Braids in Higher Dimensions - Numberphile

[@numberphile]

More videos with Dr Zsuzsanna Dancso: bit.ly/dancso_playlist Extra bit from this interview: kzbin.info/www/bejne/sGScnGangtOGiK8

[@ozdergekko]

Zsuzsanna is back! Yaaaay!

[@andraskoleszar9759]

Numberphile Could you show this to Zsuzsanna please! Szia Zsuzsi, nagyon tetszett a videó, örülök, hogy ujra itt voltál a csatornán, várom a következő videódat!😀

[@somitomi]

Jó kérdés. Amúgy üdvözlöm Zsuzsannát, ha véletlenül olvasná.

[@balintnk]

egy plusz biztosan

[@gergelybecso8288]

Hmm... Rá kell szokjak, hogy lecsekkoljam az előadó nevét... Magyarok mindenhol :D

[@fetchstixRHD]

The animations are actually amazing and really do help to visualise what's going on.

[@pmcpartlan]

fetchstix Thanks!

[@MathAndComputers]

What software do you use, Pete? Do you have a ton of custom tools/scripts you've developed specifically for Numberphile and related animations? (I've wondered for quite a while.)

[@pmcpartlan]

Hi, just seen this. Nothing specialised really just After Effects and Blender for 3D stuff.

[@spinor]

I was wondering what they meant when they said a Klein Bottle doesn't intersect itself in 4D. This cleared it up nicely, thanks!

[@antoineroquentin2297]

It's amazing listening to people who put so much effort in solving problems from which I didn't even realized that they were problems.

[@Lugmillord]

"how can you prove that we don't live in 4 dimensions?" "My braids don't untangle."

[@Dexuz]

AAAAAAAAAAAAAAA, its you.

[@KaiCyreus]

I guess today, you could say we're getting.. Braidy.

[@markcelemen]

I bet you liked your own comment ;)

[@senorgooba7360]

Cyreus I like you

[@MagicGonads]

Hey now, I made that joke first, pay me royalties!

[@KaiCyreus]

Magic Gonads So sorry sir, I owe you 0.62 internets.

[@meinbuch9458]

Cyreus How unoriginal.

[@Eetarsaurus]

I could watch Dr. Dancso all day.

[@turtles10]

WILHELM 10:11

[@JahLuvzU]

Damn, I hoped I was the first to notice, but the sound is just so recognizable.

[@sGnNPlayer]

Tristen Roddenberry what has been heard cannot be unheard

[@AntonLejon]

This is the first time I've had anything close to an intuitive understanding of 4-dimensionality. Thank you!

[@fallenasleep7247]

This was a really cool vid. I don't have anything to say, I just want to comment because I'm under the impression that commenting helps promote youtube vids.

[@KurtSchwind]

Every time I watch Numberphile I'm impressed with the animations which help narrate the idea. This is no exception. Really great work.

[@keenantroll5151]

the tangle pointed out at 2:30 CAN be undone, if you are allowed to stretch one band all the way around the plank. or rotate one plank. (considering only the two highlighted strands)

[@1verstapp]

in 4-D, no-one can have bad hair day.

[@gabrielkellar1935]

I bet my pillow can find a way around that

[@narfharder]

*_Pippi Longstocking_*_ has entered the chat_

[@PC_Simo]

Or everyone can 🤔.

[@Triantalex]

false.

[@yep_2431]

I like how excited the guests get

[@Sam_on_YouTube]

I really like the idea of picturing it as rings moving through 3 dimrnsional space over the span of a movie. It makes it much easier to picture how they are and are not allowed to move through each other.

[@singerofsongss]

This is one of the only discussions of the 4th dimension that I've really been able to visualize and understand! Good on Dr. Dancso for putting such a foreign concept into simple analogies.

[@wongmjane]

The concept sounds like that old video about how to turn a sphere inside out

[@WaveOfDestiny]

Jane Wong that video stayed in my recommanded for so long i wasn't even intrested and i regret clicking on it

[@AexisRai]

YOU MUSTN'T TEAR OR CREASE IT

[@spencerjohnson925]

​ ​ tOpoLoGy

[@davidb5205]

It's the same field: Differential Topology. Which a lot of people think is useless and theoretical but is actually practically used to describe protein folding and cell division. (Not that math NEEDS to be of practical use in order to be studied)

[@Astronomy487]

Aexis Rai IM LAUGHING TOO HARD

[@modolief]

Braid group: the braid group on n strands (denoted Bn), also known as the Artin braid group, is the group whose elements are equivalence classes of n-braids (e.g. under ambient isotopy), and whose group operation is composition of braids. Example applications of braid groups include knot theory, where any knot may be represented as the closure of certain braids (a result known as Alexander's theorem); in mathematical physics where Artin's canonical presentation of the braid group corresponds to the Yang-Baxter equation; and in monodromy invariants of algebraic geometry. (wikipedia)

[@baronDioxid]

It's Klein-bottles and Möbius-loops all over again!

[@elneutrino90]

I feel like a Commodore 64 trying to run Wolfram Mathematica, it's simply too much for my processoer

[@hindigente]

One of the most interesting Numberphile videos ever, and the animation was awesome as well. I wish to see Dancso more often!

[@Maharani1991]

Absolutely loved the tiny piece of music used for the cartoon rings flying through each other. :)

[@tonysouter8095]

Wow, great animations by Pete McPartlan.

[@MaksiZockt]

2:43 think a little bit outside the box! if you could stretch the strings infinitely, you could aswell stretch them around the wooden plank (here stretch the yellow one around the bottom plank by going counter clockwise) and untangle them

[@danieljoung5398]

Well done man. This video is so quality it's incredible.

[@colecarter2829]

Thank you for this video. I have struggled to grasp the mechanics of the supposed intersections in 4d objects like klein bottles, but the ring analogy helped me visualize what is actually happening.

[@joshpratt339]

I don't understand any of these videos but i still love them.

[@user-uz7gb7gb4v]

When you visualise the 4th dimension as stacked up timeframes and the braid as a 1d "fly", it's also easier to see how the 3d braid can be untangled in the 4th dimension: the fly just needs to wait for a short time for the other fly to move, and then it can go right through where, from its perspective, the obstacle was.

[@markcelemen]

Yay a new video from Numberphile!

[@ravneetsingh1499]

this video was filmed in 2014 but uploaded today....any reasons?

[@lucianodebenedictis6014]

Ravneet Singh you mean the fact the the other two videos that seem to be filmed together were published in 2014?

[@ravneetsingh1499]

yuuup

[@lucianodebenedictis6014]

Ravneet Singh nice one checking the playlist. By the way no clue

[@HistoricaHungarica]

It matured enough to be safely consumed. (probably the animation took some time... and then to edit together)

[@pronounjow]

Wait, what?

[@tefkah]

This video saved my thesis, thank you Brady and Dr Dansco

[@jbrady1725]

Hunh?

[@ricardo.mazeto]

So, a perfect Klein bottle may exist in the 4th dimention? Is that right?

[@kiefergreen3130]

Ricardo Mazeto yes

[@Radditz770]

If you took a 3-dimensional Klein bottle and 'moved it up' into 4d space, then it wouldn't be a perfect klein bottle. But if you made the klein bottle While in 4d space, then yes it's entirely possible :P

[@MrGokules]

Klein Bottle is not supposed to store any liquid, it was designed to be an object with ONLY one side, according to topography.

[@Mykman1]

Yes, there exists an embedding of a Klein bottle in a 4-dimensional real space.

[@BurnabyAlex]

all the 4d water would leak out.

[@mathgeniuszach]

I think its easier to imagine by creating two rooms, and the second room is a jump of one integer unit in the 4th dimension. When you move (or teleport if you want to say) into the second room, you can easily move to the other side of the second room (since the strand is not there) and then move back over into the first room, bypassing the strand entirely.

[@razlotan7504]

The teapot analogue could not be more British

[@pmcpartlan]

Raz Lotan Actually the teapot's from Utah.

[@SgtSupaman]

You're both wrong. The teapot is originally from China.

[@X_Baron]

The Utah Teapot was originally made by a Brit, while working in Utah.

[@joshuarosen6242]

Regardless of where teapots were invented or where any particular one was made, the analogy is indeed very British. We love tea. Which other country has a national sport where it stops for tea?

[@joshuarosen6242]

That looks much more like a sort of teapot common in the UK called a Brown Betty than a Utah teapot. The Utah teapot has straighter sides on its top half and the little knob on the lid is much taller.

[@SamuelHauptmannvanDam]

Great explanation, I've always wanted to be better at thinking of things in higher dimensions and this helps a lot!

[@mversantvoort]

The animations are super cool and really helpful, thanks!

[@copperdragon9286]

This ant metaphor is a very nice way for visualization. But I still had some problems to understand what it means to untangle strings when seen in 2+1 dimensional space (one dimension replaced by time to make this movie of ants crawling in a plane ). The explanation that is really missing (I would say at 9:10) is: Why cant't the paths of the ants be disentangled? Here you need to unerstand what it means to actually disentangle them. Basically you need to create another movie where the paths are just as you wish them to be (every ant sitting still in its initial position), and a huge number (actually infinitly many) intermediate movies in which the paths only differ a tiny bit from the path of the previous movie until you can make your final movie. When you try this with the simplest case of two ants just circling each other, you will find that there is no smooth transition between a movie with ants circling and a movie with ants not circling. There will always be some movie in between where the ants actually have to go through each other.

[@claytonsteele96]

I just wanted to say that adding a 4th dimension does not mean that the two strings can pass through each other [6:30]. It means that there may be a way to move around each other through the new 4th dimension. Just like adding a third dimension does not allow the ant to pass through the 2d wall. but instead allows the ant to move through the third dimension and around the wall. or am i missing something?

[@Galakyllz]

These animations are absolutely awesome. Great job.

[@asitisj]

It's like rigorous, abstract justification and intuition if not proof of topology. Super warped space-time . Very imaginative, very succinct

[@completeandunabridged.4606]

What area of mathematics does Dr Zsuzanna study/research?

[@pmcpartlan]

The World Is Logic Topology

[@wilderuhl3450]

Topology is beautiful. This tidbit is very knot theory, which is also beautiful.

[@franciscobuades3677]

I've always found it really hard to think in more than 3 dimensions, but this helped a lot! Great work, as per usual! :)

[@fredschneider7475]

That was really interesting. I was wondering: Is there an analog in 4-D for threee 1-dimensional braids interacting together so that they tangle in a way that two 1-dimensional braids can in 3-D?

[@maxmusterman3371]

this seems related to knots. very interesting video, thank you I love how mathemeticians are able to think about these concepts in diffrent dimensions.

[@PC_Simo]

This *_IS_* related to knots.

[@GordonWrigley]

So the image at 9:50, with the flies, flying in 3d over time, so 4d. Basically what we're saying is when we try and untangle the lines as well as changing where the fly flew we can change when, so if two flies fly around each other in a tangling way, can just have one fly do it's bit first and the other do it's bit later and they no longer tangle.

[@jenniferclark469]

Great video - I now think I understand the fourth dimension so much better :)

[@kmz8160]

Listening to this magic voice every year.

[@anthonyfrantz884]

I think I might solved it ,check it out .Before we start we see that the blue string is already tangled . First you take the 2nd string (from the left to the right ) and you pull it above the uper plank .Now the 2nd and third strings are untangled .What is left is the 1st and the last .What you have to do is just to pull the first above the uper plank and it is done

[@TheGuruNetOn]

11:59 reminds me of tunnels in a gophers nest. In fact it reminds me of an ancient creature that once dug tunnels and the only remaining thing that remains of its existence are those tunnels. Don't remember the video that I saw about these tunnels but I do remember the shape of the tunnels.

[@HEXVeKtAr1]

It blows my mind to see Numberphile's videos. What gets me so badly is how insanely clever these people are, and how incredibly stupid I feel compared to them.. It's a good thing to be put in your place every now and then, because no matter how good you think you are at something, there''s always someone who is better! So I applaud how vastly smart these humans are! They represent the pinnacle of our species' development! They prove that humans (while some are very stupid) can be mind-bogglingly clever too.

[@seanferney3621]

That was explained beautifully.

[@andrewriachi9222]

Wow, I'm very impressed with the animations. Well done.

[@Thorr26]

Interesting and great animations. Great job! Keep it up!

[@Robi2009]

Dr Zsuzsanna is back!

[@mathaha2922]

Does the animation at 5:46 really show the "plane in which this front strand lives"? It seems to me that the animation shows instead a plane through which the strand passes at only one point. Or have I perhaps misunderstood something?

[@Theminecraftian772]

So, if I had my headphones in my pocket, and they were all tangled up, all I would need to do is push them into a 4D space for a bit? Just jiggle them around form them to move in 4D, then pull it back to 3d? Nice.

[@SendyTheEndless]

That was beautiful and mind-expanding!

[@skeletonrowdie1768]

awesome explanation on visualizing something like a 4D möbius! thanks!

[@nemethdaniel6384]

Hi! :) Interesting video. I have a question: What is changing if you have foliation (like 3+1 dimensions). You have a foliated space+time, and 1 dimensional lines crossing the slices. In this case you can create knots of 1 dimensional lines in 3+1 dimensional spacetime. Is it right?

[@stevethecatcouch6532]

How is foliated space-time different from regular space-time?

[@nemethdaniel6384]

If you have foliation, there is no continuity in time. Im working with a theory, called Causal Dynamical Triangulations Quantumgravity, we have there triangulated spacetime with time foliation, so there are time-likely separated spatial slices, connected by timelike links. If we consider the topology, if its a sphere, it is S^3 x T^1 (instead of S^4) if torus, then T^3 x T^1 (instead of T^4 ). I was thinking about knots, if there is no continuity but discretization in time, then the effective dimension considering these knots is smaller. (Or not..this is my question)

[@IsoYear]

Wow this explained the fourth dimension. I have always struggled with visualizing (in 3d too lol) but this cleared up a lot of questions i had

[@Si-Al-Ti]

There's a pc game being developed called Miegakure that takes place in a 4 dimensional world. Where if you encounter a wall to high to climb you can scroll through the different 3d projections of the underlying 4d space to see if theres maybe a different path you can take. check it out :)

[@snazz1363]

Who else is watching this on their 5 dimensional tesseract in the year 20170

[@recklessroges]

I'm on my 4d device in 12017

[@Kanengizer]

Snazz I'm on my 2d device in 17

[@TheCivilProtector]

sorry bro but that would be physically impossible...

[@Firebolt391d]

To quote an old friend of mine: "Imagination is more important than knowledge." ;)

[@-homechord-2908]

Snazz a tesseract the is literally just a 4d cube. Man, marketing companies in the future must be really bored.

[@Nittweed]

This video shows a really nice way to imagine movement and connectivity in four dimensions which I'd never encountered before, by hiding or collapsing one of the three dimensions we can already see/imagine, and then expanding the hidden fourth one. Really gives a lot of insight into interconnectivity (or lack thereof) in 4-dimensional space. Thanks!

[@joahchewbhaka5679]

that was great! What an imagination! Love the examples, especially the one that used movie frames.

[@poutouellet]

"I'VE HAD IT WITH THESE MOTHERF****** ANTS ON THIS MOTHERF****** PLANE!" -Samuel L Jackson

[@shrimatkapoor2200]

I love that animation of how the rings fly through each other so funny

[@SerratedMilliner]

What I find most interesting is that, if you ascribe to the theory that the 4th dimension is time, and each moment of our existence is a cross section of it, you can effectively make braids or tie knots in 4 dimensions by just moving 3D objects around!

[@mydemon]

Time cannot be the fourth dimension or you wouldn't have the results in the videos. Two ropes cannot cross each other even if you add time as a dimension.

[@PC_Simo]

@@mydemon Exactly 👌🏻🎯!

[@PC_Simo]

@Serrated Miller Frankly; whenever I now hear about time being the 4th dimension, it almost makes me physically nauseous; just, because it’s so absurd a ”theory”. Time is nothing but motion and change. That’s it. I’m definitely a minimalist, when it comes to my idea of, what time is (if it even exists); but, for some reason, most people seem to consider the spacetime-nonsense a Divine Canon. Just, because Einstein said so, doesn’t mean that time is some sorta physical fabric, connected to space. It’s not; it has got nothing to do with space; and it’s not a dimension. I find it mind-boggling, how so many people don’t even dare think for themselves, or question these dogmas, like the ”Theory of Relativity”; out of fear of disproving their Divine Canon / Holy Scriptures. I can only wonder, what was Einstein smoking, when he came up with such an absurd idea. 🤯

[@Zenene-ok5el]

For understanding the basics of how the fourth dimension would work and look like, I strongly reccoment Flatland, a book by Edwin A. Abbott. It makes both for an interesting mathematical reading and a biting Victorian satire.

[@flounderflounder6833]

There's a movie about it somewhere on KZbin aswell

[@JNCressey]

At 13:30 the explanation is that you just need the dimensions to add up to at least D-1 for D-dimensional space. *But, what about ribbons!?* Can you make ribbon-ribbon braids in 4 dimensions? Ribbons are 2 dimensional right? So *2+2>3* would make you think yes. But they don't have an inside like the tubes in the example had; their cross section is just a line segment, not a circle. I'm pretty sure that you wouldn't be able to braid them; I think you could stretch them to be so thin that you can assume they are 1 dimensional, then by *1+1

[@rosePetrichor]

Unfortunately the problem here is that the 'cross section' of the braid needs to be a closed loop. Ribbons are not '2 dimensional' in the sense that a topologist would want them to be.

[@maxilexow]

Great video, very intuitive analogies

[@astro2191]

That is one hell of a good analogy for the 4th dimension!

[@PC_Simo]

Indeed; and it’s precisely that: Analogy 🎯. Sadly, many people don’t seem to realize that 😔.

[@seanp4644]

Pause the video at 13:00. This is truly a high school level maths channel.

[@allie-ontheweb]

The Minor Miner it was more showing the pattern of which 'dimensional' objects could successfully braid with which

[@MrGartenzwerg95]

I take your braids to another dimension! Pay close attention!

[@JayJay64100]

You are a prodigy for coming up with this!

[@X_Baron]

Everything, including braids, becomes clear once you jump out of (the 3D) space.

[@FrankHarwald]

Also: what will braids do when they're out of space? ;)

[@scottharbin1868]

Benetrat0r I love that song! You made it better tho...

[@wishiwasabear]

1+1=2 1+2=3 2+2=4

[@sarpkaplan4449]

Kuma Bear 2+2=1 (mod 3)

[@kent_hdd]

You sir, are absolutely correct !

[@falconhawker]

No! 1+1 =2 1+2 =0011 2+2 =0100 observerms

[@callumscott5107]

s777n base 3, noice

[@LaatiMafia]

e = mc^2

[@sam111880]

nice explaination also you probably considered the question of links can be closely related by closing up the top board with the bottom board one has a way to classify n-dimensional links to n-dimension braid theory so if you have an invariant for one you can uses it for the other.

[@PC_Simo]

Links *_ARE_* closely related.

[@annaarkless5822]

i went to a lecture at cambridge uni for a level women in maths that was basically this exact video, but somehow numberphile explained it better than the cambridge lecturer (no shade to that lecturer tho, it was fascinating even though i couldnt follow along completely)

[@jesterjames7633]

mind blown and melted at the same time.

[@Yinas]

ok this got really interesting in the second half. now my mind is all tangled...

[@whatthefunction9140]

Is there a use for this (not hate'n just ask'n)

[@recklessroges]

Not sure. Even if the answer is, "not yet" it took hundreds of years for Chinese remainder theorem to be used by you today, (to read this comment over an encrypted connection.)

[@davecrupel2817]

If we ever discover a way to go to 4th dimension, i would love to see how much, if any, of our theoretical 4th dimensional works is correct and incorrect. :D

[@corpsiecorpsie_the_original]

Someone may be able to figure out how the extra dimension that makes up 4D interacts with our 3D world. It may just answer where energy came from.

[@56kk12]

Wow. My mind is so open after that.

[@Jackcabbit]

A note about the "tangled mess" of those 4 strings between the two boards: Just rotate the boards themselves to untangle the strings. It's like a double helix.

[@robertlozyniak3661]

I have a feeling that it is against the rules to move the boards.

[@PC_Simo]

My thoughts, exactly 👌🏻🎯😅👍🏻! My OCD was literally screaming precisely that, in my mind. It’s not like the boards are nailed down to the table. Right?

[@PC_Simo]

@@robertlozyniak3661 Well, that would certainly explain, why they wouldn’t do that, or even consider it 🤔.

[@clobbopus_used_beat]

Great videos and love the new animations!!

[@DoctorCool-y8j]

These concepts are connected to string theory. I think that connection deserves another video.

[@LokNWykLeer]

This actually helped me understand the 4th dimension a bit better! :D

[@ShaunaJade96]

Brilliant animations!

[@mrpengywinz123]

She is my favorite numberphile narrator Hands down , sorry Parker sorry Grimes you're the runner ups to the real mvp

[@amandagarcia2848]

Can 4-d braids be used to compress the motion of 3-d holograms?

[@MrPSyman3]

This is the only 3+1D video I've seen that actually has a decent-looking analogy in it

[@rbnn]

Can you disentangle the braids on a board by moving the a string around the board completely?

[@LaatiMafia]

If you are allowed to stretch the lines, can you stretch them over the plank?

[@PC_Simo]

Probably 🤔.

[@legendgames128]

10:11 ah yes, a Wilhelm scream in a mathematical channel.

[@andymcl92]

Sometimes I hate my 3D mind. Those braids with rings looked really interesting, but I just can't picture them in their 4D glory!

[@kaniebr]

Isn't the animation in the 4th dimension wrong? I mean, in 3D the strings ends in the same order that they started, but in the 4D (13:29) they don't. The rule is that they should return to the same position they started. Or am I wrong?

[@pmcpartlan]

kaniebr They are still braids, just not pure braids.

[@Ecrowtist]

With the 3D braids, are you allowed to stretch the ropes around the wooden planks, or are they supposed to stretch across their entire planes (making them infinetly long)?

[@Halibali]

7:04 Looks like a sneaky ninja move :D

[@xeniaanan645]

By observing the frames and the way they move in three-dimensional space. Can we observe humans growing old in three-dimensionsal space from a perspective of a fourth dimensional person ?

[@idjles]

This is easy to visualize if the 4th dimension is time, as one braid would just move forward in time and cross where the other braid isn't and then go back in time.

[@andreadelzoppo9403]

The name "Numberphile" with the hearth on top of the "i" always made me unconfourtable...

[@infiniteplanes5775]

The four dimensional string unbraidness one works if the starting points are not in a straight line. You can move points of the strings around in the fourth dimension. But I am assuming that you can’t just go from -1 to +1 without touching 0 at some point? You can move points around one another, the lines can be undone, but the points remain in place, right? You can only move the braiding points around, but they must be there. To undo them would mean moving an intersection passed the ends of those planks. So far, as could see, it’s simply impossible without a possitive curvature in either the x y or w axis. I could be wrong though, so if I am able to get my

[@infiniteplanes5775]

Six dimensional cube viewer to show strings in four dimensions and manage to untie them, I will come back here and say how.

[@venkateshbabu5623]

So slicing through extra dimensions are possible by moving over the circumference of a circle or elliptical.