Extra on a Hole in a Hole in a Hole - Numberphile
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- Опубліковано 8 вер 2016
- Main video is at: • A Hole in a Hole in a ...
More Cliff: bit.ly/Cliff_Videos
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Cliff's glassware: bit.ly/ACME_Klein
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I loved this guy in Back to the Future.
Really, there is no place where this guy isn't lovable
yup doc....haha
+realcygnus ;) haha
great scott!
fep_ ptcp yup!
AARGH! I goofed in this video. Repeatedly. I should have said "ISOTOPIC", not "ISOMORPHIC". My apologies to all -- shows you that I'm a physicist, not a mathematician...
@Andrew Sweebe if I understood correctly, the three tubes didn't "become" the three handles by going through the edge of the sphere. What happens there is the three tubes simply change shape to become the holes of the handles.
I've just realised this is the same Cliff Stoll who wrote that infamous "Why the Web Won't Be Nirvana" article in 1995. With predictions such as "no online database will replace your daily newspaper", it makes for amusing reading in hindsight. In 2010, Cliff commented "Of my many mistakes, flubs, and howlers, few have been as public as my 1995 howler.... Now, whenever I think I know what’s happening, I temper my thoughts: Might be wrong, Cliff…"
Oh. Just read that article. Damn he was wrong.
He also wrote The Cuckoo's Egg about his hunt for some German hackers paid by the KGB to hack into US government computers.
He should be on Computerphile too.
It gets worse: he is actually an astronomer... :)
That was my introduction to Clifford. I grabbed the book at an airport kiosk on my way to a Greek resort. It was a very well-written and absorbing book that I burned through in four wonderful days on the beach. He was an astronomer who was bothered by some numbers that didn't add up. Digging into it turned him into a real-life detective and network security expert who ultimately solved an international spy mystery. This mindset of being willing to follow where the data leads and to train one's self in whatever knowledge is needed to follow wherever the data leads has become one of the most valued skills in the internet start-up community. Curiosity pays.
Next video - Cliff turns a glass sphere inside out.
i would LOVE to see that, with all of the steps in between.
(it is actually possible, but i guess you knew that already)
i dont know what he is talking about but he makes me happy somehow
His enthusiasm is absolutely infectious.
This man has never worked a day in his life.
Spook Bus
Only if you exclude mental work from your definition of working.
+Talon3000 He means that he's so passionate about what he does, that he never felt his work as a mathematician was a job (in the sense of a chore you do to earn money) but something fun instead
Ah. Well that certainly is true.
Did anyone else cheer when he showed the Klein bottle?
From inside, yeah.
Klein bottles have no inside.
***** LOL!
No. Klein bottles are scary. I was stuck inside one for a few hours as a child.
SlimThrull Klein bottles have no inside; your argument is invalid.
I'd like to hear that mug being tapped by a spoon in a few places. It's a three-handled mug!
+
THIS MUG NEEDS TO BE TAPPED!
It's so much bigger and better than the tiny one xD
#tapthemug
plus it's hollow
This NEEDS to happen.
the passion this guy has makes everything more fun to watch 😂
I love how Brady doesn't even need to say anything. Just point a camera at Cliff, and capture the magic.
This guy is amazing
The Klein bottle said to tell you that it does not fail to divide the universe, it never tried to divide the universe, it doesn't even want to divide the univers, tsk.
Cliff is truly a gift to humanity.
I'd like to hear the sound that that mug makes when tapped. #TapTheMug
Cliff Stoll ASMR?
It's from another video. He wants to hear what the notes of the different nodes would be.
Dude, I think that since it's double walled glass, it would just ring normally. You might have to tap the outside to hear it ring, and that wouldn't allow you to cause a note on the handles... well it would, but only on the handle, itself.
I love how he has so many bottles that he can't even find the one he is looking for 😂
Does Tadashi know you have a mug with 3 handles? He was looking for one!
I know it's a huge aside, but I really want to see his glass-making in action!
I love! Cli"the Klein Bottle Guy"ff -- just the right amount of craziness to be happy and make others happy.
So this is what Emmett Brown did before the discovered the flux capacitor
I wish I was as passionate about anything as Cliff is about math. Damn he gets so excited about the littlest things in math.
Unless that 3 holed button is hollow it isn't homeomorphic to the 3 holed torus.
Shhhhh
whether the volume of the torus is filled with air or with plastic doesn't matter from a mathematical point of view...
EDIT: this is not exactly true, see further discussion below. i was made to learn D:
s0me1up But the button is not a torus which just happens to be filled with plastic identical to the material of which the torus is made. It is an object which consists of a torus and the interior of the torus.The button is to a torus what a closed circular disk is to a circle.
+DrGerbils a circular disk is a three dimensional object and has a volume. a circle is two dimensional and has no volume. are you trying to say that the button has volume and the torus doesn't? because that is definitely wrong.
the problem that causes all the confusion is that we are used to thinking about the world in 3 dimensions. topology however is mostly interested in 2 dimensions, in this specific case in surfaces that happen to enclose a space. but what's inside that space is of lesser interest.
think about it this way: what if you make the walls of the glass sphere or glass torus so thick that they start touching on the inside? you effectively filled your object with glass. but for the topology involved you didn't change a thing.
homeomorphism is about being able to map one thing onto another (roughly speaking). you can definitely map a 3-holed button onto a 3-holed torus and vice versa.
+s0me1up No mathematician that I know of would refer to a disc as three-dimensional. For example, the set of points (x,y) in the plane that satisfy x^{2}+y^{2} \leq 1 is a disc; it exists in two-dimensional space (the plane \mathbb{R}^{2}) and so is not three-dimensional.
Watching these two videos and I don't really know why or what is the purpose of this hole in a hole in a hole turned into mug with three handles. The only thing I can relate is Tadashi.
The purpose is to understand interesting topology. Topology itself has quite a few practical implications; from learning how DNA works, designing barcodes, or even providing insights in quantum field theory.
It's basically an easy way to wrap your head around complex mathematical ideas. Instead of a bunch of equations and numbers, you can visualize what is happening with shapes.
Yes, but what is he trying to say? I'm pretty sure most people get what he is saying and what he is doing, but not how the solution and question are linked.
Basically, the fact that they're homeomorphic to each other means that they have the same topological properties. So if you want to know something about the quite complicated holeinaholeinahole (for instance, whether it is compact or connected or retractable), you can just look at the three-holed-torus and it will be the same.
PSL(2,7)
One of the many things you can do with that structure.
If this guy had any more enthusiasm, we would need another him, to hold all that enthusiasm. Neat!
Thanks yet again for an excellent show.
how expensive was it to make all these glass creations?
11 kg of glass - $4
a glass-blowing kiln and equipment - $900
this excitement 0:33 - priceless
skill required too.......they are quite valuable.....hundred$ to thousand$ each.......if you had to buy(if you could even find such unique items)
for reference, the klein bottles he sells are quite pricey.
however i just want to know the production cost for the sphere's
3D printer using transparent material maybe one could sell them as a modell serie ;)
if i remember correctly. glass prinitng was either impossible or in it's infancy,
also, he said it was handblown glass( because of an imperfection he mentioned)
Really appreciate Cliff!
Really cool! I appreciated the skill that went into making the glass manifolds.
Years ago I read Cliff's book "The Cuckoo's Egg" been a fan ever since.
I think there's a small mistake in the explanation. At 3:32 he crosses his arms and says that the smooth changes do not *require* passing one tube through another. While that's certainly true (as he previously demonstrated), I believe that's irrelevant because it is completely allowed for the same reason that two parts of the Klein bottle are allowed to pass through each other.
This is the way that I solved the problem by looking at the original diagram. I reasoned that the tube that shoots through the ring in the middle is not really trapped by that ring, so I mentally pulled it through the ring and moved the tubes a bit to see that the answer is a three-handled sphere. The glass transition uses an elegant trick to avoid doing that which is lovely but topologically no different.
"Air-or coffee- can't get from the outside to the inside..." Him drinking coffee from that thing explains a lot.
Shoot I wanted to see coffee in the mug.
I love this guy 😂😂😂😂😂
I want to hear more about that rug - it is amazing
I love Cliff. However, I have to correct him: When two shapes can be deformed into one another without intersecting or tearing anything, they are called 'isotopic'. Using word 'isomorphism' for such thing would actually be confusing.
Isotopy is probably derived from topos, "place". It means we can continuously, or in this case smoothly, deform from one surface to the other *in place*, that is, embedded in three dimensional space. An abstract homeomorphism (or diffeomorphism even, in this case) doesn't see how the surface is embedded in space.
I was about to post the same thing. It seems to be very common for math communicators to replace the standard definition of "two spaces being homeomorphic" by them being isotopic as subspaces of R³.
homeomrphic, not homotopic or istopic and especially not isomorphic
Well it might be isomorphic in a Category Theory way, but it's not typically called that
Studying topology while high is the best thing ever.
Cliff Stoll wrote the book "the Cuckoo's egg"
Make more videos with this guy
I understand in the context, that (an ideal) Klein bottle has no volume. But then I think, is a Klein bottle homeomorphic to any other topological shape? I really need those visuals- that made the 3 hole torus/ hole in a hole in a hole, etc. very clear to me. (No pun intended.)
nobody:
cliff: K L E I N B O T T L E
Yes
this is amazing!!!!
this guy is delightful!
I want to see how these glass objects are made, please!
Damn, I love this guy :D
Just look at that like to dislike ratio.
Mad scientist
He makes me want to do math, and I have my yearly math exam in less then a week.
What a dude
This guy is an excitement bomb
Yaaaay klein bottle!
I'd like to see the steps to get from the 3-holed torus to the 3-handled sphere. I know it's possible to turn a sphere entirely inside out, but not without passing it through itself, and I'm not coming up with any other obvious ways of inverting the handles.
I still need more holes in holes.
How do you know (mathematically) that a 3-twist Möbius strip is not isomorphic to a 1-twist Möbius strip looking only at topological data? I understand that you'd have to cut it, but what properties does that imply, or how would you know if you lived on the surface that it was different?
Also, what is the difference between homeomorphic, homomorphic, and diffeomorphic? The isomorphic example sounds pretty similar to my understanding of diffeomorphic.
This guy is better than Bill Nye
Can we all agree that that is more of a beer glass than a coffee mug :P. great little topology lesson :)
1:15 How can you use the word "inside" for a Klein Bottle? Duuuuude!
I knew a Klein bottle would eventually appear.
Love the button!
This is a "hole" lot of fun ;)
Would a left möbius loop and a right möbius loop be homeomorphic to one another? (But not, obviously, isomorphic.)
Is a mobius loop with a single twist homeomorphic to a mobius loop with a single twist in the opposite direction? i.e. one is twisted clockwise and one counter clockwise
Yes, the strip itself will have the same topological properties in both cases. (Of course, that's in topological terms: if you build it from a piece of paper then you will notice that the points of the paper are located on different points of the Möbius strip depending on which way you twist it.)
Oh man, this point he's making should have been in the original video. It's very interesting.
You have obsession on Klein Bottles.
I don't get it. How does he go from the cylinders on the inside to the "handles" on the outside without breaking or tearing? Is it akin to the turning the sphere inside out? Or is it something easier? #MathematicianInMySpareTime
You pinch off the volume between the hole and the surface to form 3 handles, then morph the sphere into a mug shape.
Klein bottles yes
I wanna see him talking about Roman surfaces!
MOREE CLIFF THE KLINE BOTTLE GUY!!!
These three statements from the video create a contradiction:
4:08 homeomorphic means they have the same mathematical properties but you can't get from one to the next.
4:15 isomorphic means they have the same mathematical properties and you can get from one to the next.
4:42 these are both homeomorphic and isomorphic to one another
The first two definitions imply that homeomorphism and isomorphism are mutually exclusive.
I suspect the first definition is inaccurate, and that homeomorphic says nothing about whether you can "get from one to the next" rather than saying that you "can't".
Is that right?
Yes. The definition of "homeomorphic" was missing a word; a more accurate definition would be as follows: "Two shapes are homeomorphic if they have the same mathematical properties but you can't _necessarily_ get from one to the next."
Isomorphic is just an "Upgrade" to homeomorphic. Every Isomorphic is Homeomorphic, but not every Homeomorphic is Isomorphic.
The video only gives an idea of the concepts of homeomorphism and isotopy (not isomorphism, he used the term incorrectly for this case) but the terms are not properly defined. Maybe sounds like he was contradicting himself because of the vague language.
Anyway, knots (in the mathematical sense) illustrate the difference between homeomorphism and isotopy:
Roughly speaking, any non-trivial knot is homeomorphic to a circle but it is not isotopic to a circle. We can't go from the knot to the circle through a continuous transition without cutting (and pasting) or without making the knot self-intersect at some instant in the transition.
I was sure he'd bring a 3 holed Klein bottle
So what is diffeomorphic ?
Yeah, I'd say that guy's a homeomorph.
is it possible to merge two handels into one and then all three into the one? Would that break some topological rules?
if you mean having them physically inside each other (like a handle with a thinner handle underneath its surface), then no, they would intersect. if you mean each handle has a smaller handle next to it and closer to the surface of the sphere/mug, then yes, because you just have to squeeze one handle and push it next to another.
no, i mean, what happens if I take two handles and push themtogether? Otherwise, what happens if I take two of the three holes in the tous and push them together? Do they merge in a larger hole?
brachypelmasmith that breaks the rules of morphing. By doing that you're removing a surface.
Cliff Stoll = like the video!
who is the glass blower?
A hole in a hole in a hole would only be homeomorphic to a 3 hole button if the button was hollow!
You know what else splits the universe into inside and outide? A sphere, or a cube, or a oyramid. YOu name it...
What about the volume of a sponge?
Yep there's the Klein bottle
Cliff should be streamer
Am I to understand he made it through all of this without breaking ANY glass?
1:20 is that a bong?
It seems like he broke some rules but I dont know enough about it to tell where.
Like the step of going from 3 holes to 3 handles???
Or splitting one tube/hole into two???
No, he didn't. For the three holes to three handles, he's just moved each hole over to the edge of the sphere and stretched the surface of the sphere out away from it and made the holes bigger so you can put your hand through them. For the three way split, the hole that leads from the surface of the sphere to the two holes inside isn't a real hole since it doesn't actually divide the sphere, it's just a deformation of the sphere surface leading to the two real holes located in the interior.
+Ryan Dean I think that it is easier to see if you think the cup is the three holes torus seen from the inside.
Ohh Cliff, you had to stir the topic to Klein bottles again ?
If they're 'isomorphic' tho, how did he take the handles through the outside of the sphere?
*x files theme*
Don't you have to disassemble the tubes to push them outside of the sphere and make the one with the handles? I admit I don't know topology, but from his definition of isomorphic, the sphere with the handles shouldn't be isomorphic to the rest
I don't know topology, either, but I think that can be done by passing through the 4th dimension.
The holes stay the holes from the handles. If you pull the tubes to the very edge a very flat handle is created, which then can be deformed. Nothing is turned inside-out
Take the sphere with 3 parallel holes, and imagine the sphere is made of toffee. If you pull those holes out perpendicular to their direction, eventually you'd have a ring or tube outside of the sphere. But since it's toffee there's still going to be a connection from out ring back to the sphere. Now you can play around with it and make something handle shaped.
NickCybert You're right man! I never saw it from that perspective. I think that the layer of glass confused me
That lamp on the chair is worrying me...
The new Einstein.
Why are n-twisted mobiüs homomorphic to each other. I thought cutting was not allowed in topology.
Air can indeed go pssht vhhwp
this guy reminds me of dr emmett brown. Maybe he should make a flux capacitor out of the three holes.
So... What is this for?
well, I think he meant isotopic instead of isomorphic. cool video anyway !
lets get him a DeLorean, unless he allready drive one, that is.
that would be über.
Can we have a short introduction to the math?
A question though isomorphism is a soft transform like a liquid flow?
But do it apply to a solids like rubber, if you would be able to weld a ballon shape like the first you would never be able to separate the two holes like suggested or?
So it somehow a dimensionless flow that just store minimal properties it can't be done with a rubber ballon or can it?
I do not mean the actual relocation moving the things/holes, i am just interested in having the one hole turned into two separate by pulling upwards it seem impossible with a solid no matter how elastic?
I thought the first video was a lot to take in
His shirt button is hollow?
I have a degree in mathematics and science
FIANLLY
I have to say that I didn't understand any of this