Damn it, punctures don't technically count as a change in the topology. That means my insurance won't cover puncture damage on my incomprehensible shapes.
No it won't. A 3D printer makes an object using three spatial dimensions, and forms a solid 3D shape. A 4D printer would need to create a 4 dimensional object that we wouldn't be able to see completely due to the fact it has created it using a geometric space with four-dimensional Euclidean space, generalizing the rules of three-dimensional Euclidean space.Time is often quoted as the 4th dimension in physics, but in reality it is a separate dimension that we can use to animate a 2D shape on a computer screen to show a 3D shape and create animated movies and the like. In modern physics, space and time are unified in a four-dimensional Minkowski continuum called "spacetime", whose metric treats the time dimension differently from the three spatial dimensions. Spacetime is not a Euclidean space. But a 4D printer, (if it could ever be created) would use 4 spatial dimensions, in Euclidean geometry, which we can only see with our human eyes 3 dimensions of, in our 3D world we are part of. There may be more dimensions, special or otherwise, but as of yet have not been proven, only theorized in string theory and so on, but, well there you are!
And 4-dimensional space to print it into. Alternatively, if you can manage to print a 3-dimensional object that can warp itself over time, you can simulate a 4-dimensional object by stretching one of its 4th dimensions through time. What a klein bottle would look like in this case, I don't know, but I'd be interested to see.
Not at all, visualization in 3D is only really useful for teaching and has been done with animations for a long time, + you can make those shapes without a 3D printer aswell, it's just not as easy.
Arthur Bernardo Coopi No, like, they're grotesque. Like something out of an artsy sci-fi film, where you're in an exhibition with every wall painted blinding white... And then the 1-sided monsters start showing up.
Cajer 1618 Not really, no. I mean, I do have the baseline gut reaction to repeating patterns, like every other human, but not to the level, that I actually notice it. Anadice Brown Yeah, it all looks so organic. No clear edges anywhere. I hate it.
I know right? This would be the perfect Christmas gift for him though - Modular Klein Bottles... Actually, I wouldn't mind a set myself, when will these be available in stores?
dude, they are all over the video, those moments when the modules he has created where explained with a 3D animation to make it easier to understand for example at 3:32
I hate topology.Why can't you discuss something nice, like division by zero, or complex and hypercomplex numbers, or matrix algebra. Why does it have to be topology, just why? Every time I see a topological transformation I know how Lovecraft's characters must have felt when they understood the noneucleading geometry of his cosmic horrors. Did I mention I hate topology?
Calm down, bra. Not much of a fan as well but this Carlo guy is very nice, at least. His video of regular polytopes in higher dimensions is among my Numberphile favorites.
stormhunter77 yeah I heard that as well, Scottish/Scandinavia (first I thought Dutch, but a Danish accent comes very close to that). I looked it up, he's from Switzerland, but there isn't any French in his speech, so that makes it even weirder.
Brady I'd love some more backstory / explanation on these videos - it seems like you're jumping into a topic assuming we all have background knowledge on the subject. I still enjoy them but it would be great to do them in levels (i.e. introductory, intermediate, advanced). Cheers!
lol... I love Carlo Séquin! His videos always have the same progression for me. "Duh." -> "Yeah, I know" -> "Oh, ok, I get it" -> "Uhhhh" -> "???" -> "He has such an interesting accent."
have just seen these innovations for the first time, do know that cardiovascular surgeons would die for them as they constantly look for different modes of connecting arteries or veins in order to prevent blood flow hysteresis, a major nemesis in their craft. Your connections could possibly alleviate blood flow pressure, damage and clotting, this art you are perfecting is the future for many different industries, am astonished and am honored to see the maturity of such a major technology, the time lapse reminences many of De Vinci or Tesla's innovations taking years to spring forth, Thank You.
A video called "BOTTLE FLIPPING EVERYTHING!!!!!" just came up in my recommendations because of this video. RUclips, that's not the kind of bottles I'm interesting, but thanks anyway.
Sebastian Cor I think genus is defined only for non negative integers. you'd need to refine the definition if you want to consider other values if genuses.
I used to play around as a kid with these types of things and drawings, was called strange and I gave up on it all. Now seeing this all I can say is that I'm sad that I didn't have someone that understood it more than myself back then.
For the mobius strip, if you cut it along the same edge that you taped together to form the strip, wouldn’t that also be a +1 to the genus of the mobius strip? Since, if assumed true, the mobius strip would have genus of 2: one cut in the middle shown in the video, then another as described above.
is there any surface with infinite genus? as in, each time you cut along the line you end up with the same surface, kinda like the natural exponential for derivatives
Oh, I do adore prof. Séquin's presentations. BTW, I do wonder, has anyone created 3-d models of Hurwitz surfaces greater than the Klein quartic? E.g. the Macbeath surface, the 1st Hurwitz triple and those of higher genus? After this video it seems that orientable surfaces need more loving attention!
Professor Sequin is like the classy evil genius with a klein-shaped armchair to Cliff's deranged mad scientist with a blown glass raygun. Both are excellent, is what I'm saying.
I'd love to play a game set on that single-surface sphere near the end. Maybe have a more complex-looking shape connected to the punctures so the player doesn't realize what's happening at first.
Hah so funny there at the end. Don't let Cliff know you're eating his prize possessions! To be honest he won't care too much if you buy another one to help him clear out his basement.
Flash Lol, it's not the holes! It's the twists in going from atrium to ventricle to atrium to ventricle with the flaps and like I said, the ability for things to pass through channels within the walls.
That's true, but you have to look at it in different paradigms. The words that explain the macroscopic scale don't do justice to the microscopic and vice versa. There is a certain element of 4 dimensionality in that on the macroscopic scale, the heart is shaped somewhat like a variant of the klein bottle but on the microscopic scale, things pass through the walls to almost make it a true klein bottle.
It makes sense. Two surfaces with one edge each glued together along that edge make one surface without any edges. Sounds simple. Now all I have to do is imagine it. And whoops. There goes my sanity.
DeadFish37 I think they are explaining that changing the genus is not the same as changing the borders. Those numbers have to be listed separately for a reason.
They explained it's only a border if it's going to another side, like with the flat paper, a hole when both "sides" are the same side, means its only a puncture
I think since they are theoretically always "filled" or "connected" they don't act as holes. If you put another structure at the point it's just extending it's surface(s) at that point (that's how I see it)
You guys just set a new record for me. I fell asleep within 4:30 into the video. I was asleep for oever 2 hours as well and I got about 8 hours of sleep last night. Well done.
mm, if you cut the moebius strip once, you get a continuous band. Then you can cut it perpendicularly, and it leaves you with a band, that is entirely connected. Does this mean that the two cuts have to happen "simultaneously" in the calculation of the genus?
I bet the University is regretting buying this professor a 3D printer.
TheAstronomyDude Especially if the students can print for free!
TheAstronomyDude I would buy him another
TheAstronomyDude yeah these are amazing learning materials, and enjoyably explained
TheAstronomyDude I did not understand you
curiousme Filament is expensive, so the university is broke now. I meant it as a joke.
if you twist a strip of bacon into a mobius strip, it'll only have one side and you won't have to flip it over when you're cooking it.
Now do that with your frying pan! 😀
Elliot Grey I can always fry it in my Klein bottle pan. (Yes IN the pan)
w8 how can it be in the pan if the pan is one sided?? shouldnt be on the pan?
john smith so you're assuming it has a top side?
Jack Le did you just assume my genus??
someone is busting the 3d printing budget.
oh, 3d printing is actually really old
used to be everywhere in universities, etc, just named a rapid prorotyping machine not 3d printer
You had a Klein bottle video without Cliff in it? Poor Cliff...
Jenny Tokumei This is exactly what I thought too.
F
Damn it, punctures don't technically count as a change in the topology. That means my insurance won't cover puncture damage on my incomprehensible shapes.
Dangit, we need a 4-dimentional 3d printer in order to get things right!
LazerLord10 do you mean a 4d printer?
No it won't. A 3D printer makes an object using three spatial dimensions, and forms a solid 3D shape. A 4D printer would need to create a 4 dimensional object that we wouldn't be able to see completely due to the fact it has created it using a geometric space with four-dimensional Euclidean space, generalizing the rules of three-dimensional Euclidean space.Time is often quoted as the 4th dimension in physics, but in reality it is a separate dimension that we can use to animate a 2D shape on a computer screen to show a 3D shape and create animated movies and the like. In modern physics, space and time are unified in a four-dimensional Minkowski continuum called "spacetime", whose metric treats the time dimension differently from the three spatial dimensions. Spacetime is not a Euclidean space. But a 4D printer, (if it could ever be created) would use 4 spatial dimensions, in Euclidean geometry, which we can only see with our human eyes 3 dimensions of, in our 3D world we are part of. There may be more dimensions, special or otherwise, but as of yet have not been proven, only theorized in string theory and so on, but, well there you are!
And 4-dimensional space to print it into.
Alternatively, if you can manage to print a 3-dimensional object that can warp itself over time, you can simulate a 4-dimensional object by stretching one of its 4th dimensions through time. What a klein bottle would look like in this case, I don't know, but I'd be interested to see.
and that one rule is....?
(Also, most of this comment chain is jokey, btw)
More like a 4d printer⚠️
So, someone dropped off a 3D printer at the maths department...
I was totally thinking how 3D printing must have revolutionized the field of topology.
Gameboygenius The Klein bottle guy took over the channel! xd
Not at all, visualization in 3D is only really useful for teaching and has been done with animations for a long time, + you can make those shapes without a 3D printer aswell, it's just not as easy.
This is what happens when you do that
True, it's very unlikely to have had a significant effect on topological research. But occasionally it's helpful.
I was waiting for Cliff to jump into the screen and take the whole video over
Britt Berg me too.
These shapes are legitimately scaring me.
The klein bottles are evolving...
Arthur Bernardo Coopi No, like, they're grotesque. Like something out of an artsy sci-fi film, where you're in an exhibition with every wall painted blinding white...
And then the 1-sided monsters start showing up.
It sounds like you might have trypophobia.
they do bear a mild resemblance to innards
Cajer 1618 Not really, no. I mean, I do have the baseline gut reaction to repeating patterns, like every other human, but not to the level, that I actually notice it.
Anadice Brown Yeah, it all looks so organic. No clear edges anywhere. I hate it.
This is a klein bottle video isn't it?
yeeeup
MrAntieMatter proabably.
MrAntieMatter proabably.
It goes beyond a mere Klein Bottle - that's for beginners!
every video is a klein bottle video.
Kerbalnaught's guide to being a topologist.
Step 1. Be obsessed with klein bottles.
Step 2. Go on Numberphile.
Kerbalnaught Step 3. Get a 3D printer.
Step one. Be obsessed with klien bottles
Step 2. ???
Step 3. Profit.
Mr Perry Do you not understand that reference? Shame
Bordermemes I don't... please explain further
Juan Coria its a southpark reference.
But I came here for the crazy bottle guy with his basement and youthful enthusiasm :o
Kaffeebohnson I thought it was going to be about the tequila bottle with a shot glass for a cap. very disappointed.
same
Kaffeebohnson the doc brown of maths
I like this geezer hes pretty cool too
I know right? This would be the perfect Christmas gift for him though - Modular Klein Bottles... Actually, I wouldn't mind a set myself, when will these be available in stores?
It's great seeing 3D printing getting used properly, creating complex form and not poorly designed toys and other scrap
The person making the models for the animations sure had their hands full this time.
puskajussi37 what animations?
dude, they are all over the video, those moments when the modules he has created where explained with a 3D animation to make it easier to understand for example at 3:32
puskajussi37 now what about that is hard?
No ants were harmed in the making of this video.
You sure about that?
Would an ant really get cut if it tried to crawl over a border of a piece of paper or something?
This is like Lego for really geeky kids. Love it
I could listen to this guy saying "border" all day
Emil Macko it's his sexy swiss accent
What? I would have sworn it is Scottish
ikr?
Dav Emil. Du er da dansk?
Or 'Tetrahedron'
Rule n°1 : The topologist must be crazy
I feel like Topology is just one massive inside joke
what? you dont get it?
i wonder what topology is used for...it looks very interesting but i can't think of any real applications
I think its about exploring the possibilities of 3 dimensional space
maybe circuit layouts?
it's used heavily in general relativity and even on advanced classical Mechanics
3 blue 1 brown. "who cares about topology?"
Give topological insulators a quick read over. actually give them a long read over :)
I hate topology.Why can't you discuss something nice, like division by zero, or complex and hypercomplex numbers, or matrix algebra. Why does it have to be topology, just why? Every time I see a topological transformation I know how Lovecraft's characters must have felt when they understood the noneucleading geometry of his cosmic horrors. Did I mention I hate topology?
AlucardNoir Because topology allows you to get donuts out of mugs.
Calm down, bra. Not much of a fan as well but this Carlo guy is very nice, at least. His video of regular polytopes in higher dimensions is among my Numberphile favorites.
well, i love it .D
Mmmmm, non-euclidean geometries...
Sirus We don't, morons do.
I wonder if all topologists 3d print these cool shapes if they have a 3d printer.
Why not?
We do
This man has the most enigmatic accent I've ever heard.
it's almost like half scottish, half... some part of scandinavia/germany
stormhunter77 yeah I heard that as well, Scottish/Scandinavia (first I thought Dutch, but a Danish accent comes very close to that). I looked it up, he's from Switzerland, but there isn't any French in his speech, so that makes it even weirder.
Well Swizz has 4 Languages - Italian, German French and Romansh(whatever this is).
baasmans Can confirm, it's swiss. He's probably from the german speaking part.
And now I know why plumbers charge so much.
You need to make a followup with one of these super bottles arranged in the shape of a larger klein bottle.
Brady I'd love some more backstory / explanation on these videos - it seems like you're jumping into a topic assuming we all have background knowledge on the subject. I still enjoy them but it would be great to do them in levels (i.e. introductory, intermediate, advanced). Cheers!
"What I would almost call a 'super-duper bottle'"..that made me laugh hard :-D
that part where the ant cut himself in half scared me
Slightly painful for the brain, but so rewarding in the end ^-^
Bob Bobson Mathematics in a nutshell.
lol... I love Carlo Séquin! His videos always have the same progression for me. "Duh." -> "Yeah, I know" -> "Oh, ok, I get it" -> "Uhhhh" -> "???" -> "He has such an interesting accent."
"wow, that's a really cool and complex shape you got there, what is it called?"
"the super duper bottle"
I Should Get A Nickname
Super duper pooper scooper bottle!
There's more to math than Klein bottles. This is like the 100000th such video.
He sounds like a computer voice :o Love it!
It looks like the obvious step would be to make a square, triangular, or hexagonal grid and construct a single-sided surface of genus infinity.
One side to rule them all
A shape to surpass the Klein Bottle...
have just seen these innovations for the first time, do know that cardiovascular surgeons would die for them as they constantly look for different modes of connecting arteries or veins in order to prevent blood flow hysteresis, a major nemesis in their craft. Your connections could possibly alleviate blood flow pressure, damage and clotting, this art you are perfecting is the future for many different industries, am astonished and am honored to see the maturity of such a major technology, the time lapse reminences many of De Vinci or Tesla's innovations taking years to spring forth, Thank You.
"Mom, I want to be an artist." - "NO, you will study a MINT subject and learn something useful!" - Challenge accepted.
I'm sure Cliff Stoll would go crazy over this video
A video called "BOTTLE FLIPPING EVERYTHING!!!!!" just came up in my recommendations because of this video.
RUclips, that's not the kind of bottles I'm interesting, but thanks anyway.
KLEIN BAGELS
I'm surprised by how truly gleeful this idea makes me XD
klein pretzels!
the klein bottles are evolving
pramitbanerjee Now it is a Groß Bottle.
(This joke makes only sense in german.
klein = small
groß = big)
I've never studied topology, but your videos are intriguing and fascinating. Thank you.
Now make a hypercube out of Klein bottles!
I think this is called going mad with power
now that's what you need in back to school section.
12:14 "inner side of the klein bottle" excuse me, what? klein bottles are one sided... aren't they?
Where's Dr. Grime gone?
When this guy inevitably dies of old age, I'll be in tears. He just looks humble.
2:08 RIP ant 😔
Nice example of ivory tower
I could listen to Prof. Séquin talk about just about anything all day.
The bottles(Klein) are beautiful.
"If I combine my 3-way parts and my 4-way parts what do we have"
does the answer start with O？
I studied topology in college 40 years ago. It was fun to hear his explainations
This guy actually perceives the fourth dimension at this point
Was waiting for that jump when Prof Cliff would pop up and say... "we have a klien super bottle"... Like if you read it in his style...
this bottle is for when you are thirsty at 4am
is it only possible to get whole numbered genus? what about 1/2 a genus?
I know that you can't, but I don't know how to explain why.
Sebastian Cor I think genus is defined only for non negative integers. you'd need to refine the definition if you want to consider other values if genuses.
I'm curious to how you perceived the definition of a surface's genus.
Diglettoss Toss its mentioned in the video. something along the lines of the number of ways you can cut a 2-manifold without it falling apart.
You'd have to be a half-hole.
I used to play around as a kid with these types of things and drawings, was called strange and I gave up on it all. Now seeing this all I can say is that I'm sad that I didn't have someone that understood it more than myself back then.
Make a bong that looks like a Klien bottle
Bradthewinner That is actually an amazing idea. I nees this in my life.
For the mobius strip, if you cut it along the same edge that you taped together to form the strip, wouldn’t that also be a +1 to the genus of the mobius strip?
Since, if assumed true, the mobius strip would have genus of 2: one cut in the middle shown in the video, then another as described above.
The cut across wouldnt be a loop
Numberphile is every scientist stereotype all condensed into one area, and it's marvellous.
2:21 both sides of a Mobious Band?
is there any surface with infinite genus? as in, each time you cut along the line you end up with the same surface, kinda like the natural exponential for derivatives
oooof
Professor: SUPER KLIEN BOTTLE
*a wild Cliff Stoll appeared*
Your scientists were so preoccupied with whether or not they could, they didn’t stop to think if they should.
This topology video has a much, much higher genus than Professor Sequin's previous topology videos.
Oh, I do adore prof. Séquin's presentations.
BTW, I do wonder, has anyone created 3-d models of Hurwitz surfaces greater than the Klein quartic? E.g. the Macbeath surface, the 1st Hurwitz triple and those of higher genus?
After this video it seems that orientable surfaces need more loving attention!
I love that voice of prof. Xavier! Oh, wait..
Professor Sequin is like the classy evil genius with a klein-shaped armchair to Cliff's deranged mad scientist with a blown glass raygun. Both are excellent, is what I'm saying.
But if you rent out the space inside a Klein bottle, you'll have a surface with no border that has a boarder.
That's okay; as soon as they move in they evict themselves.
Lol needs more upvotes
7:27 using the modular Klein bottles, we could make a new, Lego-like toy 😂
WYSI
Who designed the stl files and who printed those? it looks difficult
i want the stl files so i can print these myself
I always love your topology stuff.
6:19 It's still a loop, so can't you cut in a different direction to get a long rectangular piece of paper, which is still in one piece?
I'd love to play a game set on that single-surface sphere near the end. Maybe have a more complex-looking shape connected to the punctures so the player doesn't realize what's happening at first.
Batfan1939 What's QGame?
K1naku5ana3R1ka Typo. I meant to say *A* game.
K1naku5ana3R1ka Fixed.
...right. Makes...so much sense.
Carlo is my favorite Numberphile guest by far, and I don't even particularly like topology.
5:39 Looks like a sphere with 4 holes to me
Are they using Professor Layton sound effects?
Hah so funny there at the end. Don't let Cliff know you're eating his prize possessions!
To be honest he won't care too much if you buy another one to help him clear out his basement.
Reminds me so much of the human heart except for the 4D-ness. Things do travel through the walls of the heart though...
John T It's the holes.
Flash Lol, it's not the holes! It's the twists in going from atrium to ventricle to atrium to ventricle with the flaps and like I said, the ability for things to pass through channels within the walls.
Unless you're a quantum-size particle, you're not going to naturally pass trough anything that doesn't have a hole.
That's true, but you have to look at it in different paradigms. The words that explain the macroscopic scale don't do justice to the microscopic and vice versa. There is a certain element of 4 dimensionality in that on the macroscopic scale, the heart is shaped somewhat like a variant of the klein bottle but on the microscopic scale, things pass through the walls to almost make it a true klein bottle.
.....why is a video about bottles and strips so amazing?
It makes sense. Two surfaces with one edge each glued together along that edge make one surface without any edges. Sounds simple. Now all I have to do is imagine it. And whoops. There goes my sanity.
They say that creating punctures in a shape doesn't affect what shape it is, but surely that creates a border?
DeadFish37 I think they are explaining that changing the genus is not the same as changing the borders. Those numbers have to be listed separately for a reason.
Yeah they should have mentioned that it creates another border, might have been accidentally cut out?
DeadFish37 you are not thinking in the fourth dimension!!!, they are not talking about the shapes but properties they have
They explained it's only a border if it's going to another side, like with the flat paper, a hole when both "sides" are the same side, means its only a puncture
simsom4343 Thanks, must have missed that
Cool stuff. Did we finish with prime numbers?
I don't get why puncture don't change the classification... they said that borders ( 1:03 ) do, and adding a puncture adds that, so what gives?
@4:10 Don't the holes add borders though?
I think since they are theoretically always "filled" or "connected" they don't act as holes. If you put another structure at the point it's just extending it's surface(s) at that point (that's how I see it)
When he talks about the stand for the genius 22 structure he never fills that hole on the bottom.
+Gasduster99 Yes but I think in theory it should or is - I guess that is just for presentation/simplicity - I mean Kleinbottles are weird in 3D, too.
Holes add borders, but that doesn't affect the number of sides or the genus. It's the latter two properties that are the point of the super bottles.
No, because you're still on the same "side"
You guys just set a new record for me. I fell asleep within 4:30 into the video. I was asleep for oever 2 hours as well and I got about 8 hours of sleep last night. Well done.
klein bagels... Now, I want some 3-hole donuts. MMM
Jason Emmons And soda out of a three handle mug
It would be really great to watch a video that explains derivatives and limits! That will be helpful!
lol @ 13:47 "Wow, that's genus 14"
I mentioned it elsewhere but repeat here - I love that guy's accent
His office is a child's paradise.
I like this particular video.
Cube frame super bottle 2x2 'twisty puzzle' (Rubik's Cube) - that would be epic!
@16:12 The word is Voila, French for "There it is"
If the 15:27 object is a Klein bottle, wouldn't it need a hole to pass itself along it because of 3d limitations? Where's that said hole?
This guy or Cliff needs to get together with someone who owns an ant colony and get Möbius bands and Klein bottles inside an ant colony enclosure
mm, if you cut the moebius strip once, you get a continuous band. Then you can cut it perpendicularly, and it leaves you with a band, that is entirely connected. Does this mean that the two cuts have to happen "simultaneously" in the calculation of the genus?