only have one hole that can be looped with a string.
To answer how many holes a person has, instead of creating larger spheres around atoms, what about creating an increasing size sphere and counting how many paths through the body that sphere has? When passing a sphere through the body, only allow those paths don't cause lasting damage when the body deforms around the sphere to accommodate its passage.
@@henryseg I think I have a new approach. Take the cells from the human that are not epithelial cells. Take the set of points in these cells and the induced topology on these points from the standard topology on R^3 and apply your favourite definition for a hole. I'm afraid a precise definition that's not a torture does not exist, altough I don't have a proof. Here is a new question: this may be fine for defining the number of holes on each person. But what if I ask what's the number of holes in the concept of a human, not a specific person? If we assume there is a definition that works for every person, can you define the number of holes on The Human?
If you where to model a hole it would include void spaces, a hole is a space at which nothing solid currently occupies in an area surrounded by solid materials, you dont get liquid holes or even really gas holes, beings when you move though the space with the gas it displaces said gas, but the in talking shapes a hole should be looked at as it's individual shape. A straw has one hole, beings if you were counting holes you would fill in the already accounted holes so as an example I recommend lining up straws and pull a segment of string through each straw individually and count your strings afterwards, the ends of the straw are not the hole the entire straw is a hole, it has to entrances/exists where as a cup would still be a hole but It would only contain a single entrance/exit
I'd say a straw doesn't have holes (it is a hole itself, its like asking how many lightbulbs does this lightbulb) but if that doesnt count then it has one long hole, but not two, i think a few months ago i would have said it has two holes but i chamged my mind now for some reason so :P
@@kintsugikame hmm, my first thought is that there is the big hole, (from your head to the belly) its essentially poking two holes in a straw. so i don't know. but my answer would probably be that its a hole, with two additional holes poking into it. but then again, i don't even know my own logic. maybe its just one hole splitting into 3?. there is no answer, just my opinion, and i don't even know it so :P
The main part of confusion for people who think there is two could be because many of us are taught at a young age that cutting out a section of something would result in a hole forming. Since a straw is just a cylinder with the 2 ends cut off that could be where the idea of 2 holes is created.
The problem is that the cup is not the malleable object that you want it to be. If you dig a hole in the earth, your definition of "hole" would exclude that because "if we flattened earth out, it wouldn't go all the way through". YOU SEE!??!!? Earth isn't like that. This is the real world. A hole is a concept. That means that it is entirely based off of people's subjective opinions on the context of an indenture.
If you can follow the surface of an object and can't pass through it to the other side of it, then it is not a Mobius strip, which means it has not got a hole.
Without reading any other comments, my first thoughts are that it is a resolution problem, how big you set a radius about each atomic nucleus... Then you must decide what is person and what is not person, internal or external to person and should be ignored. Some such external objects could also create holes that would otherwise be mushed together, perhaps even weakly joined up with chemical bonds. That would make the answer depend on outside factors. I guess one must define some rules before asking the question 🤷
It's one hole with 2 "openings". The hole is the only real property of the object meanwhile openings are a virtual physiological human creation. That's how I always like to explain it.
But imagine a Y splitter in a pipe. According to you it has one hole. But if we perform the rope test in the video, the Y splitter falls it when trying to get one?
I've met a person who could put a cooked spaghetti noodle (aka food-safe rope) up their nose and have it come out their mouth. A bit gross in my opinion, but it demonstrates at least one non-lethal, topological hole in the human body. Two, if you account for both nostrils. Three, if there's a path in one nostril and out the other.
If we go the string method, a straw has one hole, but if we add one more hole by cutting one into the side of the straw, the straw now has three holes. 1+1=3
First of all did you even watch the video? Second, you are thinking of a hole as a 2D gap in space, where as the video is about 3D paths. The straw with a hole in the side has 3 "paths" top to bottom, top to side and bottom to side. There are 3 potential unique paths, and therefore there are 3 unique holes, which with an intact straw there is only 1 unique path and therefore only 1 hole
the rope trick should work again. If something can enter the body in one place and come out somewhere else then that constitutes one hole. But that means humans have several milion holes.
I guess if you have an object or a situation that is comparable to a shape, if that shape is topologically similar to another shape (usually far simpler) it has the properties of that simpler shape?
Start with a sphere. Cut a hole in it. You know what I mean by that, right? If the phrase "cut a hole" has meaning in this situation, it seems reasonable to say that you're left with "a sphere with a hole in it." If want to call these two things "a sphere" and "a (section of a) plane," then my question is: when you cut a sphere in such a way that it becomes a (section of a) plane, what is that operation called?
If you want to get rigorous, you'll have to ask a topologist; be careful when you play with words like that, this is exactly why such rigorous terms exist. Consider we define "Cutting a hole in a sphere" to leave us with a "sphere with a hole in it." So let's assign the number of holes to this object as 1. Topologically, you may deform the object into a disk, and by definition it retains the same number of holes. So therefore in your system, a flat disk has a hole in it. In this system, by definition, we performed the operation of "cutting a hole in the sphere", as morphing the object is not an operation. This is fine logically, if you are okay with a flat disk having a hole in it. Holes can be defined in many ways by many people, and it is not clear what it means without first defining it well enough. Perhaps you would benefit from watching this Numberphile video on Bertrand's Paradox, which shows that certain simple questions can be deceptive about what's getting defined: ruclips.net/video/mZBwsm6B280/видео.html.
@@RugnirSvenstarr sphere it two dimensional its embedding is three dimensional space.. People confuse the sphere with the ball it encloses, just as they confuse a circle with the disk it encloses. The casual "area of a circle is pi r squared" instead of the technically correct the area enclosed by a circle[the disk] is pi r squared... The lazy interpretation of calculus results!
My cat used to eat tinsel, like for Christmas trees, and on occasion would end up with a dingle berry bouncing along behind her on a silver tether. She would also eat shoelaces and we needed to keep shoes on a shelf out of reach. As a child I often wondered if she ate a long enough piece of string, could she be flossed safely.
There's 1. If you have a hole through a wall it doesn't matter how thick the wall is it's still A hole. Don't know what holes in the ground or cups have to do with anything they don't have holes running through them they have openings.
Holes dont need rims like the donuts shape so i wouldnt say a cup has a hole i would say a straw only has 1 hole, the thiorie that passing a line thru multiple holes makes each combination singular a is verry interesting question is it 1 hole untill the space is filled x the amount of combinations reminds me of the how many triangles are their puzzles the more you have the more combinations you can make, But i guess a holes is a representation of depth in a surface and doesnt have to go all the thru so maybe if a cup and a staw has 1 but the donut has 0 holes because their isnt any depth in the actual surface no matter how much you seemingly loop it thru you are technically wrapping it around the full object it is a good question because i cant figure out which answer i like because they both make sense
So if you dig a hole in the ground, it is not a hole, because you can reshape Earth into a plate? It seems, that you are terribly missing definition of the word “Hole”.
A "topological hole" is a tunnel. This is confusing, because the etymology of the word "hole" is "hollow" or some old german version of "hohl", which just describes a concave opening.
The discription you neglects the notion that a hole is defined by a singularity, or more generally a hold defined by curvature. An interior and exterior surfaces are knitted by these singularities. Since their two distinted places where these surfaces connect, there are two holes. These "edges" make the biscuit roll from doughnut and biscuit different from a doughnut ball. Its a different kind of topology because you have two surfaces not one.
I like how the sendoff of the video leaves you questioning not only what "holes" are, or what the object in question is, but now what it means to "have" a hole. Does the opening or closing of a mouth mean that a body no longer has that opening? Surely it still "had" it but no longer "has" it. So the question of how many holes a person "has" depends on the precise state of that person... I guess.
The mouth is still a hole. It just has a door called the jaw. Picture this: a doorframe has a hole. A blank, solid door (usually) does not. If there is one door to a room, with no other opening (door, window or vent), then the room also has no hole. Translate this to a human face: the mouth is the doorframe. The jaw is the door. The lower face is the house. Kinda. That's just my interpretation, though.
The proper answer to the question of how many holes a person has, regardless of the state of the person, is always "not enough, now where'd I put my knife?"
I follow your argument that the straw has only one hole, and it's elegant, but at the same time it seems like there must be an equally elegant definition of hole where it has two. Does the circular edge argument that you led with for the cup work? or was that just a straw man? heh. but seriously.
I'm thinking the issue is that hole also means entrance, and I want to be able to say a straw has two entrances, whereas a donut has just one because the two sides are so close to each other..
In his first example with the cup he said that the shape should be allowed to morph. If the straw is flattened it would look like a hoop and would have 1 hole, a straw is thus an extended hoop. Also, with the string method you only have one unique route through the straw, so it has one hole. By both of these definitions glasses with lenses popped out have 2 holes. But, by your logic, (which isn't unreasonable by any means) binoculars with no lenses would have 4 holes rather than 2, though they could be as short as glasses or as long as your forearm.
I think the problem is there are two ways 'hole' is commonly used. A hole in an object (like a piece of paper) goes all the way through. Holes in the ground often only have one opening and aren't topological holes. That is, holes dug by people. I think burrows are a lot more complex.
Name Brand Cereal this is a great point and I think it’s probably what’s underlying the debate. In fact, there’s even a third, more practical or normative definition of “hole” in common use: any opening that’s there by mistake or causing a malfunction. And indeed when people say “that straw has a hole in it” in everyday life, they typically mean it will leak out the side if you try to use it, so by that definition a normal straw has 0 holes
"through" and "blind" holes, respectively. If we see the human body and all its orifices and pores and consider them all to be holes, we're a manifold of uncountably many on them. To make things even more difficult, there's plenty of room to debate whether those holes are through or blind. You could say that pores are blind holes, but they secrete sweat which came from water we drank. Surface area becomes a problem, too, since those pores are breaks in the skin that blur the line between inside and out.
But the idea of a 'hole' in the ground that has been dug, is called such due to the changes in language over the years to simplify the explanation by shortening one's vocabulary. This becomes evident when you use the word 'pit' to describe the same thing.
I believe that a hole dug in the ground and a hole cut in paper are one and the same, it's just that in the paper, the hole's depth is deeper than the paper's depth, but with the one in the ground, the earth is deeper than the hole.
@@gurth-quake1627 but they’re physically different : one is a cavity water can pool into, the other is a passage with an entry and an exit, like a pipe. They’d be the same if you dug that mythical hole all the way to china. Otherwise, to a topologist, a pool or a ditch are no holes at all
Launches video about hole on day where news about a black hole comes out: genius
...which hasn't a hole at all. XD
genus*
“LAUNCHES”
This comment and it's comment comments made me realize that we've being calling "black holes" "holes" even though, as far as we know, it doesn't have a hole. You could say we found a loophole in the name.
You can figure out how many holes a person has by asking politely.
Person, would you kindly tell me how many holes you have?
🤣
@@columbus8myhw 63.996
Ah but that's the beauty of it
The intake "hole" and the exit/exhaust "hole" isn't really a "hole" it's more like a tube almost identical to a straw.
If it was mentioned, I didn't catch why the ability to loop a string through the same hole multiple times matters, since the thickness of that string relative to the size of the hole will determine how many loops are possible, unless using a mathematical line with 0 width and 0 height but infinite depth, to which it becomes infinite, but seems the same test should suffice. If you can floss the object and at no place can you pull the sting through the object when looped around, you've got a hole. All other things are like the cup and merely cavities. From there, holes can be calculated based on possible two part combinations of the easily defined mouth, nostrils, and anus and the less obvious however many tear ducts the average person has. Every other "hole" has some barrier like the eardrum. The only complication then is if we include all the microscopic channels throughout the body that allow various chemicals to selectively move through various membranes like in the lungs, kidneys, and brain. At that point, it's incalculable.
My god, we’re a manifold!
Because when you loop it there’s no way to unloop it without taking it back through the same hole.
The problem with looping the string through multiple times is that that's a loop which can't be deformed into the "go through once" loop. That ruins our plan of counting the different loops and using that as our hole count.
Upshot being we need to be a bit more careful than just calculating the different possible loops, we need to count the generators (which can be combined to make the multiply looped loops)
I would say "thread" instead of "floss"
the 'line of infinite depth' completely blew my mind for a second, had to do a hard reboot. God bless you for that :-)
A hole:
• has at least two openings that connect (3d)
• is an empty space within/through an object (2d)
A straw is inherently 3d. A straw has one hole.
A bowl is inherently 3d. It lacks a second opening. It has no holes.
A handle is inherently 3d. It has two openings. It has one hole.
A bottle is inherently 3d. It has one opening. It has no holes.
A shirt has 4 openings - it has, at least, 2 holes. Depending on how one counts holes (I'll ascribe by the rope concept), the shirt has up to 6 holes (Neck-to-L.arm, Neck-to-R.arm, Neck-to-Hips, L.arm-to-R.arm, L.arm-to-Hips, R.arm-to-Hips).
The tricky question isn't whether there is a hole. The tricky question is determining how many holes if there are more than 2 openings.
If holes are not something that can appear and disappear by stretching or bending without tearing or joining, that tricky question is where deformation comes in handy. Shirt: shorten the sleeves (reversed stretching) and widen the torso from the bottom (regular stretching) so the top eventually lies flat. A disc with 3 holes.
Interesting thing about shirt one... Is that if you take a knife, in example, and stab a shirt... Then if you count that shirt already has 6 holes, then you add 4 more with just one stab.
what happens when you dig a hole in the ground?
@@pretentiosaurus It's a hole in the sense of the relatively flat surface of the ground, as an abstract. Like if you closed your eyes and walked over it you would fall through the ground you _expect_ to be there. But in terms of the _actual_ surface of the ground it's not really a hole.
Or, it's colloquially a hole, but not technically, topologically, a hole. Different definitions for the word based on context/intent.
@@pretentiosaurus It isn't a true hole, but merely an indentation.
A hole is a boundary circle. A tunnel is two interconnected holes. A straw is a tunnel with 2 holes
I watched action lab video too, and I thought he was too philosophical and didn't explain the topological hole well at all. Thanks for doing justice to topological definition.
An idea: I think it might be interesting to geometrically define a hole instead of topologically such that it captures the intuition that "a cup has a hole and straw has 2 holes". The rough idea is to do with notion of concave geometric shape, or what kind of smallest missing volume needed to be added so the original shape becomes concave.
So if we're using the string method of determining holes, does that mean the cup/plate example actually has no holes?
correct. these would be "blind" holes, while the handle of the mug is a "through" hole. in topology, through holes are the only ones that exist
Yes. It's more of some sort of concave deformation.
In my mind, as a 3D designer, through holes and blind holes are one and the same, it's just that with through holes, the hole's just deeper than the material, and with the blind holes, the material is thicker than the hole.
@@gurth-quake1627 Sure, but the fact that the material isn't as thick affects the resulting object in a definable way. I'd agree that they're related (potentially depending on how they were made), but there's an explicit difference between the two in that a hole "reaches the other side".
@@gurth-quake1627 Also, this is entirely contingent on the fact that the majority of materials used are solid to begin with (such as wood and metal).
simple man: my straw only has 2 holes
topologist: 2 holes? mine has 1!
quantum physicist: your straws have holes?!
philosopher: what is a hole?
If a straw has 2 holes, then nothing that exists in this dimension has only 1 hole. This means that if you cut a hole into paper, then you're actually cutting 2 holes because the paper has thickness therefore there can be a hole on the top and the bottom of a paper laying on the table.
You're arguing semantics, I wanted to believe straws have one hole but it doesn't work they have to have two because thinking about clothes a shirt has 4 holes even though there are 2 pair lined up together like a straw is there are novelty straws for couples with 1 intake and 2 outputs that clearly have 3 holes but only 1 hole was added therefore a straw must have 2 holes and we all need to accept the only way too have a 1 hole item is with one end of the object being closed
You could argue that a straw and a shirt have 0 holes because they don't have any holes poked in them and function normally. They are made with some empty space but still if you tear your shirt you don't say "aw man my shirt got a fifth hole in it" its just A hole. The only hole.
Thats a lot of words. Sadly, you still are completely wrong.
@@nickd1218 let me ask you something. How deep does the hole have to be to be considered 2 holes because you're arguing that I'm wrong which means that if you cut a hole in the paper, it only has 1 hole, but straws have 2 holes. It's the same exact thing except one is deeper than the other, so tell me, how deep does the hole need to be to be considered 2 holes?
@@RobleViejo how so?
"How many holes does this object have?"
"Well. It depends on what you mean by a "hole", and it depends on what you mean by "this object""
Tonight I discovered that topology is wild
here's a good example of why it could be two holes: let's say you dug a hole in the ground. doesn't have to be very deep, just has to be a hole. then, you dig a hole 20 feet away from it. its two holes, right? now, dig deeper and connect the two holes, making a tunnel, almost like a straw. are the two holes now suddenly one hole? or are they two sides of the same hole? who knows
The "holes" you dug up aren't topological holes
Even if your analogy was using the correct definition of hole, yeah, if you connected the 2 holes then they become one because you connected then
One of the advantages of being a topologist is that your dog doesn't dig holes in the garden to bury his bones.
This is a gem 💎 of a topology exposition.
Coffe mugs to straws: "You and I are not so different..."
That man just summarized it perfectly in the first 26 seconds! Bravo
Also great videos (I came here from numberphile)
I would describe a cup as having no holes, just a sharp depression
Watching this at 4 am is fuckin weird learned so much at once but I can’t focus on stabilizing it due to sleep deprivation
Before getting too into the video, I'd like to comment my definition. A hole is a volume of space that passes through a plane/object. A cup cannot have a hole in it as the plane is deformed, but at no point can you pass through it without contacting that plane.
well if you very closely you can see it has a huge amount of holes
You've said previously that a torus is genus one because it has one hole, so my immediate thought was, "Why should a straw be considered any different than a (stretched) torus?" Does elongating an object increase the number of holes? I wouldn't think so. But I guess it's always a matter of perspective and definition (I'm certainly no expert on this).
The body has billions of holes. Water goes out through those holes all the time (sweat).
hmm but pores don't go all the way through, like the "hole" of the plastic cup
in a different frame of reference, sodium channels across your cell membranes are holes
@@amandalin4102 All the way through what ? I say it goes all the way though the skin.
But does a hole actually have to go all the way trough something ? When I dig a hole in the dirt, it doesn't go all the way trough Earth, but I still call it a hole.
@@jeremygrectethat's true-- which i think just shows that everyday use of "hole" is different from a strict mathematical definition heh
Congrats 😑
What about a tennis ball, does it have -1 hole? If you drill a "hole" in a tennis ball, then you have the same as a cup or a bowl, which have zero holes. So isn't that negative one hole if you then patch it up again?
A hole is the thickness which is removed and is completely separate from the opening. A hole in the ground, which is then covered over, is still a hole in the ground. That area is still removed. It just has no openings. It does get tricky and just falls into the category of “all words are made up” when you distinguish a whole in a glass vs. a plate, which is mostly flat, but still concave.
Interesting thought process for sure on this. But, I really believe the term hole is very loosely defined and can change very much on the situation it is used in. There are far more precise terms available to describe what a specific hole is. A straw can certainly be described much differently than a cylindrical object with a hole running though the middle of it. But, it depends on how exact someone chooses to be.
The most important point I got from this is that I have the same plates as he does.
Before I watch this video, my answer is a straw has one hole. Because a hole is a break in the material that can contain a substance or let a substance pass through. So a straw has one hole, one thick long hole.
I am starting to doubt whether holes even exist
I mean a straw is like paper rolled up. If you roll up paper you got a straw. Does the paper suddenly because it's in straw form have 2 or 1 hole, and stop having holes as soon as you unroll it?
Oh god. Help.
Henry and Liza were attempting to restore one item to topological holelessness with several items each with one topological hole. The irony.
I'll always contend that a straw has no holes being that a properly structured and functional straw is itself a tube. A hole would be a puncture in the side of the structure
That's a good point
It itself is a hole therefore making it a pipe, tube, or tunnel, so on.
But how long must it be to function as a tube? Is a finger ring a tube with no hole in it?
How many holes does a Y pipe have? Three possible inputs, two possible outputs for each input (assuming you can't output through the same input hole).
Water going in , could leave through 1 or 2 possible outputs, but generally will only use 1 output. ? What do you think?
If you go by homology, a Y-pipe has two holes.
A hole is defined as "a hollow place in a solid body or surface."
A straw has one hole that goes all the way through what would otherwise be a solid cylinder. If you take a cylinder and drill a hole 40% into the object from one end, then the other, it has two holes. If you then drill through the remaining 20% in the center you merge the holes into one hole larger hole that runs completely through the object. This is the obvious linguistic definition and what people mean. Anything else is sophistry that primarily only serves the purpose of corrupting language. Fundamentally language is a tool for communication, and corrupting it in this way ultimately leads to further difficulty using that tool. The end result of breaking definitions that are perfectly serviceable without a *serviceable* replacement is the erosion of our capacity for communication. . . Communication breakdowns often result in disaster.
at least when people bring unneccesary sophistication into conversations they typically preface it with "technically..." which is at least a good disclaimer for when people want to get annoying about details. But yes, technicalities are almsot entirely useless in day to day lives and dont serve to help the average person. Good comment.
The plate argument is why the word divot, depressed, and imprint exist
Me ordeing a coffe
The cashier: gives me a donut
Ask for a staw
The cashier: gives me a donut
I ask why they gave me donuts?
The cashier: this is not a dunken donut, we are in a math class. What are you doing here sir?
Great, video. I really enjoyed your book too.
That last part made me smile :D
imagine if every person was born with a hole from mouth to backside...
we would have very little time to digest anything XD
So due to all the definitions for a hole, simply saying "yes" to the question is kind of true lol
And this is why you shouldn't send a topologist to get coffee and doughnuts.
Technically a straw has one hole but it is an interesting rhetorical question.
Question: you say a pair of scissors has two ways of passing a rope through their loops, but isn't there at least one more? Rather than passing through one single hole or the other, couldn't you deform it so a rope passes through both holes at once? Are there ways to pass it through both holes that are non-homologous to any other sequence?
Getting a little bit more technical: the homology group is Z x Z, or the product of two copies of the integers. Another way to think about it is vectors in the plane with integer coordinates. Going right one step corresponds to going through one of the holes in the pair of scissors, and going up one step corresponds to going through the other hold. Any other way of passing rope through the loops is homologous to some number (possibly negative) of copies of each of "right" and "up".
According to a topologist, it’s 1 hole, right?
I test for holes in people the same way I test for holes in tires...just air em up and bust out the soap water in a spray bottle.
i thought 2 at first but then i was like... im thinking cercles, theres only 1 hole.
If your cup has a hole. Get a new cup
A basketball has a 2 dimensional hole, but a solid metal ball does not.
A donut shape has two 1 dimensional holes and only one 2 dimensional hole unless it is solid, in which case it has only one 1 dimensional hole.
If a straw is considered only in the limits of 3d space, it could have only one 1 dimensional hole or it could have two 1 dimensional holes plus a 2d hole that aligns with the two 1d holes. This would change if there was continuous shearing.
Then there is the possibility, if you are considering higher dimensions, that a straw is a 2d plane that approaches itself on an single axis for all points in a line, in which case it has no holes, because you can make it expand and contract infinitely without changing what its shape will be at this point in 4d.
A black hole is a hole in at least the 4th dimension, and a 3d sliver is visible in the 3d dimension, so a black hole is at least one hole of an object that exists at least in the fourth dimension.
But it is entirely possible that groups of black holes are part of a single 5+ dimensional object, like a single 2d hole of an empty donut (the ring) will look like two separate circles in a 2d cross sectional universe.
"you can also ask the question: how many holes does a person have"
0:52 that's a divit or dent. If you take a rod it is solid. Now take that rod and drill out the center. Now you have a metal tube. 1 hole, an entrance and exit.
_“After years of studying I came to the following conclusion: A straw has a hole.“_
/t- science
So... you're telling me that all of the holes that kids dig in dirt and sand, aren't really holes?
Reality is a lie.
Mathematically there’s no hole
Socially, people use hole to mean “there’s something missing in this area”
It’s pretty abstract but most language is
I overlooked the thumbnail and immediately neglected it like "what would he say in the video if it's obviously two holes", then after scrolling down a bit more realized, "wait.. It's... One long hole..." and came back up here.
Learned about hole identification, and learned that non perforated humans only have one hole that can be looped with a string.
A straw has infinite holes that get infinitely more shallow, all stacked end to end.
it seems as if everyone is a genius around here.
the dough that is removed and cooked after you make a hole in a donut is also a donut hole.
Topology: where cups are plates and mugs are straws
To answer how many holes a person has, instead of creating larger spheres around atoms, what about creating an increasing size sphere and counting how many paths through the body that sphere has? When passing a sphere through the body, only allow those paths don't cause lasting damage when the body deforms around the sphere to accommodate its passage.
Sounds painful!
@@henryseg we should define paths via fluid being able to flow without going through cell membranes.
@@zoltankurti Fluid under how much pressure...?
@@henryseg I think I have a new approach. Take the cells from the human that are not epithelial cells. Take the set of points in these cells and the induced topology on these points from the standard topology on R^3 and apply your favourite definition for a hole.
I'm afraid a precise definition that's not a torture does not exist, altough I don't have a proof.
Here is a new question: this may be fine for defining the number of holes on each person. But what if I ask what's the number of holes in the concept of a human, not a specific person? If we assume there is a definition that works for every person, can you define the number of holes on The Human?
If you where to model a hole it would include void spaces, a hole is a space at which nothing solid currently occupies in an area surrounded by solid materials, you dont get liquid holes or even really gas holes, beings when you move though the space with the gas it displaces said gas, but the in talking shapes a hole should be looked at as it's individual shape. A straw has one hole, beings if you were counting holes you would fill in the already accounted holes so as an example I recommend lining up straws and pull a segment of string through each straw individually and count your strings afterwards, the ends of the straw are not the hole the entire straw is a hole, it has to entrances/exists where as a cup would still be a hole but It would only contain a single entrance/exit
Q: How many holes this thing have?
A:Precisely define "hole" and precisely define "this thing".
Technically straws don't have any holes. They'd be hard to drink out of if they did.
The wall has thickness. So a straw is a stretched out toroid. A toroid is a Genus-1 shape and so has 1 hole.
This is a great video, very interesting discussion
9:28 Hey! Vsauce, Michael here.
My initial search was for a release trailer for a videogame coming out soon. How did I get here???
The more holes a net has, the less holes it has
I'd say a straw doesn't have holes (it is a hole itself, its like asking how many lightbulbs does this lightbulb) but if that doesnt count then it has one long hole, but not two, i think a few months ago i would have said it has two holes but i chamged my mind now for some reason so :P
here’s one for you: how many holes does a t-shirt have?
@@kintsugikame hmm, my first thought is that there is the big hole, (from your head to the belly) its essentially poking two holes in a straw. so i don't know. but my answer would probably be that its a hole, with two additional holes poking into it. but then again, i don't even know my own logic. maybe its just one hole splitting into 3?. there is no answer, just my opinion, and i don't even know it so :P
Plot twist, the cup doesn't have a hole
The main part of confusion for people who think there is two could be because many of us are taught at a young age that cutting out a section of something would result in a hole forming. Since a straw is just a cylinder with the 2 ends cut off that could be where the idea of 2 holes is created.
Before watching this video, straws have no holes. it's like a paper folded on itself.
The problem is that the cup is not the malleable object that you want it to be. If you dig a hole in the earth, your definition of "hole" would exclude that because "if we flattened earth out, it wouldn't go all the way through". YOU SEE!??!!? Earth isn't like that. This is the real world. A hole is a concept. That means that it is entirely based off of people's subjective opinions on the context of an indenture.
Thing is, since math allows for knife edge of zero thickness, no object actually qualifies as having any number of holes. Holes don't actually exist.
If you can follow the surface of an object and can't pass through it to the other side of it, then it is not a Mobius strip, which means it has not got a hole.
an application of the topologist's joke.
It's one hole functioning as two separate holes.
Without reading any other comments, my first thoughts are that it is a resolution problem, how big you set a radius about each atomic nucleus... Then you must decide what is person and what is not person, internal or external to person and should be ignored. Some such external objects could also create holes that would otherwise be mushed together, perhaps even weakly joined up with chemical bonds. That would make the answer depend on outside factors. I guess one must define some rules before asking the question 🤷
the way i see it, if you say a staw has two holes, then thats the same as saying a piece of paper stabbed with a pencil has 2 holes and not 1.
It's one hole with 2 "openings". The hole is the only real property of the object meanwhile openings are a virtual physiological human creation. That's how I always like to explain it.
But imagine a Y splitter in a pipe. According to you it has one hole. But if we perform the rope test in the video, the Y splitter falls it when trying to get one?
Nope, a Y arrangement has 2 holes and 3 external openings. The openings again, are the primary misleading factor.
I've met a person who could put a cooked spaghetti noodle (aka food-safe rope) up their nose and have it come out their mouth. A bit gross in my opinion, but it demonstrates at least one non-lethal, topological hole in the human body. Two, if you account for both nostrils. Three, if there's a path in one nostril and out the other.
Well if you consider the spaces between atoms I’d say several trillion.
If we go the string method, a straw has one hole, but if we add one more hole by cutting one into the side of the straw, the straw now has three holes.
1+1=3
First of all did you even watch the video?
Second, you are thinking of a hole as a 2D gap in space, where as the video is about 3D paths. The straw with a hole in the side has 3 "paths" top to bottom, top to side and bottom to side. There are 3 potential unique paths, and therefore there are 3 unique holes, which with an intact straw there is only 1 unique path and therefore only 1 hole
@@JackdotC Maybe you should go back and re-read the comment you just replied to.
the rope trick should work again. If something can enter the body in one place and come out somewhere else then that constitutes one hole. But that means humans have several milion holes.
One hole. Only logical answer. Else everything has to have two holes.
So fascinating and well explained. Does anyone know what the real world applications of this kind of mathematics are?
Counting holes , of course
I guess if you have an object or a situation that is comparable to a shape, if that shape is topologically similar to another shape (usually far simpler) it has the properties of that simpler shape?
Keeping track of bodies I've buried
@Sky Luke I find that tattoos are good for that
If you put a hole through earth and you travel to both sides you would say that’s a hole to both holes
its a tube with 1 hole going the long way
Start with a sphere. Cut a hole in it. You know what I mean by that, right? If the phrase "cut a hole" has meaning in this situation, it seems reasonable to say that you're left with "a sphere with a hole in it." If want to call these two things "a sphere" and "a (section of a) plane," then my question is: when you cut a sphere in such a way that it becomes a (section of a) plane, what is that operation called?
If you want to get rigorous, you'll have to ask a topologist; be careful when you play with words like that, this is exactly why such rigorous terms exist. Consider we define "Cutting a hole in a sphere" to leave us with a "sphere with a hole in it." So let's assign the number of holes to this object as 1. Topologically, you may deform the object into a disk, and by definition it retains the same number of holes. So therefore in your system, a flat disk has a hole in it. In this system, by definition, we performed the operation of "cutting a hole in the sphere", as morphing the object is not an operation. This is fine logically, if you are okay with a flat disk having a hole in it.
Holes can be defined in many ways by many people, and it is not clear what it means without first defining it well enough. Perhaps you would benefit from watching this Numberphile video on Bertrand's Paradox, which shows that certain simple questions can be deceptive about what's getting defined: ruclips.net/video/mZBwsm6B280/видео.html.
You might be looking for one-point compactification? Sounds like the reverse of what you're doing
I have no idea what you're talking about, a section of a plane is 2 dimensional and a sphere is 3 dimensional
@@RugnirSvenstarr sphere it two dimensional its embedding is three dimensional space.. People confuse the sphere with the ball it encloses, just as they confuse a circle with the disk it encloses. The casual "area of a circle is pi r squared" instead of the technically correct the area enclosed by a circle[the disk] is pi r squared... The lazy interpretation of calculus results!
@@carly09et a section of the shell of a sphere is 3 dimentional. Even if it has 0 thickness
Accordingly to topologists the only way to dig a hole is to go all the way through the earth
And mountains don't exist.
My cat used to eat tinsel, like for Christmas trees, and on occasion would end up with a dingle berry bouncing along behind her on a silver tether. She would also eat shoelaces and we needed to keep shoes on a shelf out of reach. As a child I often wondered if she ate a long enough piece of string, could she be flossed safely.
I swear, topologists are way too bored and have the most absurd way of considering things.
next vid : light speed engine, a simple concrete proposition feasable in a few year
There's 1. If you have a hole through a wall it doesn't matter how thick the wall is it's still A hole. Don't know what holes in the ground or cups have to do with anything they don't have holes running through them they have openings.
Holes dont need rims like the donuts shape so i wouldnt say a cup has a hole i would say a straw only has 1 hole, the thiorie that passing a line thru multiple holes makes each combination singular a is verry interesting question is it 1 hole untill the space is filled x the amount of combinations reminds me of the how many triangles are their puzzles the more you have the more combinations you can make,
But i guess a holes is a representation of depth in a surface and doesnt have to go all the thru so maybe if a cup and a staw has 1 but the donut has 0 holes because their isnt any depth in the actual surface no matter how much you seemingly loop it thru you are technically wrapping it around the full object it is a good question because i cant figure out which answer i like because they both make sense
We went from "how many holes does my straw have" to "how many atoms does your straw have" really fast ey..
This made me think of a 👕. There are several ways to count the holes. Either 2,4, or 6 holes.
So if you dig a hole in the ground, it is not a hole, because you can reshape Earth into a plate? It seems, that you are terribly missing definition of the word “Hole”.
Cups do not have a hole. It does not go through or deep enough. A straw is nothing but a hole with plastic around it
one its a cylinder with a long hole in it
A "topological hole" is a tunnel. This is confusing, because the etymology of the word "hole" is "hollow" or some old german version of "hohl", which just describes a concave opening.
i believe that there are multiple definitions of a hole, being through holes, and what people commonly consider holes, any concave surface
in the second case there a straw would have no holes
It has one hole through it
Are we counting the sweat glands and adrenal glands as "holes" in the human body?
The discription you neglects the notion that a hole is defined by a singularity, or more generally a hold defined by curvature. An interior and exterior surfaces are knitted by these singularities. Since their two distinted places where these surfaces connect, there are two holes. These "edges" make the biscuit roll from doughnut and biscuit different from a doughnut ball. Its a different kind of topology because you have two surfaces not one.
I like how the sendoff of the video leaves you questioning not only what "holes" are, or what the object in question is, but now what it means to "have" a hole. Does the opening or closing of a mouth mean that a body no longer has that opening? Surely it still "had" it but no longer "has" it. So the question of how many holes a person "has" depends on the precise state of that person... I guess.
The mouth is still a hole. It just has a door called the jaw.
Picture this: a doorframe has a hole. A blank, solid door (usually) does not. If there is one door to a room, with no other opening (door, window or vent), then the room also has no hole. Translate this to a human face: the mouth is the doorframe. The jaw is the door. The lower face is the house. Kinda.
That's just my interpretation, though.
The proper answer to the question of how many holes a person has, regardless of the state of the person, is always "not enough, now where'd I put my knife?"
I follow your argument that the straw has only one hole, and it's elegant, but at the same time it seems like there must be an equally elegant definition of hole where it has two. Does the circular edge argument that you led with for the cup work? or was that just a straw man? heh. but seriously.
I'm thinking the issue is that hole also means entrance, and I want to be able to say a straw has two entrances, whereas a donut has just one because the two sides are so close to each other..
In his first example with the cup he said that the shape should be allowed to morph. If the straw is flattened it would look like a hoop and would have 1 hole, a straw is thus an extended hoop.
Also, with the string method you only have one unique route through the straw, so it has one hole.
By both of these definitions glasses with lenses popped out have 2 holes. But, by your logic, (which isn't unreasonable by any means) binoculars with no lenses would have 4 holes rather than 2, though they could be as short as glasses or as long as your forearm.