im more impressed by the glass blowing cuz forging a knife is already hard. but thats a whole 'nother level. if it were made of metal it would be even more impressive.
@@meekrab9027 Oddly enough, he does not drink. The story goes that his dad told him that he would let him know when he is ready to drink alcohol. His dad died, never telling this mathematician when he was ready to drink, and so he never did. He tells it better.
I doubt anyone else in the world has this kind of sentence in the dedicated wiki page: " He stores his inventory in the crawlspace underneath his home, and accesses it when needed with a homemade miniature robotic forklift." en.wikipedia.org/wiki/Clifford_Stoll It's a bit like Alice in Wonderland or Being John Malkovich.
@@youngflamer1667 I saw that comment coming. 😂 Suppose the absolute majority of us are interested in holes of some kind, especially since both sexes have them. 😏🤭 Unfortunately (?) they are not real topological holes, but merely "caverns".
So what I got from this is that a regular donut is topologically equivalent to a regular coffee mug (assuming the handle was hollow). That must be why they go so well together.
One of the top 10 reasons for not doing your Math homework is that you took a break for a cup of coffee and a doughnut and spent the rest of the evening trying to figure out which one to dunk.
I know many have already mentioned how wholesome his enthusiasm is, but at 5:21 when cliff goes "hee hee hee" excitedly is the most wonderful thing ever
The 3 holed taurus mug was in the video @ 8:10. Take the sphere with 3 handles, push the top down into the bottom and it actually makes a vacuum mug - in that it has an empty space between the inside and outside walls of the mug... 3 holed doughnuts, also totally do-able.
"Morty! Morty I *urp* I made a hole in a hole in a hole Morty, its *urrrp* hole-inception up in here Morty!" "Aww jeez Rick, w-why did you make so many holes, Rick?" "You ever try to drink from a mug with 3 handles? Its *urrrp* Its amazing Morty. Its topological magic Morty!"
I just imagined offering that first object to a pre-industrial King. If you weren't executed as a magician you'd probably be able to trade one of those for a whole life of luxury.
Why would you be executed as a magician ? A pre-industrial King would just be amazed by the fact it is transparent glass, and also probably a lot harder to break than ordinary glass. Otherwise, it's really not so weird. It's not because it has funny topologic properties that it would look like magic... (and it really doesn't).
You mean an item like this one? en.wikipedia.org/wiki/Cage_cup sure it was a luxury item fit for a king, but its maker was probably a slave and in no way entitled to a "life of luxury". Topologically it's also much more complex.
homeo meaning same, morphic meaning shape, literally its definition is something of the same shape, and in this case, possessing all the same features; three holes.
Homeomorphic simply means you have a function that is continuous both ways. Continuous functions are something you learn about first thing in calculus.
@Bl00drav3n after watching the video im sure youre right but it wasn't obvious to me as i was watching the video. i thought for sure i knew where it was headed.
This is mathematical art! You should display these as a sculptural art piece, and when people ask what it means, you can blow their minds by pointing at the last one in the sequence and saying, "This one with the handles is exactly/technically the same as," -points to the first one in the sequence- "this one"! Then you can go into this explanation to both blow their minds even more as well as educating them!
Then cultivate your mathematical interests, do some hobby research, and date an enthusiastic mathematician! 😄 We do have a childlike enthusiasm for beauty and life, and lots of us have charm, humor and some seductive skills. 😊
111SPIDERPIG999111 Yeah, but it's a lot easier to make lots of the same object than a few of many objects thanks to mass production. The whole reason he has so many is because he needed enough that it would be profitable to make them en masse.
This is my best proof about Flt Define Sx=1+2^2+3^2+4^2+....+x^2.=x(x+1)(2x+1)/6=(2x^3+3x^2+x)/6 Sy=1+2^2+3^2+4^2+....+y^2=y(y+1)(2y+1)/6=(2y^3+3y^2+y)/6 Sz=1+2^2+3^2+4^2+....+z^2=z(z+1)(2z+1)/6=(2z^3+3z^2+z)/6 So 2x^3=6Sx-3x^2-x 2y^3=6Sy-3y^2-y 2z^3=6Sz-3z^2-z so x^3=3Sx-3/2x^2-x/2 y^3=3Sy-3/2y^2 - y/2 z^3=3Sz -3/2z^2-z/2 supose x^3+y^3=z^3 3Sx-3/2x^2-x/2+3Sy-3/2y^2 - y/2 - (3Sz -3/2z^2-z/2)=0 or 2Sx-x^2-x/3+2Sy-y^2 - y/3 - (2Sz -z^2-z/3)=0 or 2Sx+2Sy-2Sz-(x^2+y^2-z^2) =(x/3+y/3-z/3) So Sx+S(x-1)+Sy+S(y-1)-Sz-S(z-1)=(x+y-z)/3 So (x+y-z)^2=[3Sx+3Sy+Sy+3S(y-1)-3Sz-3S(z-1)]^2 because (x+y)^2=x^2+y^2+2xy (x+y-z)^2=(x+y)^2+z^2-2z(x+y)=x^2+y^2+2xy+z^2-2z(x+y=x^2+y^2-z^2+2xy-2z(x+y )+3z^2 So (x^2+y^2 - z^2)=(x+y-z)^2+2xy-2z(x+y )+3z^2=[3Sx+3Sy+Sy+3S(y-1)-3Sz-3S(z-1)]^2+(2xy-2zx-2zy )+3z^2 And (x^2+y^2-z^2)=(2Sx+2Sy-2Sz)-(x+y-z)/3 So [3Sx+3Sy+Sy+3S(y-1)-3Sz-3S(z-1)]^2+(2xy-2zx-2zy )+3z^2=(2Sx+2Sy-2Sz)-(x+y-z)/3 So (2xy-2zx-2zy )=(2Sx+2Sy-2Sz) - [3Sx+3Sy+Sy+3S(y-1)-3Sz-3S(z-1)]^2-(x+y-z)/3 Named (2Sx+2Sy-2Sz) - [3Sx+3Sy+Sy+3S(y-1)-3Sz-3S(z-1)]^2 is a Note a is an integer. So (2xy-2zx-2zy )=a-(x+y-z)/3 So 6xy=[3a-(x+y-z)+6zx+6zy.] So x=[3a-(x+y-z)+6zx+6zy.]/6y Because x is integer so [3a-(x+y-z)+6zx+6zy.]=K6y This is the first condition that to be added And 6zx=6xy+-2zy - 3a-+x+y-z So x=[6xy+-2zy - 3a-+x+y-z]/6z This is second condition that to be added And 6zy=6xy-6zx-3a+x+y-z So y=[6xy-6zx-3a+x+y-z]/6z Thie is third condition that to be added And z=[6xy-6zx-3a+x+y-z]/6y This is fourth condition to be added Total have 4 conditions that to be added 6zx=6xy+-2zy - 3a-+x+y-z x=[6xy+-2zy - 3a-+x+y-z]/6z y=[6xy-6zx-3a+x+y-z]/6z z=[6xy-6zx-3a+x+y-z]/6y [3a-(x+y-z)+6zx+6zy.]=K6y [6xy+-2zy - 3a-+x+y-z]=h6z [6xy-6zx-3a+x+y-z]=g6z [6xy-6zx-3a+x+y-z]=f6y Attention! they are different With 4 conditions added It will definitely hinder the problem solving.
If someone told me about that folklore without naming any names or other easily recognizable facts, and then asked me: What country is this legend from? I would guess it's somewhere from Germany or Belgium or Netherlands. You just seem like a holey-type of people. I like that.
on his first klein bottle video he said he had a friend that had made a klein bottle out of glass with him, the thousands of klein bottles were ordered to a company, but these... these are just too specific, I'm pretty sure he's his own glass blower now, and a master at that
@@Xthis1s4youX I always assumed he enjoyed doing glass-blowing and never stopped. Just decided that he couldn't make a 1000 Klein bottles by himself, so outsourced. There's no wáy that anyone else would make all these for him :P
2011 - Welcome to my new channel, Numberphile, a channel about numbers. The first video will be about the number 11. 2016 - Let's talk about a hole in a hole in a hole.
Tadashi Tokieda, one of the other video series on Numberphile. His videos focus mostly on physical interaction between objects. I found some videos interesting, eg cutting Mobius paper band and merging paperclips
I neglected to mention that these glass models are not just homeomorphic to each other, they are also isotopic to each other. Isotopic is a strong property than homeomorphic: Homeomorphic means that there's a mapping between them that preserves all their topological information. Isotopic goes further and demands a continuous deformation that is a homeomorphism at each step.
[repeated from elsewhere] Two objects that are homeomorphic but not Isotopic? Sure ... You usually make a Möbius band by giving a strip of paper a half twist, and taping their ends together ... the result has one edge, one side, one hole, and is non-orientable. If you give the strip of paper three half twists ( = 3/2 twists), the result has the same properties: one side, one hole, and non-orientable. So these two things (the 1/2 twist Möbloop and the 3/2 twist Möbloop) are homeomorphic to each other -- they have the same topological properties. However, they are not isotopic to each other ... you can't continuous deform a 1/2 twist Möbloop into a 1 1/2 twist Möbloop. (You gotta cut the 1/2 twist guy, add a full twist, and tape it back together).
This made me whish to become a glassworker myself. Just for the minute chance, that someone like him pops in my shop, demanding me making such amazing things :D
first of all, this is beautiful. the simplicity in explanation and the passion this guy has. second, imagine how weird out the glassmaker must feel to make all these shapes lol
Taking this a step further, you could stretch the two other holes, and spiral them round the central hole in a double helix, then pull the whole thing out to the side to create a one handled coffee mug with a double helix filigree patternation.
It's not THAT expensive. He (or atleast one of his friends) works in a glas shop so they got enough glas around to basicly produce 2 handfull or glas balls.
I highly recommend his book, The Cuckoo's Egg. It's fantastic and a really great insight into Cliff Stoll's life. Never know where life will take you. ;)
I'm listening to this without seeing the video. The voice sort of reminds me of Kermit the frog I think it would be awesome to have an episode of numberphile with Kermit the frog.
Be careful where you put the handles, otherwise you'll end up with a two-handled mug with a hole in the bottom. Or if you're crafty, you could instead fashion a one-handled mug with two drinking straws built-in, perfect for sharing a milkshake.
@@TyneMint Yeah kind of. The important answer is that all these three holed shapes are topologically "the same", because they are homeomorphic. For 2dim surfaces (bended in 3dim or 4dim space, higher surrounding space makes no difference), the number of handles and the orientability are the only topological variants for compact (finite) real manifolds, except when we have linked holes (like an external hole going through the center of an internal hole), which means there are several surfaces disconnected from each other. Linking is only relevant for 2dim manifolds embedded in 3dim space, because the holes/surfaces can always be unlinked in 4dim space. Topology is trickier for bended 3dim bodies/"surfaces" though, and it gets worse in higher dimensions. And non-compact manifolds are also different, especially if they have an infinite number of handles going through each other. Also there are topological spaces that are not real manifolds, and some that are not even chain complexes composed of real manifolds of different number of dimensions.
I was taking my topology final today and the last proof required me to invoke _that_ one classical theorem, which I couldn’t remember for the love of my life. Then suddenly I thought of this video, and at that very moment, my mind went: “Ah yes, a classical theorem of topology states that *blblblbrblbrlrbrlrbrbbbbrrrr* .” This video saved my GPA!
Please go to Cliff's site and buy some of his manifolds. You are not just buying manifolds, but an entire unique experience of doing business with this man. Highly recommended!
I can't decide if I'm more impressed by the math or the glass blowing skills.
me too hahaha
sithlordmaster181 there was no math....
patsdude54 Topology has a lot to do with math. its like saying geometry isnt math
sithlordmaster181 that deserves a million likes brah
im more impressed by the glass blowing cuz forging a knife is already hard. but thats a whole 'nother level. if it were made of metal it would be even more impressive.
“ Is it okay I’m pointing with my nose?” And this man is amazing.
Such wholsomness.
3:25
No, it's weird, but please keep doing it :)
😂I want a 3 handled coffee mug
??.
Started with an advanced topology problem. Spent hours making multiple complicated glass spheres. Made them into a mug. Worth it.
Any advanced maths problem will invariably end with the mathematician at the pub.
@@meekrab9027 Oddly enough, he does not drink. The story goes that his dad told him that he would let him know when he is ready to drink alcohol. His dad died, never telling this mathematician when he was ready to drink, and so he never did. He tells it better.
@@bonniejeandominguez656 is that really true? If so, I applaud him for sticking to his morals. That's a hard thing to find nowadays.
@@TheGauges420 It is. It's in a Ted Talk he gave, if memory serves me right.
5:21
This guy is great. He's always smiling and you can tell he really enjoys what he's doing
Bread doing something you love so much will do that to you
I think he's used the same method to make a 3 holed bong...;-)
So adorable
Awesome
And he put so much work into this
at the beginning of the video i thought "oh nice, he actually made 2-3 things out of glass to show us something"
and then it just escalated!
You should watch his klein bottle video, he's got hundreds of them!
+Renate vd Bent hundreds? HUNDREDS?!?! he has THOUSANDS if not TENS OF THOUSANDS
I doubt anyone else in the world has this kind of sentence in the dedicated wiki page: " He stores his inventory in the crawlspace underneath his home, and accesses it when needed with a homemade miniature robotic forklift."
en.wikipedia.org/wiki/Clifford_Stoll
It's a bit like Alice in Wonderland or Being John Malkovich.
Renate vd Bent don't worry, i've seen them all
that escalated quickly
The conclusion is topologists can't distinguish a donut and a cup of coffee
grats on almost paying attention in topology 101
But they both go great together in the morning👍
I bet they get dizzy from the view of a spaghetti bowl.
(G)old
Three handled coffee mug with a three-holed donut
I love how excited the people on this channel get about math
*meth
Lovely.
Giggity
especially that one guy that looks like oscar isaac
Math is awesome when it's not shoved down your throat and taught in the most boring way possible
"at 1200 degrees Fahrenheit, it is flexible glass"
I love you Cliff, you're so silly.
That smile and excitement signify a magician level mathematician.
TheDiscoMole Everytime we go into Math class, I'm hyped.
You mean a mathamagician
This man is precious.
He makes topology, a notoriously daunting subject, sound whimsical.
Vsauce also makes boring subjects interesting.
@@elliottwilkinson5741 should they do a collab?
@@youngflamer1667YEEEEE
This guy gets so excited, I can't help but smile :)
( ͡° ͜ʖ ͡°)
I'd be happy too if my job was tearing up holes
Remember the rules... you can't tear only stretch
H
THIS MAN JUST TURNED A HOLE-CEPTION INTO A 3 HANDLED COFFEE MUG.
THE GUY IS ABSOLUTELY AMAZING
in addition the mug is also insulated because it has 2 walls
You should thank topology for being able to do that.
@@username_not_found6926 thank
I'm surprised he didn't turn it into a beanie
Definitely not coffee goes in there
this person's dreams must be fantastic!
A dream, in a dream, in a dream, in a Klein Bottle !
Ikr, I need me some of that
Imagine applying rule 34
No kidding! LOL
darrick steele he’s just so fun
I love how this guy takes interesting topological concepts and turns them into useful everyday items like hats and mugs
Where on earth would we be without three-handled coffee mugs?
You can more easily win some beer-drinking competition, you would have more grip on the glass
Not if the handles are hollow!
OMG a recent post on an old video.
Grimlock - and this makes u happy?
I love this man, his enthusiasm is just fantastic. If I had a teacher like that back in school maybe I wouldn't hate math so much.
Yes, indeed. Made a mistake.
Our world geography teacher is like this and it's awesome. I definitely think it's helping our grades
***** Well I meant that kind of enthusiasm that makes you happy to learn something. Not many teachers like that.
Everybody loves Klein Bottle Guy :D
We had a Chemistry teacher in secondary school like him and a math teacher as well made you want to learn.
The enthusiasm is infectious, never have I ever been so interested in so many holes in all my life.
Yeah, one time I saw my step sis get stuck in the washing machine, let's just say I was busy explorin' some holes too.
So.... you stopped being an asexual?
@@youngflamer1667 I saw that comment coming. 😂
Suppose the absolute majority of us are interested in holes of some kind, especially since both sexes have them. 😏🤭
Unfortunately (?) they are not real topological holes, but merely "caverns".
So what I got from this is that a regular donut is topologically equivalent to a regular coffee mug (assuming the handle was hollow). That must be why they go so well together.
Policemen are geniuses!
Underrated comment! LikeLikeLike
One of the top 10 reasons for not doing your Math homework is that you took a break for a cup of coffee and a doughnut and spent the rest of the evening trying to figure out which one to dunk.
It gets better. Humans are topological donuts as well.
+Drecon84 President Kennedy said "Ich bin ein Berliner." A Berliner is a donut. JFK knew some topology.
5:20 is why Cliff Stoll is the best person on Numberphile
It makes one so happy ^^
Exactly what I thought haha
+PlexyPanda Get a life!
I agree. His enthusiasm and charm is infectious :)
+Penny Lane You deserve 1-up vote
I love this guy, he's like the crazy old professor I never had.
Chloe Dropbear I know, he is the perfect image of the crazy old genius who everyone thinks is insane!
Great Scott!
Yes I agree, you now have 360° of likes ☺👍🏽
Reminds me of Doc from back to the future (an old movie)
I know many have already mentioned how wholesome his enthusiasm is, but at 5:21 when cliff goes "hee hee hee" excitedly is the most wonderful thing ever
Now put coffee into your 3 handled mug and dunk a 3 holed doughnut into it.
Second best comment so far, only to the Rick and Morty comment lol
hahaha creativity peaked. getting flashes of the situation.
why not dunk a 3 handled doughnut? it can be done
RNG How are you going to make a 3 holed taurus mug work? It won't be able to hold anything.
The 3 holed taurus mug was in the video @ 8:10. Take the sphere with 3 handles, push the top down into the bottom and it actually makes a vacuum mug - in that it has an empty space between the inside and outside walls of the mug... 3 holed doughnuts, also totally do-able.
"Is it okay if I'm pointing with my nose?"
The important questions in life
Yes
Voldemort triggered 🙃
You should use everything in your disposal....
This man makes the sun shine.
The moment I saw his hairstyle, I knew he meant buiseness.
I refuse to believe someone actually disliked this video.
+Alexey Saranchev it happens
Ikr
You are seeing things. Maybe an eyecheck and glasses will make the dislike bar that magically appeared go away?
Probably theists.
+joseph crosby mecham So what you are saying is this is like a Turing test, but based on topology… a "Torus test"?
He either made those all himself, or we have a VERY confused glass worker somewhere.
+Dritto1010 sizes? Like 0, 0.0, 0.00, 0.000, etc.?
+Dritto1010 a Klein bottle as no volume (v = 0), that's the joke
*but* this only goes for true four-dimensional klein bottles, not the ones he actually has, right?
Doesn't a Klein bottle have infinite volume minus all other actual volumes?
but it it would have infinite volume then as 'all other actual volumes' are
I was waiting to see thousands of these spheres under his house :(
Well he has a couple (hundred) of Klein Bottles lying there already :P
+BloggingLP That's why I said that xP
He can't enter or leave his house.
Diru
I suspected so. Sry if it came off that way, wasn't intended.
+BloggingLP I don't speak too much english, so was my fault, I didn't understand very well
Find someone who looks at you the way this guy looks at a three-holed taurus.
TheOfficialCzex torus, you mean. Taurus is a whole scarier thing.
@@angelmendez-rivera351 I would be more scared by a three holed torus charging at me at 35 mph
🐂😣😃
@@angelmendez-rivera351 I'm a Taurus and I can confirm this statement 😏
*torus. Not the bull.
"Morty! Morty I *urp* I made a hole in a hole in a hole Morty, its *urrrp* hole-inception up in here Morty!"
"Aww jeez Rick, w-why did you make so many holes, Rick?"
"You ever try to drink from a mug with 3 handles? Its *urrrp* Its amazing Morty. Its topological magic Morty!"
HAHAHA! He does look like Rick!
You just made my day, sir.
I love you.
rick and morty is trash
Pablo Griswold Its' the oURRPnly way to hide it from space cops, Morty!
I just imagined offering that first object to a pre-industrial King. If you weren't executed as a magician you'd probably be able to trade one of those for a whole life of luxury.
+Tristan Ridley love that thought
you have to imagine that if enough people attempt hit like some will accidentally dislike it.
you have to imagine that if enough people attempt hit like some will accidentally dislike it.
Why would you be executed as a magician ? A pre-industrial King would just be amazed by the fact it is transparent glass, and also probably a lot harder to break than ordinary glass.
Otherwise, it's really not so weird. It's not because it has funny topologic properties that it would look like magic... (and it really doesn't).
You mean an item like this one?
en.wikipedia.org/wiki/Cage_cup
sure it was a luxury item fit for a king, but its maker was probably a slave and in no way entitled to a "life of luxury".
Topologically it's also much more complex.
This guy must make the coolest bongs ever
And this is *EXACTLY* why I’ve loved topology and mathematics since I was a child!
Had no idea what "homeomorphic" meant until he said "This is homeomorphic to *this!*"
Then the whole video clicked into place.
homeo meaning same, morphic meaning shape, literally its definition is something of the same shape, and in this case, possessing all the same features; three holes.
Homeomorphic simply means you have a function that is continuous both ways. Continuous functions are something you learn about first thing in calculus.
And the function needs also to be a bijection
Federico Mangano A function has no inverse in the first place without being a bijection
A 3 handled mug?! Let Tadashi tap it with his spoon
Ike!
*ikr
My thoughts exactly!
I'd tap that
+ oh yeah, wanted to say that
I wish my math professor teach me that lessons with that enthusiasm and energy.
Vapor Wave - sama taught*
That's stretching things a bit, wouldn't you say?
3buffalo13 badum tss
the day he stretches things a bit is the day we move into the second dimension
I'm obligated to say I am aware of the pun. I liked it 🙂
I see what you did there... lol
Gee, Doc, that's wild. Can't we get back to saving our future though? --McFly
This made my day.
how does this not end as Klein bottle???
watch the extra footage!!!
Wild guess: A torus with 3 holes has an inside and an outside, while the Klein bottle doesn't - so it's impossible for the two to be homeomorphic.
@Bl00drav3n after watching the video im sure youre right but it wasn't obvious to me as i was watching the video. i thought for sure i knew where it was headed.
make more of these!
There are times in life when you have to put away the bottle and start a hole new life.
The world is seriously lacking more of Cliff. What an energetic guy!
This is mathematical art! You should display these as a sculptural art piece, and when people ask what it means, you can blow their minds by pointing at the last one in the sequence and saying, "This one with the handles is exactly/technically the same as," -points to the first one in the sequence- "this one"! Then you can go into this explanation to both blow their minds even more as well as educating them!
The typical "mad scientist" laugh :D
If he smoked he'd get the mad scientist laugh. Now he has the merry old elf laugh.
5:22
VidkunQL jjajajajjajaja
Dan thanks for sharing such a passion.
That’s how Professor Emmett Brown invented his time machine :)
Came to the comments looking for a Doc Brown reference. I think the character was based on this guy.
ikr
someday an archeologist is going to find these and be fully confused.
ahahha indeed
i'm not sure if they would resist the test of time...
Glass does not decay, so as long as they are in a safe place they will last forever
They would probably say it had ritualistic use, just like todays archaeologists say to cover up what they don't understand.
soufang what if it's more than 1 million years into the future? then it'd decay
At least once in my life, I want someone to look at me the same way this guy looks at math.
Then cultivate your mathematical interests, do some hobby research, and date an enthusiastic mathematician! 😄
We do have a childlike enthusiasm for beauty and life, and lots of us have charm, humor and some seductive skills. 😊
I cannot imagine how long it took to make all these examples.
Those are the videos I want to see
This man has thousands of klein bottles under his house
MarioFanaticXV that's what I'm wondering. It looks like it would take awhile.
111SPIDERPIG999111
Yeah, but it's a lot easier to make lots of the same object than a few of many objects thanks to mass production. The whole reason he has so many is because he needed enough that it would be profitable to make them en masse.
so are you suggesting that I might buy one somewhere!?
This is distracting me from my video-games.
Games? What video games? xD
TheGanamaster
I was trying to play GTA V but I kept turning my head to watch this video
BigMobe lol
same
This is my best proof about Flt
Define
Sx=1+2^2+3^2+4^2+....+x^2.=x(x+1)(2x+1)/6=(2x^3+3x^2+x)/6
Sy=1+2^2+3^2+4^2+....+y^2=y(y+1)(2y+1)/6=(2y^3+3y^2+y)/6
Sz=1+2^2+3^2+4^2+....+z^2=z(z+1)(2z+1)/6=(2z^3+3z^2+z)/6
So
2x^3=6Sx-3x^2-x
2y^3=6Sy-3y^2-y
2z^3=6Sz-3z^2-z
so
x^3=3Sx-3/2x^2-x/2
y^3=3Sy-3/2y^2 - y/2
z^3=3Sz -3/2z^2-z/2
supose
x^3+y^3=z^3
3Sx-3/2x^2-x/2+3Sy-3/2y^2 - y/2 - (3Sz -3/2z^2-z/2)=0
or
2Sx-x^2-x/3+2Sy-y^2 - y/3 - (2Sz -z^2-z/3)=0
or
2Sx+2Sy-2Sz-(x^2+y^2-z^2) =(x/3+y/3-z/3)
So
Sx+S(x-1)+Sy+S(y-1)-Sz-S(z-1)=(x+y-z)/3
So
(x+y-z)^2=[3Sx+3Sy+Sy+3S(y-1)-3Sz-3S(z-1)]^2
because
(x+y)^2=x^2+y^2+2xy
(x+y-z)^2=(x+y)^2+z^2-2z(x+y)=x^2+y^2+2xy+z^2-2z(x+y=x^2+y^2-z^2+2xy-2z(x+y )+3z^2
So
(x^2+y^2 - z^2)=(x+y-z)^2+2xy-2z(x+y )+3z^2=[3Sx+3Sy+Sy+3S(y-1)-3Sz-3S(z-1)]^2+(2xy-2zx-2zy )+3z^2
And
(x^2+y^2-z^2)=(2Sx+2Sy-2Sz)-(x+y-z)/3
So
[3Sx+3Sy+Sy+3S(y-1)-3Sz-3S(z-1)]^2+(2xy-2zx-2zy )+3z^2=(2Sx+2Sy-2Sz)-(x+y-z)/3
So
(2xy-2zx-2zy )=(2Sx+2Sy-2Sz) - [3Sx+3Sy+Sy+3S(y-1)-3Sz-3S(z-1)]^2-(x+y-z)/3
Named
(2Sx+2Sy-2Sz) - [3Sx+3Sy+Sy+3S(y-1)-3Sz-3S(z-1)]^2 is a
Note a is an integer.
So
(2xy-2zx-2zy )=a-(x+y-z)/3
So
6xy=[3a-(x+y-z)+6zx+6zy.]
So
x=[3a-(x+y-z)+6zx+6zy.]/6y
Because x is integer so
[3a-(x+y-z)+6zx+6zy.]=K6y
This is the first condition that to be added
And
6zx=6xy+-2zy - 3a-+x+y-z
So
x=[6xy+-2zy - 3a-+x+y-z]/6z
This is second condition that to be added
And
6zy=6xy-6zx-3a+x+y-z
So
y=[6xy-6zx-3a+x+y-z]/6z
Thie is third condition that to be added
And
z=[6xy-6zx-3a+x+y-z]/6y
This is fourth condition to be added
Total have 4 conditions that to be added
6zx=6xy+-2zy - 3a-+x+y-z
x=[6xy+-2zy - 3a-+x+y-z]/6z
y=[6xy-6zx-3a+x+y-z]/6z
z=[6xy-6zx-3a+x+y-z]/6y
[3a-(x+y-z)+6zx+6zy.]=K6y
[6xy+-2zy - 3a-+x+y-z]=h6z
[6xy-6zx-3a+x+y-z]=g6z
[6xy-6zx-3a+x+y-z]=f6y
Attention! they are different
With 4 conditions added It will definitely hinder the problem solving.
I wish i could be as enthusiatsic in school, as this guy is about topology.
About everything, actually.
Your glass creations belong in a museum. Far better than any computer graphic morphology. Absolutely mind-blowing.
In Belgium we actually have a folklore about a 3-eared pot (google for "Pot van Olen")
Thought it might be interesting for you people.
thats definitely interesting for tadashi!
If someone told me about that folklore without naming any names or other easily recognizable facts, and then asked me: What country is this legend from? I would guess it's somewhere from Germany or Belgium or Netherlands.
You just seem like a holey-type of people. I like that.
Topologist: someone who can't tell the difference between a donut and a coffee cup.
I remember my teacher making that joke on my first year as an undergrad student Hahahaha
The newest one now is : a topologist is a person who can't tell a three handled mug from a hole in a hole in hole.
If I'm honest, I can't either before 10am
If a 3-hole donut translates to a 3-handle cup, a normal donut translates to a normal cup!
Buy does a hyper-donut translate to a hyper-cup?
(I asked in jest, but upon thinking about it, I really have no idea. Does anyone know?)
I wonder what that three handled mug sounds like when you hit with a spoon everywhere
a 6-fold symmetry I would guess? he should definitely try it
yeah, but the handles have holes in them, wouldn't that change things? at least musically speaking.
It would change the pitch but I don't think much else
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You could say this takes topology to a HOLE new level!
No?
Okay, I'll leave...
WHY DOESN'T THIS COMMENT HAVE MORE LIKES
@jokar jokar I think you mean a "hole-some" pun
While I read this pun, the only hole I'm interested in making is a BULLET HOLE through my skull
Hole-y me!
In the end all mathematicians just want coffee and donuts
xD
Great video numberphile
Andy Chamberlain That'a actually a beer glass
I think you can only dip 3 holed donuts in it though.
Damn you guys got doc brown to star in one of your videos that's impressive
More then one
*than
Andrew Korkush no firts he wants more and then he wants 1
I do not understand...
Watch all their "Klien bottle"-videos! This guy is so wonderful! He's so excited by maths.
Thumbs up if you want Klein bottle man for president!!
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YES!
For some odd reason though, Trump Klein bottles are unusually small.
It's not the size that matters. It's how you use it.
Isaac Briefer Shut up and take my upvote!
Glass Blower: I'm sorry Mr Klein, you want *WHAT* in *WHERE* in *WHAT* , _this time?_ Maybe you should take over.
on his first klein bottle video he said he had a friend that had made a klein bottle out of glass with him, the thousands of klein bottles were ordered to a company, but these... these are just too specific, I'm pretty sure he's his own glass blower now, and a master at that
btw fyi he is Cliff Stoll not Felix Klein but It doesn't matter anyways
@@Xthis1s4youX I always assumed he enjoyed doing glass-blowing and never stopped. Just decided that he couldn't make a 1000 Klein bottles by himself, so outsourced.
There's no wáy that anyone else would make all these for him :P
2011 - Welcome to my new channel, Numberphile, a channel about numbers. The first video will be about the number 11.
2016 - Let's talk about a hole in a hole in a hole.
You DIDNT get him to tap it with a spoon in 12 spots?! But... but.... 3 handles!!!
RIGHT?!
RIGHT?!
RIGHT?
LEFT?
(doublepost happened because of a youtube error)
I really hope this is the second part of a trillogy
This man is really cool! I love that he talks so excitedly about topography. And all that glass shapes!
*topology, topography is something else
omg I'm so embarrassed. ^^yeah
Pro tip: You can edit comments
_Yes_ I agree! It is very •cool• and the theory holds up
I love how he giggles and enjoys what he has done. It gives so much life to a dry subject.
He's freaking awesome! He's crazy, but clearly loves what he does.
At 5:22 I wish I could get that excited about math
The man likes to stretch holes. Who can blame him?
You win
This comment is underrated.
1:35
W
( ͡° ͜ʖ ͡°)
HOW MANY OF THESE THINGS DOES HE HAVE??? 😆👍
He has a basement full of about 1000 Klein bottels
just 1, hes just pulling on holes like magic
Yes.
I thought he put the Klein bottles under his house
@@nickramirez5310 But is the house outside the Klein bottles or inside them?
omg get one of those three handled mugs to Tadashi!!
Exactly what I thought.
I though of Tadashi too! What sounds does it make at the nodes?
He already did that.
What's a tadashi?
Tadashi Tokieda, one of the other video series on Numberphile. His videos focus mostly on physical interaction between objects. I found some videos interesting, eg cutting Mobius paper band and merging paperclips
I'm always amazing that this guy isn't surrounded by broken glass and covered in cuts. Gotta love him. :)
why does that make you amazing?
I neglected to mention that these glass models are not just homeomorphic to each other, they are also isotopic to each other. Isotopic is a strong property than homeomorphic: Homeomorphic means that there's a mapping between them that preserves all their topological information. Isotopic goes further and demands a continuous deformation that is a homeomorphism at each step.
Very interesting! Is there an example of two objects that are homeomorphic but not isotopic?
I wanna know that too
oops - I answered your question, but it shows up elsewhere.
[repeated from elsewhere]
Two objects that are homeomorphic but not Isotopic? Sure ...
You usually make a Möbius band by giving a strip of paper a half twist, and taping their ends together ... the result has one edge, one side, one hole, and is non-orientable. If you give the strip of paper three half twists ( = 3/2 twists), the result has the same properties: one side, one hole, and non-orientable.
So these two things (the 1/2 twist Möbloop and the 3/2 twist Möbloop) are homeomorphic to each other -- they have the same topological properties. However, they are not isotopic to each other ... you can't continuous deform a 1/2 twist Möbloop into a 1 1/2 twist Möbloop. (You gotta cut the 1/2 twist guy, add a full twist, and tape it back together).
I thought isotopic was just a standard used to define chocolate bar hazelnut content.
Just when I wouldn't think it possible, he'd pull out yet ANOTHER glass construct. That was the most mind-blowing part of this video, if you ask me.
Tadashi this guy's got your cup!
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jeez, how big is this guy's glass collection? xD
Why do you think he's so happy all the time?
He has a robot to sort his thousands of glass models.
I want a Klein bottle bong so I can get fourth-dimensionally high.
He has 1000 Klein Bottles on his basement. There's a video about it.
If I would think, his house is probably one of those kienbottle-houses they showed. Möbius buildings I think they're called
Cool! I have no idea what any of that means lol
It must be great fun to be able to just "go into the glass shop" and *make* these things. :-)
This made me whish to become a glassworker myself. Just for the minute chance, that someone like him pops in my shop, demanding me making such amazing things :D
That must be one happy glass shop down the street.
I absolutely love the enthusiasm of this man and everything he does. He reminds me of back to the future.
5:21 is unquestionably the best part
Read the comment yesterday and came back just to see if it's still there and to click on the time-stamp again.
first of all, this is beautiful. the simplicity in explanation and the passion this guy has.
second, imagine how weird out the glassmaker must feel to make all these shapes lol
whyuugly he makes them himself.
He looks like a mad scientist.
I think that has a reason. :D
he is
All scientists are mad, but this one is the madest
I feel the need to point out that maths isn't a science. I love him none the less 😂
A happy one.
Topology: everything is coffee cups.
...
(occasionally with Klein bottles and cross-caps glued on)
This proves that coffee is the secret of the universe. You can beat the Borg with it.
There's coffee in nebula, I can smell it.
Ah, it all comes back to _Star Trek,_ everytime. :-)
Sara Llewellyn After all, the average color value for the universe is the same as latte.
Taking this a step further, you could stretch the two other holes, and spiral them round the central hole in a double helix, then pull the whole thing out to the side to create a one handled coffee mug with a double helix filigree patternation.
Apache ooooooh, I want one of those...
"Patternation" 😊
Apache you can make 1 more hole if you put it in a hole
Beyond the number/math interest, this guy is a fantastic glass blower! Superb! Well done Sir!
But how will you hold the coffee if your cup has three holes?
checkmate, science
from the bottom
You can just hold it like a regular cup...
The Joke
Your heads
The holes are self contained, i.e. There can be a vacuum between the filled part of the vessel and the handles
But how will u pour ur coffee if the spout and the handle are not in a straight line?
Checkmate, mediocrity
He should contact Dr. Tadashi , he needed a 3 handled cup last time :D
yeah lol
Does the original sphere ring the same way the mug does?
I was thinking the same thing; but give him the original sphere and tell him you found one to see what he does
TroZ Well the simmetry is the same so it should behave in a similar way around the edge
a colaboration between the 2 isn't wanted or needed, it is REQUIRED
i want to see what crazy math examples/experiments they can come up with.
Wow! How much glass has this guy got?
Ikr
He makes them himself!
you ought to see what's hidden under his house...
All of it.
This man is amazing! I love the enthusiasm, this video just made me think 'woah, that's totally a thing! COOL!'
How to get your teacher to *stop* asking you to show your work: a summery.
Won't work with an english teacher (for a couple reasons).
@@Milkmans_Son : Why? Just show your hand painted diorama of the most importand battles of the War of Roses or so...
@@robertnett9793 summery vs. summary
@@Milkmans_Son *sigh* this one whooshed over my head. In my defence, I am not a native English speaker :D
@@robertnett9793 Defense ! Haha, are you french by any chance?
That, my friends, is a very expensive proof. oO
I mean, he works in a glass shop lol
It's not THAT expensive. He (or atleast one of his friends) works in a glas shop so they got enough glas around to basicly produce 2 handfull or glas balls.
I want to be just like him when I grow up
you're going to grow up? brave, i'm not
dito
I highly recommend his book, The Cuckoo's Egg. It's fantastic and a really great insight into Cliff Stoll's life. Never know where life will take you. ;)
One of the best books I've ever read. You can't stop reading, you just have to know what happens next.
My alltime favourite professor in numberphile. Like his explanations so much.
How much did all that glass cost?
Edit: Also where will he put all of them?
Fill them with glitter, never buy another Christmas tree decoration
Under his house
e4htysybvgtjyevbdehybrj eb48 ihuter IDK, he makes them himself.
Under his house with the 1000 Kline bottles
He puts them in a hole. Duh.
Under his house with the hundreds of Klein bottles he has.
I'm listening to this without seeing the video.
The voice sort of reminds me of Kermit the frog
I think it would be awesome to have an episode of numberphile with Kermit the frog.
Does this guy make Bongs ?
This man made 18 different glass globes to prove a solution theorized some 100 years ago
Bless his heart.
this guy owns the most amazing junk i wish i owned this level of awesome stuff. hes like let me pull out my 11 ball set of a hole in a hole
I need a T-shirt of him, I don't care how much it costs!
I have 3 tshirts with holes
my t-shirts already have holes in their holes...
I have a t-shirt with 3 holes!
t-shirt with 3 holes ? What sorcery is this ?
And that T-shirt should be made of glass.
A wonderful demonstration of glass blowing and topology
Be careful where you put the handles, otherwise you'll end up with a two-handled mug with a hole in the bottom.
Or if you're crafty, you could instead fashion a one-handled mug with two drinking straws built-in, perfect for sharing a milkshake.
Milkshake from under the bottom of the mug...
Gotcha
...so as long as it is a shape with 3 holes. The answer will be correct?
@@TyneMint Yeah kind of. The important answer is that all these three holed shapes are topologically "the same", because they are homeomorphic. For 2dim surfaces (bended in 3dim or 4dim space, higher surrounding space makes no difference), the number of handles and the orientability are the only topological variants for compact (finite) real manifolds, except when we have linked holes (like an external hole going through the center of an internal hole), which means there are several surfaces disconnected from each other. Linking is only relevant for 2dim manifolds embedded in 3dim space, because the holes/surfaces can always be unlinked in 4dim space.
Topology is trickier for bended 3dim bodies/"surfaces" though, and it gets worse in higher dimensions.
And non-compact manifolds are also different, especially if they have an infinite number of handles going through each other.
Also there are topological spaces that are not real manifolds, and some that are not even chain complexes composed of real manifolds of different number of dimensions.
@@henrikljungstrand2036 Huh... That is a lot to process... Woah
I was taking my topology final today and the last proof required me to invoke _that_ one classical theorem, which I couldn’t remember for the love of my life.
Then suddenly I thought of this video, and at that very moment, my mind went: “Ah yes, a classical theorem of topology states that *blblblbrblbrlrbrlrbrbbbbrrrr* .”
This video saved my GPA!
Please go to Cliff's site and buy some of his manifolds. You are not just buying manifolds, but an entire unique experience of doing business with this man. Highly recommended!
The absolute burning passion for maths that made this guy blow several dozen intricate glass tubes and spheres