A Hole in a Hole in a Hole - Numberphile

Audible: www.audible.com/numberphile
More links & stuff in full description below ↓↓↓
Cliff's glassware: bit.ly/ACME_Klein
Extras for this video: • Extra on a Hole in a H...
More videos featuring Cliff Stoll: bit.ly/Cliff_Videos
Cliff references A Comprehensive Introduction to Differential Geometry by Michael Spivak: amzn.to/2cwmdB7
Animation: Pete McPartlan
Note from Cliff:
Original ideas & suggestions from Peter Nguyen of Coker College and Danny Stoll of U/Chicago
Amazing glasswork by Tom Adams, George Chittenden, Lucas Clarke, and Xing Wei
Support us on Patreon: / numberphile
NUMBERPHILE
Website: www.numberphile.com/
Numberphile on Facebook: / numberphile
Numberphile tweets: / numberphile
Subscribe: bit.ly/Numberphile_Sub
Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): bit.ly/MSRINumberphile
Videos by Brady Haran
Brady's videos subreddit: / bradyharan
Brady's latest videos across all channels: www.bradyharanblog.com/
Sign up for (occasional) emails: eepurl.com/YdjL9
Numberphile T-Shirts: teespring.com/stores/numberphile
Other merchandise: store.dftba.com/collections/n...

[@Cr42yguy]

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!

[@RosesAndIvy]

You should watch his klein bottle video, he's got hundreds of them!

[@s0me1up]

+Renate vd Bent hundreds? HUNDREDS?!?! he has THOUSANDS if not TENS OF THOUSANDS

[@lxathu]

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.

[@Cr42yguy]

Renate vd Bent don't worry, i've seen them all

[@augustorodriguez743]

that escalated quickly

[@sithlordmaster181]

I can't decide if I'm more impressed by the math or the glass blowing skills.

[@urquix96]

me too hahaha

[@patsdude54]

sithlordmaster181 there was no math....

[@kuco30]

patsdude54 Topology has a lot to do with math. its like saying geometry isnt math

[@Aspri987]

sithlordmaster181 that deserves a million likes brah

[@RobertMinsk]

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.

[@CraftQueenJr]

“ Is it okay I’m pointing with my nose?” And this man is amazing.

[@kasai0555]

Such wholsomness.

[@pranavlimaye]

3:25

[@rich1051414]

No, it's weird, but please keep doing it :)

[@That_One_Gaming_Guy]

😂I want a 3 handled coffee mug

[@Triantalex]

??.

[@aghaanantyab]

The conclusion is topologists can't distinguish a donut and a cup of coffee

[@evlredsun]

grats on almost paying attention in topology 101

[@Twinspinner]

But they both go great together in the morning👍

[@VibratorDefibrilator]

I bet they get dizzy from the view of a spaghetti bowl.

[@AndresHernandez-lo9uy]

(G)old

[@MrEdrftgyuji]

Three handled coffee mug with a three-holed donut

[@RealClassixX]

He either made those all himself, or we have a VERY confused glass worker somewhere.

[@evildude109]

+Dritto1010 sizes? Like 0, 0.0, 0.00, 0.000, etc.?

[@Maltanx]

+Dritto1010 a Klein bottle as no volume (v = 0), that's the joke

[@The0Skeleton123]

*but* this only goes for true four-dimensional klein bottles, not the ones he actually has, right?

[@DerBullgod]

Doesn't a Klein bottle have infinite volume minus all other actual volumes?

[@abyssaljam441]

but it it would have infinite volume then as 'all other actual volumes' are

[@marbleswan6664]

Started with an advanced topology problem. Spent hours making multiple complicated glass spheres. Made them into a mug. Worth it.

[@meekrab9027]

Any advanced maths problem will invariably end with the mathematician at the pub.

[@bonniejeandominguez656]

@@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.

[@TheGauges420]

@@bonniejeandominguez656 is that really true? If so, I applaud him for sticking to his morals. That's a hard thing to find nowadays.

[@bonniejeandominguez656]

@@TheGauges420 It is. It's in a Ted Talk he gave, if memory serves me right.

[@Bread-yx2eg]

5:21 This guy is great. He's always smiling and you can tell he really enjoys what he's doing

[@AHappyCub]

Bread doing something you love so much will do that to you

[@maxnullifidian]

I think he's used the same method to make a 3 holed bong...;-)

[@arcalter6075]

So adorable

[@danielclaresmyth]

Awesome

[@TheRealFlenuan]

And he put so much work into this

[@thegeneralissimo470]

"at 1200 degrees Fahrenheit, it is flexible glass" I love you Cliff, you're so silly.

[@tristanridley1601]

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.

[@numberphile]

+Tristan Ridley love that thought

[@finnberuldsen4798]

you have to imagine that if enough people attempt hit like some will accidentally dislike it.

[@finnberuldsen4798]

you have to imagine that if enough people attempt hit like some will accidentally dislike it.

[@Ezullof]

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).

[@SebBrosig]

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.

[@GamesFromSpace]

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.

[@OMGLYKEWOW]

Policemen are geniuses!

[@pedrocavalcanti46]

Underrated comment! LikeLikeLike

[@MerrillMath]

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.

[@Drecon84]

It gets better. Humans are topological donuts as well.

[@colt4667]

+Drecon84 President Kennedy said "Ich bin ein Berliner." A Berliner is a donut. JFK knew some topology.

[@stuberosum1]

THIS MAN JUST TURNED A HOLE-CEPTION INTO A 3 HANDLED COFFEE MUG. THE GUY IS ABSOLUTELY AMAZING

[@a_common_weeb]

in addition the mug is also insulated because it has 2 walls

[@username_not_found6926]

You should thank topology for being able to do that.

[@asheep7797]

@@username_not_found6926 thank

[@mk_rexx]

I'm surprised he didn't turn it into a beanie

[@steelstriderq7801]

The enthusiasm is infectious, never have I ever been so interested in so many holes in all my life.

[@braxtonbarfriquaorme8703]

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.

[@youngflamer1667]

So.... you stopped being an asexual?

[@henrikljungstrand2036]

@@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".

[@AgentMidnight]

"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!"

[@Snakeyes244]

HAHAHA! He does look like Rick!

[@NickiRusin]

You just made my day, sir.

[@misteratoz]

I love you.

[@Linkedblade]

rick and morty is trash

[@NickiRusin]

Pablo Griswold Its' the oURRPnly way to hide it from space cops, Morty!

[@alexeysaranchev6118]

I refuse to believe someone actually disliked this video.

[@numberphile]

+Alexey Saranchev it happens

[@dlee645]

Ikr

[@nicholaslau3194]

You are seeing things. Maybe an eyecheck and glasses will make the dislike bar that magically appeared go away?

[@TheGreatSteve]

Probably theists.

[@Abdega]

+joseph crosby mecham So what you are saying is this is like a Turing test, but based on topology… a "Torus test"?

[@TheMartian11]

The moment I saw his hairstyle, I knew he meant buiseness.

[@TheOfficialCzex]

Find someone who looks at you the way this guy looks at a three-holed taurus.

[@angelmendez-rivera351]

TheOfficialCzex torus, you mean. Taurus is a whole scarier thing.

[@Adahn99]

@@angelmendez-rivera351 I would be more scared by a three holed torus charging at me at 35 mph

[@yazajag]

🐂😣😃

[@elysia4982]

@@angelmendez-rivera351 I'm a Taurus and I can confirm this statement 😏

[@tanveer_badar_]

*torus. Not the bull.

[@Inertia888]

this person's dreams must be fantastic!

[@TiO2_]

A dream, in a dream, in a dream, in a Klein Bottle !

[@gisforgeetpack7299]

Ikr, I need me some of that

[@frillneckedlizard8529]

Imagine applying rule 34

[@idorandomstuff2853]

No kidding! LOL

[@RibbittIII]

darrick steele he’s just so fun

[@avihenriksen3947]

I love how excited the people on this channel get about math

[@Aleksandar0100]

*meth

[@Ivan_1791]

Lovely.

[@IETCHX69]

Giggity

[@stipser1]

especially that one guy that looks like oscar isaac

[@KordintheHusky]

Math is awesome when it's not shoved down your throat and taught in the most boring way possible

[@Mel-bq4jd]

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

[@3buffalo13]

That's stretching things a bit, wouldn't you say?

[@MrCubFan415]

3buffalo13 badum tss

[@fidelio6311]

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 🙂

[@lkkoller]

I see what you did there... lol

[@thepoopenator7341]

5:20 is why Cliff Stoll is the best person on Numberphile

[@TheSweBoo]

It makes one so happy ^^

[@guyhommeki]

Exactly what I thought haha

[@unvergebeneid]

+PlexyPanda Get a life!

[@Madsy9]

I agree. His enthusiasm and charm is infectious :)

[@Madsy9]

+Penny Lane You deserve 1-up vote

[@TheDiscoMole]

That smile and excitement signify a magician level mathematician.

[@zerogenic]

TheDiscoMole Everytime we go into Math class, I'm hyped.

[@ThousandStars1000]

You mean a mathamagician

[@gaarathesandslayer]

At least once in my life, I want someone to look at me the same way this guy looks at math.

[@henrikljungstrand2036]

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. 😊

[@rannov.4707]

You could say this takes topology to a HOLE new level! No? Okay, I'll leave...

[@Sztefa001]

WHY DOESN'T THIS COMMENT HAVE MORE LIKES

[@ChaoticTeen16]

@jokar jokar I think you mean a "hole-some" pun

[@pranavlimaye]

While I read this pun, the only hole I'm interested in making is a BULLET HOLE through my skull

[@gallium-gonzollium]

Hole-y me!

[@jaredhoeft2832]

This man is precious. He makes topology, a notoriously daunting subject, sound whimsical.

[@elliottwilkinson5741]

Vsauce also makes boring subjects interesting.

[@youngflamer1667]

@@elliottwilkinson5741 should they do a collab?

[@unfig3034]

This guy gets so excited, I can't help but smile :)

[@t1glistenerelf]

( ͡° ͜ʖ ͡°)

[@JeyPeyy]

I'd be happy too if my job was tearing up holes

[@mothcatcher893]

Remember the rules... you can't tear only stretch

[@cs4all670]

H

[@DeannaEarley]

And this is *EXACTLY* why I’ve loved topology and mathematics since I was a child!

[@GweiTheLeafChild]

Glass Blower: I'm sorry Mr Klein, you want *WHAT* in *WHERE* in *WHAT* , _this time?_ Maybe you should take over.

[@Xthis1s4youX]

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

[@hamiltonianpathondodecahed5236]

btw fyi he is Cliff Stoll not Felix Klein but It doesn't matter anyways

[@omikronweapon]

@@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

[@aidanfuge2108]

I love how this guy takes interesting topological concepts and turns them into useful everyday items like hats and mugs

[@sender1496]

Where on earth would we be without three-handled coffee mugs?

[@VAFFANFEDE18]

You can more easily win some beer-drinking competition, you would have more grip on the glass

[@chronosorion6911]

Not if the handles are hollow!

[@razpootis5802]

OMG a recent post on an old video.

[@IronSmeltery]

Grimlock - and this makes u happy?

[@inwemeneldur2025]

"Is it okay if I'm pointing with my nose?"

[@nomekop777]

The important questions in life

[@EvonixTheGreatest]

Yes

[@desmondperis1457]

Voldemort triggered 🙃

[@robertnett9793]

You should use everything in your disposal....

[@DreckbobBratpfanne]

HOW MANY OF THESE THINGS DOES HE HAVE??? 😆👍

[@radulescubg]

He has a basement full of about 1000 Klein bottels

[@brandonwithnell612]

just 1, hes just pulling on holes like magic

[@eurickevardone2976]

Yes.

[@nickramirez5310]

I thought he put the Klein bottles under his house

[@clickrick]

@@nickramirez5310 But is the house outside the Klein bottles or inside them?

[@JK-wt7rc]

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!

[@canyousmoketho]

He's so proud of himself, he seems so happy.😊

[@HosamSherif]

Yeah, it's great to see someone so passionate about a subject.

[@jacobmock6682]

I Lost My Halo yeah and he just wasted 13 pounds of glass on this video

[@Assultmaster1]

Jacob Mock wasted?

[@Assultmaster1]

Jacob Mock i mean, have you Seen techrax?

[@mikesekulovski7966]

Word has it he's been playing with glass for the last six months, this is his first contact with the outside world in over a year.

[@JamesRevan]

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.

[@JamesRevan]

Yes, indeed. Made a mistake.

[@TheRedstoneTaco]

Our world geography teacher is like this and it's awesome. I definitely think it's helping our grades

[@JamesRevan]

***** Well I meant that kind of enthusiasm that makes you happy to learn something. Not many teachers like that.

[@cellina.starfire]

Everybody loves Klein Bottle Guy :D

[@MrRobinhalligan]

We had a Chemistry teacher in secondary school like him and a math teacher as well made you want to learn.

[@e4htysybvgtjyevbdehybrjeb469]

How much did all that glass cost? Edit: Also where will he put all of them?

[@jameshughes3014]

Fill them with glitter, never buy another Christmas tree decoration

[@aedile2819]

Under his house

[@Alex-wi1mx]

e4htysybvgtjyevbdehybrj eb48 ihuter IDK, he makes them himself. Under his house with the 1000 Kline bottles

[@EricH928]

He puts them in a hole. Duh.

[@IndigoGollum]

Under his house with the hundreds of Klein bottles he has.

[@chrischiesa3253]

It must be great fun to be able to just "go into the glass shop" and *make* these things. :-)

[@robertnett9793]

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

[@diegofonseca5096]

I was waiting to see thousands of these spheres under his house :(

[@The_Blog]

Well he has a couple (hundred) of Klein Bottles lying there already :P

[@diegofonseca5096]

+BloggingLP That's why I said that xP

[@OwenPrescott]

He can't enter or leave his house.

[@The_Blog]

Diru I suspected so. Sry if it came off that way, wasn't intended.

[@diegofonseca5096]

+BloggingLP I don't speak too much english, so was my fault, I didn't understand very well

[@craigroberts7013]

how does this not end as Klein bottle???

[@numberphile]

watch the extra footage!!!

[@Bl00drav3nz]

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.

[@craigroberts7013]

@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.

[@Theraot]

make more of these!

[@lxathu]

There are times in life when you have to put away the bottle and start a hole new life.

[@subven69]

Bless his heart.

[@whyuugly]

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

[@CraftQueenJr]

whyuugly he makes them himself.

[@tommythai1043]

A 3 handled mug?! Let Tadashi tap it with his spoon

[@eduardocortez476]

Ike!

[@eduardocortez476]

*ikr

[@gcwarkenyout]

My thoughts exactly!

[@johannaguerrero1790]

I'd tap that

[@adityakhanna113]

+ oh yeah, wanted to say that

[@dan339dan]

The typical "mad scientist" laugh :D

[@VidkunQL]

If he smoked he'd get the mad scientist laugh. Now he has the merry old elf laugh.

[@Roth-kana]

5:22

[@zlsoypalomocojo1659]

VidkunQL jjajajajjajaja

[@MichaelRWolf]

Dan thanks for sharing such a passion.

[@BritishBeachcomber]

Your glass creations belong in a museum. Far better than any computer graphic morphology. Absolutely mind-blowing.

[@slinky6481]

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.

[@tally9542]

I love this guy, he's like the crazy old professor I never had.

[@CraftQueenJr]

Chloe Dropbear I know, he is the perfect image of the crazy old genius who everyone thinks is insane!

[@hirokatsuvictor8755]

Great Scott!

[@yazajag]

Yes I agree, you now have 360° of likes ☺👍🏽

[@madide3978]

Reminds me of Doc from back to the future (an old movie)

[@oceanblue2358]

This man makes the sun shine.

[@Kapin05]

This man is amazing! I love the enthusiasm, this video just made me think 'woah, that's totally a thing! COOL!'

[@Speireata4]

I love how he giggles and enjoys what he has done. It gives so much life to a dry subject.

[@BigMobe]

This is distracting me from my video-games.

[@TheGanamaster]

Games? What video games? xD

[@BigMobe]

TheGanamaster I was trying to play GTA V but I kept turning my head to watch this video

[@Chelsea2009FC]

BigMobe lol

[@SamuelHabekotte]

same

[@prajnaprajna1923]

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.

[@ilghiz]

That’s how Professor Emmett Brown invented his time machine :)

[@ezequielblanco8659]

Came to the comments looking for a Doc Brown reference. I think the character was based on this guy.

[@HideoMatsuoka]

ikr

[@argonman1]

Beyond the number/math interest, this guy is a fantastic glass blower! Superb! Well done Sir!

[@Thomas-hv1xh]

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.

[@XenophonSoulis]

Milkshake from under the bottom of the mug...

[@leddaudet2350]

Gotcha

[@TyneMint]

...so as long as it is a shape with 3 holes. The answer will be correct?

[@henrikljungstrand2036]

@@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.

[@TyneMint]

@@henrikljungstrand2036 Huh... That is a lot to process... Woah

[@adamtaylor1739]

Thumbs up if you want Klein bottle man for president!!

[@mecca4521]

+++

[@isaacbriefer193]

YES!

[@isaacbriefer193]

For some odd reason though, Trump Klein bottles are unusually small.

[@deralmighty8011]

It's not the size that matters. It's how you use it.

[@MrCubFan415]

Isaac Briefer Shut up and take my upvote!

[@fawnjenkins5157]

Gee, Doc, that's wild. Can't we get back to saving our future though? --McFly

[@bibasik7]

This made my day.

[@TheZANCAT]

This guy seems like he's about to pop into a hole himself, taking Alice to wonderland

[@chilling_at_pontiff]

This man made 18 different glass globes to prove a solution theorized some 100 years ago

[@bstoppel1]

someday an archeologist is going to find these and be fully confused.

[@valeriobertoncello1809]

ahahha indeed

[@iota-09]

i'm not sure if they would resist the test of time...

[@soufang]

Glass does not decay, so as long as they are in a safe place they will last forever

[@Hedning1390]

They would probably say it had ritualistic use, just like todays archaeologists say to cover up what they don't understand.

[@MercurySG3M]

soufang what if it's more than 1 million years into the future? then it'd decay

[@metleon]

Now put coffee into your 3 handled mug and dunk a 3 holed doughnut into it.

[@strafeae4618]

Second best comment so far, only to the Rick and Morty comment lol

[@143TYAGI]

hahaha creativity peaked. getting flashes of the situation.

[@HanibiSW]

why not dunk a 3 handled doughnut? it can be done

[@metleon]

RNG How are you going to make a 3 holed taurus mug work? It won't be able to hold anything.

[@samajammy]

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.

[@tparadox88]

This video answered a question I had since I was a kid and I saw a traveling topology exhibit at a children's museum. Has Numberphile ever covered turning a sphere inside out? I thought I remembered seeing it on this channel, but I couldn't find it. Might not be as impressive as the legendary Outside In, but also wouldn't be as creepy.

[@corncolonel9171]

This is the most beautiful man I've ever seen

[@leddaudet2350]

BoBoMosasaur then u haven’t saw me yet 😉sorry I’m too high a bit of comprehension

[@martiddy]

I wish my math professor teach me that lessons with that enthusiasm and energy.

[@zenny3859]

Vapor Wave - sama taught*

[@CrashperM]

Damn you guys got doc brown to star in one of your videos that's impressive

[@user-ux2dd9pp8x]

More then one

[@thecrazylovelyboy]

*than

[@CrashperM]

Andrew Korkush no firts he wants more and then he wants 1

[@thecrazylovelyboy]

I do not understand...

[@LeCaNiVideos]

Watch all their "Klien bottle"-videos! This guy is so wonderful! He's so excited by maths.

[@Jaojao_puzzlesolver]

I love how added some sound effects by his mouth to illustrate like 2:28 😂

[@asylumarcade]

王兆成 I love the way he says the phrase "were gonna straighten it out" right afterwards

[@hammond1994]

I would love to see a video of these things being constructed. The glass blower must be a Grand Master of his craft.

[@ptt619]

This guy must make the coolest bongs ever

[@firelow]

I wonder what that three handled mug sounds like when you hit with a spoon everywhere

[@starofcctv94]

a 6-fold symmetry I would guess? he should definitely try it

[@iota-09]

yeah, but the handles have holes in them, wouldn't that change things? at least musically speaking.

[@starofcctv94]

It would change the pitch but I don't think much else

[@fossilfighters101]

+

[@siddharth_desai]

+

[@nakamakai5553]

Having read (and been absorbed / amazed by) The Cuckoo's Egg in my youth, it's been just a blast to discover all the Cliff videos all these many decades later. I never knew what he'd gone on to do, and now I enjoy every minute of his passions on-screen. I was in zoology at the time, with a heavy emphasis on math and theoretical ecology, and I must conclude that his book contributed heavily to the fact that I've now spent 30+ years in IT, nearly all of it in security. What a great inspiration he is.

[@duckmeister5385]

"But why?" --The Glassblower

[@calpitruzzello3544]

At 5:22 I wish I could get that excited about math

[@mrvolans7862]

I wish i could be as enthusiatsic in school, as this guy is about topology.

[@oz_jones]

About everything, actually.

[@Lynn-rv4ty]

it is so adorable how excited he is about not only the math but also about his glass creations

[@theoverseer393]

I love the enthusiasm in this. That 3 handled mug is awesome

[@MarioFanaticXV]

I cannot imagine how long it took to make all these examples.

[@markgriz]

Those are the videos I want to see

[@Cragnathar]

This man has thousands of klein bottles under his house

[@coolbrotherf127]

MarioFanaticXV that's what I'm wondering. It looks like it would take awhile.

[@MarioFanaticXV]

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.

[@NoahWenzel]

so are you suggesting that I might buy one somewhere!?

[@carlacamargo4158]

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.

[@n124lp]

I really like the idea of making the solids out of glass. Excellent conception and execution.

[@Linkophere]

When someone teaches with passion you can't help but wanna hear what they have to say. Great work

[@lirothen]

You DIDNT get him to tap it with a spoon in 12 spots?! But... but.... 3 handles!!!

[@NemosChannel]

RIGHT?!

[@NemosChannel]

RIGHT?!

[@fofolp1213]

RIGHT?

[@NemosChannel]

LEFT? (doublepost happened because of a youtube error)

[@fafetico]

I really hope this is the second part of a trillogy

[@AgentMidnight]

The world is seriously lacking more of Cliff. What an energetic guy!

[@Otoma]

Cliff is a legend.

[@RaindropsBleeding]

I love your videos. Your fascination with glass work and pushing the boundaries of three dimensional topology is intoxicating.

[@trevorhunt]

Tadashi this guy's got your cup!

[@GerSHAK]

+

[@d3j4v00]

+

[@ethanferguson9530]

+

[@kurtbruneau]

+

[@pendrife]

+

[@atklobas]

Topologist: someone who can't tell the difference between a donut and a coffee cup.

[@KleinGrass]

I remember my teacher making that joke on my first year as an undergrad student Hahahaha

[@josevillegas2721]

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.

[@dojix96]

If I'm honest, I can't either before 10am

[@L4Vo5]

If a 3-hole donut translates to a 3-handle cup, a normal donut translates to a normal cup!

[@SlimThrull]

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?)

[@xexpaguette]

*Yay! It's Doc from Back to The Future again!*

[@inujosha]

I wish I could have had this man as a professor for anything in college.

[@therealteal620]

Had no idea what "homeomorphic" meant until he said "This is homeomorphic to *this!*" Then the whole video clicked into place.

[@WeAreGRID]

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.

[@TosiakiS]

Homeomorphic simply means you have a function that is continuous both ways. Continuous functions are something you learn about first thing in calculus.

[@VAFFANFEDE18]

And the function needs also to be a bijection

[@TosiakiS]

Federico Mangano A function has no inverse in the first place without being a bijection

[@aircraftcarrier6789]

In the end all mathematicians just want coffee and donuts xD Great video numberphile

[@forloop7713]

Andy Chamberlain That'a actually a beer glass

[@peteman1000]

I think you can only dip 3 holed donuts in it though.

[@CaptDF97]

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!

[@8bitbulldog]

seeing the excitement of this guy is so damn wholesome

[@ortipianinho]

I want to be just like him when I grow up

[@JackSwatman]

you're going to grow up? brave, i'm not

[@fofolp1213]

dito

[@SpeedyRK]

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. ;)

[@henrikkoberstein]

One of the best books I've ever read. You can't stop reading, you just have to know what happens next.

[@PsychologyStud]

He looks like a mad scientist.

[@OhThatsWhoYouRepliedTo]

I think that has a reason. :D

[@M4S73RN19H7M4R3]

he is

[@Bram06]

All scientists are mad, but this one is the madest

[@Cyrusislikeawsome]

I feel the need to point out that maths isn't a science. I love him none the less 😂

[@GottfriedLeibnizYT]

A happy one.

[@thegaminpianistccc3580]

The fact that this guy has the patience to make all these models out of glass, HAND-BLOWN glass, deserves some respect in itself.

[@srenhaandbk7904]

The absolute burning passion for maths that made this guy blow several dozen intricate glass tubes and spheres

[@zackferguson1661]

jeez, how big is this guy's glass collection? xD

[@bigmike8586]

Why do you think he's so happy all the time?

[@Heligoland360]

He has a robot to sort his thousands of glass models.

[@bootblacking]

I want a Klein bottle bong so I can get fourth-dimensionally high.

[@Aletaire]

He has 1000 Klein Bottles on his basement. There's a video about it.

[@davidhansson7041]

If I would think, his house is probably one of those kienbottle-houses they showed. Möbius buildings I think they're called

[@bm-br3go]

He's freaking awesome! He's crazy, but clearly loves what he does.

[@hikaru-live]

Should have ended up with a nice glass of drink in that mug. (And it being double layered means it can have fantastic thermal insulation, perfect for both chilled and heated drinks to be kept for longer.)

[@martinmonath9541]

My alltime favourite professor in numberphile. Like his explanations so much.

[@Dalroc]

omg get one of those three handled mugs to Tadashi!!

[@vnen]

Exactly what I thought.

[@all_time_Jelly_Fish]

I though of Tadashi too! What sounds does it make at the nodes?

[@OctagonCookies]

He already did that.

[@Th4w]

What's a tadashi?

[@Xeverous]

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

[@adaharrisonn]

5:21 is unquestionably the best part

[@UCyYuGkOkjrLqRgKh-_j1oNA]

Read the comment yesterday and came back just to see if it's still there and to click on the time-stamp again.

[@jamspoon]

A wonderful demonstration of glass blowing and topology

[@roccoleader279]

This man makes me so happy he is so full of wholesome energy i love it