Could Flatland Be CURVED?!

Measuring the curvature of a space takes parallel transport and the Riemann curvature tensor (General Relativity). Let's explore this using Flatland and Lineland, an old tool involving lower-dimensional spaces. Brilliant for 20% off: brilliant.org/ScienceAsylum
________________________________
VIDEO ANNOTATIONS/CARDS
The Ultimate Guide to Relativity:
• The Ultimate Guide to ...
What the HECK is a Tensor?!?
• What the HECK is a Ten...
How Elastic is the Fabric of the Universe?
• How Elastic is the Fab...
________________________________
RELATED KZbin VIDEOS
PBS Space Time on Space Curvature:
• Can a Circle Be a Stra...
Flatland by Carl Sagan:
• Video
Flatland by "What The Bleep?"
• Dr. Quantum Flatland
________________________________
SUPPORT THE SCIENCE ASYLUM
Patreon:
/ scienceasylum
Advanced Theoretical Physics (Paperback):
www.lulu.com/shop/nick-lucid/a...
Advanced Theoretical Physics (eBook):
gumroad.com/l/ubSc
Merchandise:
shop.spreadshirt.com/scienceas...
________________________________
HUGE THANK YOU TO THESE PATRONS
Warden of the Asylum:
[Available]
Asylum Counselors:
Matthew O'Connor, Nikolaos Vasiloglou II
Asylum Orderlies:
William Morton, Fabio Manzini
Einsteinium Crazies:
Wacky, Eoin O'Sullivan, Ken Davis, LT Marshall, Ilya Yashin
Plutonium Crazies:
Al Davis, Compuart, Ellis Hall, Kevin MacLean, Randall Sylvia
Platinum Crazies:
Olga Cooperman, Kyle Bowles, Zein Kassim, Rick Dinning, Eugene Boone, Jon Adams, Mr. Orn Jonasar, Vittorio Monaco, Stephen Blinn, Mikayla Eckel Cifrese, Jonathan Reel
________________________________
OTHER SOURCES
www.physics.ucc.ie/apeer/PY411...
web.stanford.edu/~oas/SI/SRGR...
www.asc.ohio-state.edu/mathur...
www.physics.usu.edu/Wheeler/Ge...
www.preposterousuniverse.com/...
________________________________
LINKS TO COMMENTS
• How Elastic is the Fab...
• How Elastic is the Fab...
• How Elastic is the Fab...
________________________________
IMAGE CREDITS
Earth by GOES-16:
www.nesdis.noaa.gov/content/g...

[@Uncle-Mike]

I haven't heard anyone recommend Edwin Abbott's "Flatland" in a while. So I'm recommending it. It's a book. It's flat.

[@aniksamiurrahman6365]

Great suggestion. I enforce it.

[@phamminhduc0609]

yeah I heard it in a TED-ED video

[@alexandertownsend3291]

Yeah great recommendation.

[@raghu45]

How many pages? Is it hard bound or paperback? Are each lines to be read left to right or right to left or top to bottom? Is it also in Kindle & PDF version ?

[@donwald3436]

Aren't most books flat?

[@crouchingtigerhiddenadam1352]

Paralax is very noticable after too many beers. Great video Nick, I'll be sharing this with the family.

[@bierrollerful]

I'm not getting drunk. I am just measuring the distance between my eyes.

[@wesshepard]

I bet they’re excited about that

[@Samu2010lolcats]

It's also very noticeable when you're seeing something very close to your face.

[@dans6127]

Easy to notice if you focus on a distant object and then touch your fingers together in front of your face. The old floating finger trick

[@mitseraffej5812]

The other thing I notice when I drink beer is how doing a stupid thing seems like a really good idea.

[@anti-troll-software6151]

Everyone: Flatten the Curve! ScienceAsylum: Could Flatland Be CURVED?!

[@stefansauvageonwhat-a-twis1369]

Anti-Troll-Software Can the Curve be flattened ? Can the Flat be curved ?

[@uwose]

I see, I am way too late to ask if the pun was intended ... you already got a heart...

[@TheExoplanetsChannel]

Haha

[@dhayes5143]

Mathematicians: So here's a flat curve.

[@the4spaceconstantstetraqua886]

@@TheExoplanetsChannel The translator translated Haha to lol, the weirdest translation, it's correct though.

[@GraveUypo]

i love how you poke them without mentioning them a single time.

[@ScienceAsylum]

😈

[@yomumma7803]

poke what?

[@lauranceberiya1314]

@@yomumma7803 proponents of alternative astral geometry

[@frankquinn7061]

Taurus 😈 What about half a taurus 🙀

[@mattdowds8505]

@@yomumma7803 Give me an F, Give me an L, Give me an E, Give me an R, Give me an F!

[@skmaurya19]

So we are basically THREE dimensional beings living in a FOUR dimensional world where our eyes makes TWO dimensional images. And the only way to transcend is MATHEMATICS.

[@adamqazsedc]

MIND BLOWN

[@ruthlessadmin]

And technological adapters/interfaces

[@ffggddss]

Well, not exactly. We are (3+1)-dimensional beings living in a (3+1)-dimensional spacetime that is locally Minkowskian, not Euclidean, in which mass-energy and stress impose Gaussian curvature on that spacetime, in a way dictated by Einstein's Field Equations. Our eyes make (2+1)-dimensional images that our brains turn into (3+1)-dimensional perceptions. And our only hope of understanding how that all works, is MATHEMATICS. Fred

[@skmaurya19]

@@ffggddss That makes more sense Fred.

[@Graeme_Lastname]

I don't know if it's the only way. Have you heard of LSD? 😱

[@tom_something]

So, on a sphere, two paths that seem like parallel lines to the surface-dwellers will converge. In our universe, if you have two objects with mass floating in space with zero relative velocity between them, those objects' path through 4D spacetime would be projected as two parallel lines. Their paths will converge, which Newton would explain with the force of gravity. But those who see a little further by standing on his shoulders say that gravity manipulates the relationship of the three spacial and one temporal dimension, effectively curving the universe. Is the curvature that we describe as gravity the same as the intrinsic curvature of the universe itself?

[@nibblrrr7124]

Afaiu spacetime is always curved in the presence of masses, but space itself seems to be very flat (based on those CMB triangle measurements),

[@thedeemon]

@@nibblrrr7124 These are two different things: how space looks locally near a star or a black hole, and how it looks globally on intergalactic scale. It looks pretty flat globally but locally near massive objects it seems as curved as GR predicts. Of course numerically time rate change is the biggest factor, space parts of the metric tensor don't change that much, but still change.

[@ScienceAsylum]

Yeah, that all sounds good Tom.

[@tom_something]

@@ScienceAsylum oh, neat! And something with zero relative velocity to us is observed to travel through time at the speed of light, and we know what the speed of light is, so that's how we derive the _extent_ of the universe's curvature, right? It's geometry... though a little more complicated than I could handle (stopped after AP calc).

[@tom_something]

@@nibblrrr7124 We travel through space at some speed that we're constantly monitoring, like how fast we're running, or how fast Earth is going around the Sun. But we're also traveling through time. We're aging. And based on the relationship between space and time, we're essentially going through time at very close to the speed of light. So I think that's where the challenge comes from in reconciling the curvature. The speed at which we watch stuff travel through space is many, many orders of magnitude slower than we watch them travel through time. Physical distances between human-made devices are also very, very small by comparison for the same reason. Remember that mass isn't the only thing that causes gravity and is affected by gravity. Energy is also in the club. So when you get into "mass and/or energy", in practice that's similar to "existance". Things that exist cause gravity and are subject to it. And if you look at gravity as a force, two falling objects of different mass fall at the same rate, meaning gravity happens to scale up with inertia, the very thing that resists it. It's too convenient.

[@gardenhead92]

That was a great explanation of Rayman curvature

[@jskratnyarlathotep8411]

Roman curvature!

[@alancrabb]

Wo-man curvature!

[@ConnorMcCormick]

It's Rayman culture. You clearly wouldn't understand Rick and Morty

[@ShiftyMcGoggles]

I know, roymain curvature is pretty neat!

[@nou4898]

ps2

[@manuelcheta]

I've seen another explanation: triangles on the surface of a sphere can have 90 degrees in each of the three angles.

[@ScienceAsylum]

That is mathematically equivalent to this 🤓

[@harrygenderson6847]

Another related point: The ratio between the circumference and diameter of a circle on the surface of a sphere will be less than pi. The expected value for this ratio if the surface were flat (pi itself) can be calculated independently of measured values using infinite series or other methods. I had to derive one of these for an engineering exam last year :).

[@power2go3]

I just like to say that as a physics student I appreciate the way you simplify things so that everyone understands, really shows a solid and deep grasp of physics which I'd like to have

[@thedeemon]

Some time ago I made a program where you can see how it looks when you're inside such curved space. I took a curved 2D surface like the surface of a sphere or a torus or a wormhole and added one more orthogonal dimension to make it a curved 3D space where you can walk around and where light follows the geodesics. Video here: kzbin.info/www/bejne/qZCzf4ycadaJgcU

[@ScienceAsylum]

What did you make this in?!

[@thedeemon]

@@ScienceAsylum github.com/thedeemon/curved Coded it from scratch.

[@realcygnus]

cool ...... would make a hell of a shadertoy

[@GraveUypo]

your video is way cooler than this one.

[@nicolascalandruccio]

The "trip" is really cool

[@kylebowles9820]

This was the longer video you were talking about on the live stream! Next level "flatland" animations better than others I've seen; unique that you touched on the Plato's cave element when they get back to flatland haha!

[@TheRealGuywithoutaMustache]

I learn so many things from your channel, please keep these uploads up

[@kornsuwin]

Just Some Guy without a Mustache ok daddy

[@AxionSmurf]

Love this guy. He's the good kind of mad scientist.

[@MrPink-cn5rr]

Yehh haha

[@grimwatcher]

You know KZbin algorithm is doing something right when you discover a new science channel and it talks about flatland. You got a new subscriber sir!

[@dronillon2578]

Love your content. I find these topics hard, but I'm happily chewing through it. Thanks for cuting it into easily manageable bits.

[@mixpick138]

Yet another marvelous video! One of the best/most approachable explanations for conceptualizing space dimensions I've seen in a long time. Thanks!

[@shempincognito4401]

While I'm not sure about the gimmicks (amusing clones and stuff aren't my thing, but to each their own...), the PEDAGOGICAL skill and DEPTH of this channel are very impressive! Even if I've watched countless videos on a topic, and even taken classes or read about it, I often find this channel adding a point or clarifying something important. And in doing so succeeding where other educational creators fail. One should never judge a book by its cover (or a channel by its gimmicks); This channel, along with PBS Spacetime and a few others, are among the best KZbin has to offer! Update: Ok, I did laugh about some gags in this video.

[@eden4292]

Love your content, help in learning and in explaining what has been learnt to others.

[@jasoncollins1011]

I was so hoping you were going to do a “rainman curvature “ joke!! Great work sir, you make science funny and interesting. I absolutely love your work. 😊

[@asymptoticspatula]

This channel is seriously amazing. It just gets better and better!

[@averagemilffan]

Nice video. U deserve so many more subscribers. Btw can u do a video on optical tweezers?

[@antonnym214]

Hyper-interesting and well-presented! I liked and subscribed.

[@saubhagyasharma9933]

I love you bro... Never stop uploading videos on such curious topics... I am inspired by you.

[@nokian9005]

I get excited every time I stumble upon a video of yours I haven't seen yet. Keep up the good work. We're always thirsty for more knowledge.

[@ScienceAsylum]

This video didn't perform as well as I thought it would. I'm glad you found it 🙂

[@galaxy_apollo13]

Your videos are awesome

[@MatheMagiX]

Zach Star had a video on that today too, 1 hour earlier. Well done to both of you. I repeat my idea for an episode: please explain (liquid) paint on molecular level - colors, mixing colors, how come green mixed with yellow is always the same and what happens to the surface of a paint after mixing so that the bounced light is always the same, not yellow, nor green etc.

[@andysmith1996]

Wouldn't that just be that your eyes can't resolve that level of detail? Just like colour printing or TV screens are made up of dots of separate colours but to us they appear blended.

[@MatheMagiX]

@@andysmith1996 So there is like a mix of red-reflecting and green-reflecting atoms separately, but our eyes see it blended as brown?

[@nokanol45]

paint color is subtractive, meaning they absorb some color of light and reflect some color of light. Red paint, for example, reflects red light and absorbs other colors of light. So when you mix 2 colors of paint together, say red and green, the resulting paint in theory will absorb every color of light, since green paint absorbs red light, and red paint absorbs green light, thus it would be black. however, in practice paints aren't perfect and thus in reality mixing red and green paint gives brown.

[@Wetefah]

You have a great talent making complicated stuff digestible. Awesome video, thank you so much.

[@bhavyananda3193]

WAS EAGERLY WAITING FOR YOUR NEW VIDEO

[@inuka6969]

5:21 See, they also have two balls.

[@ScienceAsylum]

That's a brain.

[@narfwhals7843]

@@ScienceAsylum there's a metaphor here somewhere.

[@WillToWinvlog]

One thing I've always wondered is how do you know you are keeping the angle steady as you travel? There is no universal grid for reality...

[@jskratnyarlathotep8411]

gyro

[@FlyingOctopus0]

@@jskratnyarlathotep8411 gyros don't work in flatland.

[@WillToWinvlog]

@@jskratnyarlathotep8411 My issue with the gyro is that it would follow the curvature of space time rather than remain neutral to some universal grid lines. There has to be some clever solution to this, I'm curious what Nick would say.

[@nithyadavuluri7287]

Do the grids need to be universal though? Can't we map it relative to the initial vector? Just asking..

[@onradioactivewaves]

Nick, you're videos are top notch for taking these advanced topics and breaking them down into laymen perceptions.

[@flannn6]

Another video from my favorito channel! Thank you so much for making my monday better :D

[@akankshagautam7740]

It warms my heart to see that we apes can even understand the cosmos this much. It gives me hope for the future of humanity. I hope human ingenuity overcomes all the sociopolitical mess that stands against us.

[@shrikant8446]

3:57 "om Nom Nom Nom! 👌

[@michaelgrinter448]

you got the best and most interesting science videos on youtube hands down.

[@chrisz6860]

Great channel Nick! Have been sharing your videos with my kids

[@ScienceAsylum]

That's wonderful! 🤓

[@chrisz6860]

Can you do more videos on Lagrangian Mechanics with examples? Thanks!

[@RevoLee]

I love your videos so much. I wish everyone had the passion and time to watch videos like these instead of chasing clout on tik tok.

[@krishnagupta2777]

Totally agreed.

[@mojocore]

Does this mean there are a bunch of 4-dimensional shapes we might be living in but we would have absolutely no way of measuring it?

[@ScienceAsylum]

Yep!

[@eswing2153]

I’m not so sure we would have no way of knowing it.

[@kapilellawadi]

Mind blown everytime! Thankyou for putting the music back though.

[@Roberto-REME]

Great video Nick!

[@conoroneill8067]

Great video! I do have one question, though - How does parallel transport work in our 4d universe if you can't create a sub-light speed path to get back to the same point in space-time where the parallel transport started from?

[@ScienceAsylum]

Right, so parallel transport (with one vector) will work for our 3 dimensions of space... but it isn't going to work for _time._ That's another video (which I'm already working on).

[@materiasacra]

Instead of going around, you follow two timelike (or lightlike) halves of the itinerary, starting with the same vector. When arriving at the same endpoint via two different paths, you compare the resulting transported vectors looking for any mismatch. In fact, this idea of comparing two paths, mathematically leads to a very nice way to express the Riemann tensor as the commutator of two derivatives-that-take-into-account-parrallel-transport (called 'covariant derivatives'), which is often useful. It's a good way to think.

[@conoroneill8067]

@@materiasacra Thanks! I think that makes sense.

[@JorgenewtonB]

I'm sorry about my ignorance, but has anyone tried to measure this?

[@andrewpatton5114]

@@JorgenewtonB Yes, and the conclusion is that the universe's global curvature is zero to within one part in 10^62. Significant curvature is, however, present near massive bodies.

[@danfg7215]

If we parallel transport while orbiting the Earth, will we measure the spacetime curvature of Earth's gravity?

[@kylefillingim9658]

What should we use as reference points? If points on earth are used we are likely to get a false positive. All stars in the sky seem to be moving relative to each other, eaven the galaxy's seem to be moving around, although they are slower. We have no fixed points therefore we cannot draw any conclusions.

[@EternalSilverDragon]

@@kylefillingim9658 Your reference point is always the last vector. The idea is that when you return to your starting position, your final vector will either be the same or different from your first vector. If you used outside things as references, the test would be meaningless, like travelling around the Earth's surface using a compass arrow for the vectors; the final vector would always be the same as the first vector even if all the other vectors aren't parallel.

[@sunrazor2622]

Nice and challenging presentation. I like how you start with the simple and work towards the complex.

[@dolphintech1249]

7.34, Love how Pacman is put into its natural habitat

[@chalecoflash]

alert. brain overheating. take a nap.

[@James42_]

1:00 ohhhhh!! So that’s why the do that in theater to make 3D movie?

[@s3cr3tpassword]

Bach-James they used to. It’s a little different now. That why 3D glasses are different.

[@SpacyNG]

as s3cr3tpassword mentions, this is oldschool. The downside is, you loose a bit of color because of the colored lenses. The upside still is, it is ridiculously cheap. Nowadays there's two different kind of techniques: 1. Let the TV/Projecter emit double the framerate so 2x the pictures per second and then let the glasses basically alternate visibility. It's like like holding a hand in front of one eye, looking at a picture, then you hold the hand in front of the other eye, while someone changed the picture. Was mostly used in home TV sets but is on the decline. Usually called "active glasses" 2. Actually put two pictures on the screen simultaneously. With different light polarization. And then the glasses are polarization filters to only let one of the two pictures through to your eyes. This makes for nearly disposable, very cheap glasses. Which is why the theaters mostly use it. Usually called "passive glasses"

[@rarra]

Your videos are well worth the wait. Some of it went over my head though

[@rc5989]

Love it! Another very high quality video.

[@feynstein1004]

I find it absolutely incredible that math, in this case algebra, has ways to describe things that our brains are fundamentally unable to comprehend. Hmm could we build a machine that could "see" 4 dimensions and thus could appreciate spacetime in all of its glory?

[@jamestheotherone742]

It'll be on the post-singularity AI's short list.

[@feynstein1004]

@James the Other One Fair enough 😂

[@eliyasne9695]

2:58 For the average person no, but for me, the fact that we can literally constrain the unimaginable using math is one of the happiest phenomena in the universe.

[@jimivey6462]

I read Abbott’s “Flatland” 50 years ago. He would have enjoyed your presentation, as I did. Nicely done!

[@dragonfly3402]

Very entertaining and informative. Thanks! 👍😊

[@DigGil3]

Now here's the kicker: the curvature in the time dimension is what causes gravity.

[@Nulley0]

3:58 eat each other lol

[@byronvega8298]

Him: that's a bit dark Me: that's kinda hot

[@olmostgudinaf8100]

Me: that's inevitable.

[@martingamauf4725]

This is so interesting, please do more videos on this topic :)

[@scienceandknowledgearchive8197]

Thats so great and informative. Thanks The Science Asylum

[@DJ_Force]

Nerd clone wanted me to mention you can't have crossed neurons between the eyes and brain of flatlanders. Left eye must route to the left hemisphere.

[@ScienceAsylum]

..........dang it! You're right 🤦‍♂️

[@gamerrfm9478]

You also probably couldn’t have human level brain complexity without a huge area, due to the amount and geometries of neuron connections being limited.

[@johnfarris6152]

You should write a book, Oh yeah you did. I wish I read it.

[@jlpsinde]

Great as always, thanks Nick!

[@kitflash97]

Oh my god I had been thinking about this for so long! Thanks for the video! I've got new questions now!

[@luudest]

Maybe I too stupid for this: But the fact things fall onto earth doesn‘t this mean that space is curved? Or is this video refering to a different curving?

[@cosmicwakes6443]

Change Gamer He is talking about manifolds, ambient space is excluded, it's an intrinsic geometry.

[@chriskirk3670]

Things falling to earth is almost entirely caused by curvature in the time dimension not space curvature.

[@nibblrrr7124]

Well, if you assume objects in freefall take the shortest path, and the path they take doesn't turn out to be a straight line, then yes. General relativity does that. But classical mechanics can explain this just as well by saying there is a gravitational force that pulls objects away from a straight line path - no spacetime curvature needed. It just turns out that the relativistic theory can also explain other things which Newton's classical gravity can't. (Also, afaiu it's usually spacetime that's curved, not so much just the space part itself.)

[@TNaizel]

He's talking about the overall shape of the universe. The Earth causes a bump on the fabric of spacetime, but we don't know if it is an infinite flat sheet or a sphere with an extremely small curvature

[@ronenshtein7083]

Well, two things - first, you say that because it's already "well known" that mass curves spacetime and that this curvature is felt by mass as gravity... Before Einstein no one played around with the connection between gravity and curvature, so the idea isn't too obvious. Once you understand what causes a certain force, only then it becomes "fictitious". Prior to that, the understanding was that gravity is just "is". Second, I think he's referring to the curvature of the entire universe regardless of small local masses and curvatures (at universal scales the earth is "local"). So the question would be whether the universe is an infinite (hyper)plane with curvature 0, or forms a finite size (hyper)sphere, or an infinite hyperbolic (hyper)surface, or some other weird curvature. This is what we are trying to figure out by measuring giant triangles in space, and so far it seems that the curvature is very very close to, or is exactly, 0. To produce this measurement, the lower bound on the size of the universe last I checked was something like 1000 times the size of the observable universe. So if the universe is a (hyper)sphere it's at least very very big...

[@marlin_rtbt3114]

thought ur going to talk about my mom

[@OmniGuy]

Absolutely love your videos, Nick

[@navneet1464]

Since when did we start objectifying space too!

[@materiasacra]

1915

[@gvibanhez1]

Awesome as always!

[@ManojSubramani84]

Thanks Nick! Brilliant video!

[@hhlavacs]

I got your book today, Nick. Looks great, cant wait to read it!

[@ScienceAsylum]

Enjoy! 🤓

[@RocketLR]

AAAAH THANK YOU! I've been obsessing over this for such a long time!

[@ScienceAsylum]

Glad I could help 😊

[@illogicmath]

Excellent video as always. Thanks

[@ScienceAsylum]

You're welcome 😊

[@rkn8109]

Thanks Nick for making video on this, I always wished the same.

[@lazaraza]

Thanks for the entertaining and informative video. Do you have one where you explain curvature?

[@mithsaradasanayake3211]

Nice work

[@saumitrachakravarty]

Wow! I didn't know what parallel transport is. Now I love it.

[@daverei1211]

Well done Nick, well done. Thank you.

[@michaelfarrell6448]

I'm sure I'm not the first : You are better at teaching (quality science ) than any I've seen so far on my sons homewrork thanks

[@HouseClubber75]

Very clear explanation thankyou

[@philochristos]

This is a very good explanation of something I've wondered about for a while--how we can measure the curvature of space since any instrument we used would curve WITH space.

[@Qrexx1]

You're certainly among the most entertaining online educators.

[@ChrisWalshZX]

Yay! My fix for the week! Thanks Nick

[@ScienceAsylum]

You're welcome 😊

[@viers7040]

good work man a great video i like it

[@idan37]

I admire your videos!! Can you do a video on E8 theory?

[@johnclark8359]

I think this just might be your very best video, and that's saying a lot because they're all great!

[@ScienceAsylum]

Really? It didn't do very well at first. I'm hoping it's what I like to call a "slow burn."

[@johnclark8359]

I'd love to see you do a video on the difference between curved space and curved space-time@@ScienceAsylum

[@ARBB1]

Good examples and presentation as always. In tensor calculus, I remember a example where the Riemann curvature and scalar where associated as generalizations of Gaussian curvature in regular shapes, I think that'd be worth mentioning too.

[@taw3e8]

At first i thought that tensors are scary, then spinors but now i know they're both

[@0cgw]

It sounds like you are referring to the Gauss-Codazzi equations.

[@CarolynFahm]

Edwin Abbott - Flatland! You are a superb teacher.

[@dankestofmemers]

Great video! I did want to point out a subtlety regarding parallel transport that's often elided in GR videos for understandable reasons. The change in a vector after it's parallel transported around a loop is called holonomy. Holonomy doesn't necessarily imply curvature! For example, consider a cone with the vertex removed. This is a flat surface, but parallel transport around the missing vertex will rotate a vector by an amount based on the cone angle. The holonomy comes from the nontrivial topology (a loop around the missing vertex can't be shrunk to a point) as opposed to curvature. Another example is the flat Möbius strip: parallel transport around around the center circle multiplies vectors by -1. However, holonomy around loops that can be shrunk to a point (the fancy term is "restricted holonomy") does imply curvature. Any loop will do in a simply connected universe, such as a sphere. If we're worried about the global topology of the universe, then we should build our curvature detection laboratory in a region small enough to ensure that it's simply connected. As @ffggddss points out, a contractible loop on a curved manifold may not have holonomy. This can happen on a surface, for example, if the loop encloses equal amounts of positive and negative curvature. In higher dimensions, we may get unlucky and have the loop traveling in "flat directions" only. However, if the Riemann curvature tensor is nonzero at a point, then there will be a rectangle based at that point which is small enough and oriented in the correct directions so as to have holonomy. Going back to the case of surfaces, if the Gaussian curvature is, say, positive at a point, then a small enough loop based at that point will only enclose positive curvature. In a precise sense, the RCT measures infinitesimal holonomy.

[@alexandrebatalha7253]

I LOVE YOUR VIDEOS! It's the general relativity class I wish I had taken in college.

[@BlackFiredDragon]

Beautifully made illustration and explanation of that illustration. I just wish you brought it home by applying what we learned from the illustration to our 4D world, and explained what vectors we have used to measure our curvature

[@ScienceAsylum]

The problem is that the 4th dimension is _time._ That makes it complicated enough that it requires an entire other video to explain... which is coming.

[@2x1e2]

Loved it. In fact i was brainstorming this same thing after watching pbs studio video few days back. One more thing. It would have been nice to explain "why" the vector deviates after it's parallel transport in a curved space. I tried many simulations in my head and it does, and i understand that. But would have been better if someone showed the difference between parallel transporting step by step in a flat vs a curved space side by side. Maybe pointing out how a square on flat grid is different from one on curved grid

[@salimhassansiddiki4613]

Great video...

[@lucidzfl]

So find the flattest part of space and project three lasers at each other. If the angle exists there is a tensor. It’s really neat. Cool vid

[@realcygnus]

Good one !

[@user-uu7sk8bz5l]

Sir eventhough I dont understand any of this your teachings and knowledge is massively attractive and wonderful

[@princevegeta5907]

Awesome!

[@fontainenick]

If you really wanna go crazy with dimensions, Cixin Liu's 'Three body' problem series (especially Baoshu's fourth book) is pretty amazing stuff.

[@ScienceAsylum]

You're not the first to recommend it. People have recommended that series to me several times over the years.

[@balajisriram6363]

Didn't understand much maybe because i only have an electrical background , but I truly appreciate the efforts for this video. Love them as i have always

[@ScienceAsylum]

One of these days, I'll get back to electronics. I have half of a script written about Kirchhoff's rules that I can't ever seem to finish.

[@kalevprayah3148]

I love these videos so much. The math is way over my head but everything else is too much fun!

[@DeconvertedMan]

very well ROUNDED video. :D

[@ScienceAsylum]

😂

[@DeconvertedMan]

@@ScienceAsylum Time is said to be the fourth dimension - but I've heard there are other possible dimensions, but I do not recall and google has failed me to find (plus I don't trust it) what they are or thought to be, how they work, why we need them and so on. If you have not yet covered that, would you cover it? And if you did, please link me to that video. Thanks for your content I love learning more about science. Also, on a tangent - lets say the whole earth was small enough to eat - would eating it kill a person - due to all the things that the earth has - (lava comes to mind) OR would it be that at the point that its small enough to eat that your body would be able to do so without being killed instantly - (but maybe would be killed by something, say radiation) and as a side, side note since I'm thinking about eating the earth, if humans could eat anything - how long would it take humanity to eat the earth in years? (assuming we tossed our waste into space) also at what point in eating it would we need to have space suits as it no longer has atmosphere? I'm not sure that you would/could or will cover such odd thought experimental questions ... but if you did it would give us FOOD for thought... also what would it taste like?

[@flexico64]

I love your little 2D critters~

[@ScienceAsylum]

Me too 🙂

[@Tokhaar]

You are simply amazing no matter what you're explaining, everything becomes so extremely fascinating and exciting to learn 😁

[@hussiendaeeh]

Thanks, you are the best.