Hi all! Thanks for your patience for part 2 - hope it was worth the wait. Just addressing a few points here: THE STACK GAME: I feel like this went over some people's heads - I had to edit it down for length so it might have felt rushed. The idea is that the stacks are exactly the same as the colored bars tracking orientation and rotation in the 3 (and later 4) directions. Making it concrete with chips takes the rotation context out of it - we can 'rotate' by exchanging chips. This allows us to add a 4th dimension (the 2nd hidden box) in a way that doesn't break our brains, preparing us to understand 4D rotation. The big takeaway is that we can rotate a 3D object into the 4th dimension, which brings the w-axis into view (as in 18:18). If that's all you got out of this, then that's really all you need. ISN'T THIS ALL JUST 3D? Actually no, it's all just 2D. The projections I'm using are quite standard, and are usually regarded as '3D shadows' of 4D objects. But we can't literally see 3D, we simply interpret 3D information from 2D images. We can also learn to interpret 4D information from 2D images. But that means treating the 'space' we're looking at as 4D, not 3D. This interpretation involves learning a new 4D visual framework, which is [Doc Brown voice] what makes 4D visualization possible. This will be the focus of part 3. WHAT'S THE PLANE OF SIGHT?: In this visual framework, we're projecting down from 4D to 2D, which means there are now two hidden dimensions. There's no longer a single direction that's 'in front of us', but two perpendicular directions that are hidden from our 2D view. It's weird at first, but becomes easier to understand with more exposure. WOULDN'T THIS BE BETTER IN VR? Maybe? I haven't tried it yet, though there are videos of this on UA-cam. I'm wary for two reasons - one, VR doesn't change the fact that we have 2D vision, so we're in the same boat. Secondly, VR, or a Star Trek-style hologram, reinforces the 'realness' of the 3D space around us. Learning to think in 4D means recognizing that our line of sight is hidden from our view. On a screen, this line of sight is purely virtual. Accepting a 2nd virtual hidden dimension is easier when it's on the same footing as the first one. The more 'real' the 3rd dimension is, the more suspect the 4th is. I think that it's better to learn this on a 2D screen before attempting VR, but that's just my opinion. (This video is one of the better ones using VR: ua-cam.com/video/S-yRYmdsnGs/v-deo.html) CAN I PLAY WITH THESE 4D MODELS? The wonderful thing about virtual 4D models is that they give us the 4D experience we can't get in the real 3D world. Creating and playing with my own interactive models is how I learned much of this. Your brain starts to learn this 4D visual framework naturally, as it's the only way to make sense of where objects are in relation to each other. So I do plan on making interactive teaching models available to play with in some form, either with GeoGebra files or simple Unity game-like demos. Stay tuned! HOW CAN I SUPPORT THIS SERIES? If there's enough interest, I will likely be setting up a Patreon. I'm definitely putting a lot of time into creating this series, and it's becoming my main focus for the near future. Financial support would help me create more videos, with a faster timeline and better production values. I'm pretty new to UA-cam-ing, so I'm figuring out the financial side as I go. Please let me know if you'd be interested in supporting to help this series continue! Thanks, and see you in part 3!! -HC
Hey dude - you probably won't see this comment, but if you do, I've got some extremely useful tips on how to represent a 4D plane for your future video. I've put a comment somewhere explaining the way I arrived to the conclusion a while back, in a very intuitive way using the behavior of black holes. Black holes cause time dilation and more importantly, length contraction of spatial planes, which is precisely the 4th dimensional phenomenon that you're trying to explain in this video. Instead of using squares to represent planes, the 4th dimensional plane is best represented as a sphere, who is able to contract all of the other planes into a singularity where it would intersect the X/Y/Z axes at the origin. As a "black hole" sucks the object in, the planes get more and more "smaller" as the 4th dimensional plane distorts the 3D object towards the center. This is extremely helpful because it visually provides a way for people to understand the concept of orthogonality, where a 4D spherical plane would not intersect any other 3D plane except at its origin. This is an extrapolation of the famous Schwarzchild metric visual with cones intersecting a flat 3D space. Length contraction IS the 4th dimension, which was nailed perfectly in this video. Edit 2 - Another way to understand length contraction is to think about the concept of a flat mirror versus a curved mirror. A flat mirror allows for a 3D image projected on a 2D surface. When this mirror is curved away from the page (towards you), your 3D image is distorted in the same exact way a 4D plane would 'bend' your 2D mirror. On a curved / wavy mirror, you're looking at a 4D projection onto a 3D surface.
Best video ever about abstract concept. Mega like , amazing job 👍🏼 I agree that 18:18 is “aha” moment as watched this part 5 times. For next videos you may just assume that the viewer is 4D-minded nerd 🤓
I don't think this is as much a challenge as you might think for many people (unless I'm a very unique person)... I've always had quite an easy grasp on visualizing 3D environments and objects. Adding the 4th dimension slowly as you did here DID shatter the shackles of my 3D mind but I find it quite easy to acclimate to... Keep the great work! Your videos are a paradigm shifting revelation!
3blue1brown practically simultaneously released video about 2D Holograms, where 3D-objects are like packed in 2D, not just it's projected shadows. Maybe playing with it in some way will help to unwrap 4th dimension for viewer?
@@pavlovrublevskyi2960 I saw that but only watched a bit. I do have a fun idea in mind to see something similar and get a more 'physical' representation of 4D space. Later video.
Interestingly, this video did not add much to the stack of cards mental model. In fact, upon multiple occasions, I had to imagine the edges of that card-shaped screen around the drawings, so that I can mentally follow rotations that were shown. Towards the end, it was mentioned again, and it came back for the last minutes. It helps a lot in visualising 4D.
@@sztellanora The thing is, the deck of cards is a special case of the larger framework. It represents a specific angle (specifically a 90 degree rotation) of the w-axis into 'our' 3D space. It's a great introduction, but ultimately we will have to deal with 3-planes that appear to overlap in 3D space. The way to understand this is to accept that they are all 'behind' each other along the 4th axis, which is usually hidden or partially hidden from view. We'll look at this 4D framework more in part 3.
Seeing a model of 3d being projected into that weird shape really took some getting used to, but when you extended the w axis so we could see it was parallel it all just clicked. This is actually crazy.
The way I'm conceptualizing it is that essentially "hiding an axis" is similar to performing a 3D "crop." So as you rotate in 4 dimensions you are only seeing the 3D visible portion "crop". Which is how hiding a 3D into 2D leads from a cube to a square. We are only seeing one face. Now we are seeing one volume.
@@jeffreychandler8418 Yes but I would really love to see how we are actually going to visualize ourselves the whole 4D object. Or rotating the 3D object into 4th dimension. The method in this video is helpful but it's not the real stuff. I am skeptical but looking forward to being fully wrong! :D
Are you kidding me? I just watched the 20+ minutes of the first part, and when I checked the channel, this one has been published 7 minutes ago. Which means, it got released while I was watching the first part. Well, that's a nice coincidence, I'm able to watch the second part without any waiting time. :D
@@t.n.h.ptheneohumanpatterna8334 Well, now that I'm through with this, I'm gonna have to join you all waiting for the future parts. :D Well, one thing's for sure, I found another subscribe worthy channel today. :>
Part 2 actually existed already, but it was in the fourth dimension. You did such a good job with part one visualization, that you were able to see part two! (Meanwhile I'm stuck here with part 2 unable to visualize the fourth dimension well enough to see part one.)
Again, my mind has been enlightened. Part 3 is going to be fabulous. Hoping to see it sooner than later, but, no pressure; quality >>> quantity. Best of luck!
hands touching hands reaching out touching me touching you oh sweet caroline oh oh oh good times never felt soo good ba ba ba bu bu ba bda bbda bb also hae a good one
The Stack game analogy was mindbending. You esentially offered a hand to walk us from our old perception to new in a blink of an eye. I hope your content will be as much helpful as it was to me to anyone watching
From 18:15 that’s when it all clicked, what a genius lmao bro you need to be recognised for this work in the science field. You’re gifted, never seen anyone make 4D so easy to visualise like wtf
You again have represented a concept I've never seen anyone else visualize before; the moment you rotated the "flat" cube to reveal the W axis to show the perpendicular angle of the 4th dimension. That's always been one of the hardest things to wrap my head around, but I thought you worked up to it nicely! Great videos.
This is insane and I adore it. Flattening out the 3 dimensions and stacking them into a fourth one is what completely blew my mind. I can almost *see* it happening and that means we need a 5d sequel 😂 thank you so much, brilliant visuals!
I really appreciated how, at around 2:30, you clearly explained your graph-a clarity that's often missing in other presentations. It was great to see that you identified your axes as vectors and then went the extra step to convert them into distances. That level of attention to detail is refreshing! I haven't finished the video yet, but I’m looking forward to seeing how you maintain that consistency throughout. Based on what I’ve seen so far, I’m confident you will!
Having the gauges/disks represent the _squares_ of the vectors in those directions rather than the actual length also makes them much easier to work with since they'd have a constant sum. Why that's not used more often is beyond me.
This feels like some kind of forbidden ritual meeting. Like there is a city of math and you are this math dealer who lives inside of a hidden alley sharing secret info that no one else knows about. Edit: Holy $&@/! This blew up so much! Thank you all!
So far the models have been with straight-edge, cubic figures, which makes sense. But when the 4th dimension really gets interesting in my opinion is with non-uniformity (or a "curved line") in the fourth dimension. For example, a 4D object which on one end of the 4D length is a cube, but morphs into something else (like say, a sphere) as you move along the 4th dimension. Or a figure which only exists along one segment of the 4th dimension (i.e. a sphere that appears and gets larger, then shrinks back into nothing as you scroll along). I would love to see that kind of thing covered at some point in this series.
@@cezarcatalin1406 Here's a twist: it's impossible to depict a Möbius strip in 2D without self-intersections. Similarly, the "3D cubes" being shown in this video are absolutely _full_ of self-intersections, because they're actually 2D images, and the only times they _don't_ have self-intersections are when they're seen directly face-on, or edge-on in 4D, at which point they're unlikely to be recognized as cubes.
21:53 You don't need to tell us to like and subscribe. Anyone with a SOUL is slamming that bell as hard as they can to learn more. This series is incredible so far, and I have deep respect for you refraining from making the "whole new dimension of possibilities" pun.
The human brains ability to perceive higher spacial dimensions is like a muscle that needs to be strengthened and worked on everyday. It is 100% possible that one day this will be second nature to us. Thanks to resources like this, produced in such a concise and immaculate way, more and more people will learn. Great video 👍
The problem is that we have no reference for it in the physical world and therefore have no experience to draw from. But interactive 4D computer models can give us that experience and teach our brains to deal with 4D information. It's a weird sensation, you feel your brain rewiring the more you play. Kind of like when you study a foreign language and you start to actually think in that language.
@ob3ythee.t.128 Yes but since we see in 2D (a 2D version of our 3D world), our brains can really only visualise one dimension above that, therefore visualising in 4D which is two dimensions past our 2D sight will never be second nature. It’s like a flat organism trying to picture 3D, when his vision only consists of a 1D version of his 2D world which is just a line. However I do agree that with practice you’ll be able to better understand the logistics of the 4th dimension, but you would never be able to truly perceive/visualise it.
Watching these two videos has changed my whole perception on dimensions (dare I say reality). I love the way you explain everything, really easy to understand and has led me down some wonderfully deep thought experiment rabbit holes. Waiting for the next episode like it’s my next binge worthy series.
Part 1 felt like a great intuitive way to visualise what I already knew and understood about 4D, but this one just went above and beyond in terms of showing some of the fundamental logic and tricks you can do once you accept the framework, and all the consequences that come with it
This is single-handedly the best explanation I’ve ever seen for how the W axis can be perpendicular to parallel 3D planes. Your analogies are genius and I look forward to the third video in this series
The effect of rotation with the emergence of the fourth axis is truly mind-bending. It feels like something in my brain has been reorganized. I wonder how the same rotation would look like in VR headset, where the depth is perceived much better because of binocularity. Looking forward to the 3rd part.
Ok , the only reason that you don't have that much subscribers is that you don't upload that much. And I dont mean that in a bad way. You put sooo much effort in all of your videos! Keep up the good work!
@@HyperCubist maybe, one day, you'll release another video about spinors🥴😊 Also too tough subject as for my brains😵😬 Great videos! You're doing great job! Waiting for the next series 💪🏼
Great series, I've been working on visualizing 4D for a while now. Honestly I get that the stack game was an analogy for the vectors, but it came across very muddled and offered no insight (for me at least). The partial rotations into the w-axis are much clearer and actually improved the 4D conceptual space in my brain. Thanks, and keep up the excellent work!
To me it’s flattening a picture of a cube but with frames, like the film on a movie, and stretching it to see it different times but being able to still see them in 3D when you turn them around 360 degrees. This is awesome!
gosh yout videos make my brain tingle in a way i hadn’t have in a while. thank you so much! i hope i manage to grasp this entirely new and exiting concept of understanding a reality i can only see in my imagination.
Brilliant way to help us visualize 4D. Instead of adding a fourth dimension to our 3D reality, you converted our 3D reality to 2D, thus making the fourth dimension a much more familiar axis for our understanding. Absolute genius! Thanks for sharing!
Now to take it to the next step. If you set your total chip stack equal to “c” and make your mechanism of rotation total 4 velocity you have a visual representation of Special Relativity where “length contraction” an “time dilation” are simply the amount of chips in the hidden stacks relative to the observation point. Of course our experienced observation point is from within the 4D object like the LEGO man instead of from the outside perspective (god view) we saw in the video. I have been working on a grand unifying hypothesis for a few years now and I always struggled to explain what you have so eloquently shown in this video. Someday when I am finished with what I call the Time-Force Hypothesis I will have to find a way to make a UA-cam explanation video like yours. Excellent work 10 out of 10! Keep it up!
I'm having a Hypothesis about a new form of a universal cycle and therefore explored the world of 4D among other things like black holes and bubbles. I would really love to see and understand a visual representation of Special Relativity. Unfortunately what you just said did not connect with me. Can you point me in the right direction where i can find more info on it? ( @HyperCubist , will you touch on this somewhere in the future series? ) edit: oh my bad, i'm actually looking to visualize general relativity from god view. 😶🌫 not special.
If I do touch on Special Relativity (let alone GR), it will likely be in a separate series. I'm trying to keep this as purely math/geometry as possible, without any physics baggage. Partly because I don't have quite the depth I'd like to have in physics, and partly because you don't need any physics to understand four spatial dimensions, just geometric intuition and maybe some basic math. That said, I do use time as a crutch now and then for conceptual purposes, and may talk about light cones and the potential hyperspherical shape of our universe as physical examples.
@HyperCubist honestly you should skip that. Its 2024 and generalist content is a waste of time. You should focus on your thing. Its the audience job to be motivated or not.
The stack model is just perfect! Suddenly everything starts to fit. When you asked about difficulty to perceive it, I was not at 25/10 but may be at 3. It's easy and very logical this way. Thank you so much!
I love how you slowly build up the visualization technique, it's very crazy to think you were able to make visualizing 4D space take close to no effort. I specially liked the part at 18:18 where it all made sense. Thanks for the videos!
I understood every single thing at first time. Not only I am proud of myslef but also i am intimidated by how fast can process things. You're doing awesome job. Your way of explaining is great. English is not even my first language but I just needed to watch two of your videos about 4D one time with no repeats to understand. Again, awesome job. Please keep doing what you're doing!
I've never been very good, or interested in maths, but always loved the idea of the 4th dimension. Thanks for this series and making it quite understandable!!! Can't wait for part 3.
I really have to thank you for this video. You not only explained to me how I can imagine 4 dimensions but also that complex concepts are often taught worse than they could be taught. In the past, I have watched several videos on the concept of the fourth dimension but they never actually explained how I can imagine it. They always made a comparison to how the second and third dimension work but never explained how that can work between the third and fourth dimension as you did. Now for context: I live in Germany and have Maths "Leistungskurs (LK)" and Physics LK. These are special subjects were you dive deeper into the topics. Especially in Physics LK we never actually learn the concepts that are taught. We're always taught about the formulas and all the mathematical stuff but to this day I have not understood a significant part of the given topics. For example the theory of relativity. We were taught the concepts but I have never fully wrapped my head around it. The teacher really tried to do so but due to the heavy focus on actually all the mathematical stuff it's never been possible in the way as you explained the fourth dimension. If you could do another video series on the theory of relativity that would be really cool! 🙏
The moment you turned the cube into the 4th dimension and caused the other dimensions to flaten out i got this weird detached from reality feeling for a split second. Thanks for that!
Just came back to look through the channel to make sure there wasn't a video I missed. I don't trust YT's Algorithm. I see that Part 3 still isn't up. Will wait 👌.
During episode 1 I was a bit dubious on whether or not this would actually allow me to visualise the 4th dimension, but this part helped me make such huge strides that I'm pretty sure this series will deliver on its promise.
It's a gradual process. Especially once you get to play with models, you just start to get accustomed to certain things and grow to accept them as normal.
I'm a math teacher and have been fascinated with 4D visualizations since I was 12 or so, but never saw a visualization so understandable and tangible as this one. To realize that our perception of a cube on a screen is analogous to looking down from further up in 4D space is fantastic. This is amazing! Thanks!
This just hit me and now I can take objects around my room and just imagine the different shapes they make because of how they sidestep into 4D space! Thanks so much!!
Great video. Just one note: Something I've missed about the "Adding The Fourth Dimension"-Part of the video, is the discussion of lighting, which is important, because the way light travels to the camera determines the geometric 2D-projection that reaches our screens and helps us understand the 4th dimension. In the previous 3D case, the lighting was quite straight-forward, behaving in more or less the same way as in real 3D space. In your 4D demonstration however, it appears that the light rays can actually travel through the 4th dimension, which is fair enough, but not trivial, and really important to mention, because now the appearing "rectangles" or "cuboids" with shortened edges (3D-projections of objects being tilted into 4D space) are NOT the objects that ARE actually in the "3D slice" viewers are accustomed to. If we just want to see what ACTUALLY IS LOCATED in our 3D slice of the 4D world, than we would only have lines instead of squished squares and areas instead of squished cubes. Also a demonstration of a hypercube (4 pairwise perpendicular edges per vertice) in this visualisation would've been fitting.
I think I understand what you're saying. Is that like, if you rotate a 3d cube along the WX-plane, then a human who sees only 3d will just see the entire cube disappear except the middle of it as a flat YZ plane?
@@anipodat394 Short answer: Yes, at least I think so. If you are rotating a 3D cube along the WX plane, it should "disappear" in the sense that the observer limited to light rays traveling exclusively in his 3D slice would only see a 2D area. Note, that this area doesn't have to be a rectangle, if you would rotate along something nontrivial, like along the x=y=z=w line e.g. Also note, that the points you can see by rotating in the 4th dimension don't have to be limited to the boundary of your 3D cube in 3D space, so the area could also display points from inside the cube. One rather morbid demonstration of that is the rotation of a human body in 4D, which would yeald 2D cross sections displaying slices of his inner organs.
@@padgames8179 Yes, it's fascinating to me how you could see inside things this way. Check out the UA-cam video on how player skins look in the 4D game "4D Miner"! As I understand it, a 4D being who sees in 3D (as we 3D beings see in 2D) would see all our 3D "surfaces" at the same time. A human would be "flat" to them. They'd see the whole body; nose and back of the head and I think soles too (sounds quite strange), at once. Same for everything inside the body.
@@padgames8179 For a demonstration of this "looking inside", check out how player skins look in the 4D game, 4D Miner. I guess a 4D being, actually seeing in 3D, would see us as "flat". They'd see inside us, every single level of depth. Also, they'd see all our sides at once. Like face, back of head, and even soles? all visible at the same time (hard to imagine)
I understand the concept of this kind of, but it's blown my mind. I think this is the absolute best way to understand and visualise the fourth dimention. Absolutely outstanding video, look forward to the next one.
The stack game really helped me accept that a 3D object rotating into the 4th dimension and blinking out of existence from the 3rd can make logical sense. That was always something that I felt I had to suspend my disbelief going into videos like these.
Hello! Independently discovered my roomate had seen your last vid too. We at Ilum dorm 19 are avid fans. Unironically clapped when this showed up in his recs. Thanks for the great new vid!
I just feel like that is on the cusp of opening me up to something. Theres points everynow and then when I just feel like Im about to push through into something, or Im just im sensing somethint that my senses cant register but im absolutely feeling. These videos feel like theyre about to provide me with some way of understanding this feeling. Really greatful for these being made
Your visual expressions of this complex subject open a, dare I say, new dimension toward explaining so much of what we know is real, but are baffled with how to explain. Your two-part series are an absolute masterpiece, and should be mandatory viewing for anyone entering a class on geometry - or anyone with even the slightest curiosity about anything at all. I'm left with the sensation that, "Something just changed."
I don't see it as complicated. 4D emerged quite naturally when hiding the other "3D" vector. The stacks concept is super easy to grasp. Kudos. Crazy how a cube now looks like a triangle. It has always been there, right in front of us... Kudos and thanks.
I've waited so long for this. I always knew ppl were wrong to say "our brains cant do it" like neuroplasticity doesn't exist lmao. The bit you said about vr was extra special to me, no matter what we only have 2 eyes EVERYTHING ELSE IS JUST THE BRAIN.
Imagine figuring out all the freaking linear algebra to do this... It's already freaking hard with 3 dimensions, lust image the rotation matrix in 4D🤯🤯🤯🤯
Oh my god. I can see it!! I'm starting to combine my knowledge of the rank-nullity theorem with this video and I'm starting to be able to picture objects rotating "away" from us into the w dimension, causing comprehension. This is insane!!! I can't wait for more videos. You've made this mathematician really happy. The stack game as a visualization for the effects of a view point (so an orthogonal "perspective") was such a smart choice. EDIT: I think I'm starting to get it because I know what linear transformations look like so well, so when I see them violated here, such as parallel lines no longer looking parallel, I realize that some dimension must be lost into another dimension, which is the stack game!!! EDIT2: 20:00 And this is why 3D would look flat to 4D creatures!!!!
Yes a lot of Linear Algebra concepts come alive when you start to see this way. Later we'll see a visualization of how 2D planes can intersect at a point that would make Gilbert Strang's jaw drop ;)
All the stack game does is track rotation by visualizing the various components of an object's dimensions. The idea is to take the context out and strip rotation down to it's fundamentals, making it easier to accept a fourth dimension as just another place to store information.
Finally! I've been day dreaming about this for ages but my computer visual skills were just not up to par to make it visually happen! Tysm! Finally seeing it as imagined is so gratifying!!
This stack model is so good. I must say I'm not disappointed by episode 2 and now I can't wait for the rest of your course. Thanks a lot and keep up the great content
Yo I think you're a genius. Like a literal genius. So many people have tried to explain this and I've watched so many videos. This is amazing and so clear! I totally get it. Also I think I understand how we could achieve time travel.
I feel enlightened after watching this, just like after the first part. I love mathematics content on youtube, but this is the first time Ive ever learned something to do with math on youtube that will actually stick with me. One of the only things out there that I'd actually support on patreon, if only I wasnt a broke university student
I cannot express how much this has helped me At least, not with a 3-dimensional explanation :D (Thank you so much, each and every demonstration led up to a click in my brain, and more wows as each elaboration further enhanced how I could view the 4th dimension, the peak being at 21:15 when I completed the sentence with HyperCubist out of pure eureka-style excitement, simply amazing!)
I've tried to bring this concept of "squashing axis" into VR and oh boy, that looks way more amazing (and maybe even terrifying) than on a flat 2D screen. 😮😮😮 Actually SEE 3D things from every side of a 3D plane is breathtaking.
There is so much love in this explanation I was able to grab the concept and even made it to a 4D imagination room in my mind. It feels like starting to see things from more than one perspective like use more imagination fuel every minute and simulate other perspectives simultaneously. We can also relate to the infinity mirror to help the mind gap. It would mean a 3rd point of view when talking about a aspect that has maybe many different point of views on it helping to shift the mind more easy around and avoid sticking to only one idea. :D
omg! i'm struggling to conceptualize 4d space for few years already and have watched few dozens of videos on this subject but this is by far the best series to explain and visualize 4d. but i'm still not there yet. i feel like i'm 95% away from the top of the mountain but not 100% yet and remaining 5% are the steepest and the hardest to climb... thank you for your work and i'm waiting for the next video in this series. liked and subscribed!
![Visualizing 4D Geometry - A Journey Into the 4th Dimension [Part 2]](http://i.ytimg.com/vi/4URVJ3D8e8k/mqdefault.jpg)
![Visualizing 4D Geometry - A Journey Into the 4th Dimension [Part 2]](/img/tr.png)
Hi all! Thanks for your patience for part 2 - hope it was worth the wait. Just addressing a few points here:
THE STACK GAME: I feel like this went over some people's heads - I had to edit it down for length so it might have felt rushed. The idea is that the stacks are exactly the same as the colored bars tracking orientation and rotation in the 3 (and later 4) directions. Making it concrete with chips takes the rotation context out of it - we can 'rotate' by exchanging chips. This allows us to add a 4th dimension (the 2nd hidden box) in a way that doesn't break our brains, preparing us to understand 4D rotation. The big takeaway is that we can rotate a 3D object into the 4th dimension, which brings the w-axis into view (as in 18:18). If that's all you got out of this, then that's really all you need.
ISN'T THIS ALL JUST 3D? Actually no, it's all just 2D. The projections I'm using are quite standard, and are usually regarded as '3D shadows' of 4D objects. But we can't literally see 3D, we simply interpret 3D information from 2D images. We can also learn to interpret 4D information from 2D images. But that means treating the 'space' we're looking at as 4D, not 3D. This interpretation involves learning a new 4D visual framework, which is [Doc Brown voice] what makes 4D visualization possible. This will be the focus of part 3.
WHAT'S THE PLANE OF SIGHT?: In this visual framework, we're projecting down from 4D to 2D, which means there are now two hidden dimensions. There's no longer a single direction that's 'in front of us', but two perpendicular directions that are hidden from our 2D view. It's weird at first, but becomes easier to understand with more exposure.
WOULDN'T THIS BE BETTER IN VR? Maybe? I haven't tried it yet, though there are videos of this on UA-cam. I'm wary for two reasons - one, VR doesn't change the fact that we have 2D vision, so we're in the same boat. Secondly, VR, or a Star Trek-style hologram, reinforces the 'realness' of the 3D space around us. Learning to think in 4D means recognizing that our line of sight is hidden from our view. On a screen, this line of sight is purely virtual. Accepting a 2nd virtual hidden dimension is easier when it's on the same footing as the first one. The more 'real' the 3rd dimension is, the more suspect the 4th is. I think that it's better to learn this on a 2D screen before attempting VR, but that's just my opinion. (This video is one of the better ones using VR: ua-cam.com/video/S-yRYmdsnGs/v-deo.html)
CAN I PLAY WITH THESE 4D MODELS? The wonderful thing about virtual 4D models is that they give us the 4D experience we can't get in the real 3D world. Creating and playing with my own interactive models is how I learned much of this. Your brain starts to learn this 4D visual framework naturally, as it's the only way to make sense of where objects are in relation to each other. So I do plan on making interactive teaching models available to play with in some form, either with GeoGebra files or simple Unity game-like demos. Stay tuned!
HOW CAN I SUPPORT THIS SERIES? If there's enough interest, I will likely be setting up a Patreon. I'm definitely putting a lot of time into creating this series, and it's becoming my main focus for the near future. Financial support would help me create more videos, with a faster timeline and better production values. I'm pretty new to UA-cam-ing, so I'm figuring out the financial side as I go. Please let me know if you'd be interested in supporting to help this series continue!
Thanks, and see you in part 3!!
-HC
Hey dude - you probably won't see this comment, but if you do, I've got some extremely useful tips on how to represent a 4D plane for your future video. I've put a comment somewhere explaining the way I arrived to the conclusion a while back, in a very intuitive way using the behavior of black holes.
Black holes cause time dilation and more importantly, length contraction of spatial planes, which is precisely the 4th dimensional phenomenon that you're trying to explain in this video.
Instead of using squares to represent planes, the 4th dimensional plane is best represented as a sphere, who is able to contract all of the other planes into a singularity where it would intersect the X/Y/Z axes at the origin. As a "black hole" sucks the object in, the planes get more and more "smaller" as the 4th dimensional plane distorts the 3D object towards the center.
This is extremely helpful because it visually provides a way for people to understand the concept of orthogonality, where a 4D spherical plane would not intersect any other 3D plane except at its origin. This is an extrapolation of the famous Schwarzchild metric visual with cones intersecting a flat 3D space.
Length contraction IS the 4th dimension, which was nailed perfectly in this video.
Edit 2 - Another way to understand length contraction is to think about the concept of a flat mirror versus a curved mirror. A flat mirror allows for a 3D image projected on a 2D surface. When this mirror is curved away from the page (towards you), your 3D image is distorted in the same exact way a 4D plane would 'bend' your 2D mirror. On a curved / wavy mirror, you're looking at a 4D projection onto a 3D surface.
Best video ever about abstract concept.
Mega like , amazing job 👍🏼
I agree that 18:18 is “aha” moment as watched this part 5 times.
For next videos you may just assume that the viewer is 4D-minded nerd 🤓
I don't think this is as much a challenge as you might think for many people (unless I'm a very unique person)... I've always had quite an easy grasp on visualizing 3D environments and objects. Adding the 4th dimension slowly as you did here DID shatter the shackles of my 3D mind but I find it quite easy to acclimate to...
Keep the great work! Your videos are a paradigm shifting revelation!
3blue1brown practically simultaneously released video about 2D Holograms, where 3D-objects are like packed in 2D, not just it's projected shadows. Maybe playing with it in some way will help to unwrap 4th dimension for viewer?
@@pavlovrublevskyi2960 I saw that but only watched a bit. I do have a fun idea in mind to see something similar and get a more 'physical' representation of 4D space. Later video.
The long awaited sequel
fr and he did not disappoint at all
Interestingly, this video did not add much to the stack of cards mental model. In fact, upon multiple occasions, I had to imagine the edges of that card-shaped screen around the drawings, so that I can mentally follow rotations that were shown. Towards the end, it was mentioned again, and it came back for the last minutes. It helps a lot in visualising 4D.
Damn right. 4 times the D, 4 times the excitment
Absolutely
@@sztellanora The thing is, the deck of cards is a special case of the larger framework. It represents a specific angle (specifically a 90 degree rotation) of the w-axis into 'our' 3D space. It's a great introduction, but ultimately we will have to deal with 3-planes that appear to overlap in 3D space. The way to understand this is to accept that they are all 'behind' each other along the 4th axis, which is usually hidden or partially hidden from view. We'll look at this 4D framework more in part 3.
Seeing a model of 3d being projected into that weird shape really took some getting used to, but when you extended the w axis so we could see it was parallel it all just clicked. This is actually crazy.
The way I'm conceptualizing it is that essentially "hiding an axis" is similar to performing a 3D "crop." So as you rotate in 4 dimensions you are only seeing the 3D visible portion "crop". Which is how hiding a 3D into 2D leads from a cube to a square. We are only seeing one face. Now we are seeing one volume.
@@jeffreychandler8418That’s exactly it
Exactly
@@jeffreychandler8418 Yes but I would really love to see how we are actually going to visualize ourselves the whole 4D object. Or rotating the 3D object into 4th dimension. The method in this video is helpful but it's not the real stuff. I am skeptical but looking forward to being fully wrong! :D
For me was the rotating figurine, all of the sudden i *THINK* i got it!
Are you kidding me? I just watched the 20+ minutes of the first part, and when I checked the channel, this one has been published 7 minutes ago. Which means, it got released while I was watching the first part.
Well, that's a nice coincidence, I'm able to watch the second part without any waiting time. :D
Bruh I had to wait a month 😢 your lucky
@@t.n.h.ptheneohumanpatterna8334 Well, now that I'm through with this, I'm gonna have to join you all waiting for the future parts. :D
Well, one thing's for sure, I found another subscribe worthy channel today. :>
Lucky you!
Part 2 actually existed already, but it was in the fourth dimension. You did such a good job with part one visualization, that you were able to see part two!
(Meanwhile I'm stuck here with part 2 unable to visualize the fourth dimension well enough to see part one.)
@@t.n.h.ptheneohumanpatterna8334 BRUH I HAD TO WAIT 4 MONTHS UR LUKY
Again, my mind has been enlightened. Part 3 is going to be fabulous. Hoping to see it sooner than later, but, no pressure; quality >>> quantity. Best of luck!
hands touching hands reaching out touching me touching you oh sweet caroline oh oh oh good times never felt soo good ba ba ba bu bu ba bda bbda bb also hae a good one
it depends on what the quantity is for if quality > quantity or not.
The Stack game analogy was mindbending. You esentially offered a hand to walk us from our old perception to new in a blink of an eye. I hope your content will be as much helpful as it was to me to anyone watching
From 18:15 that’s when it all clicked, what a genius lmao bro you need to be recognised for this work in the science field. You’re gifted, never seen anyone make 4D so easy to visualise like wtf
I mean science is for empirical stuff, and we have spacetime but that's different. This is more for pure mathematics
This was the lightbulb moment for me as well. It feels familiar and new all at once it's crazy.
Seeing the W axis as 90degree to all XYZ axises made that clicked for me as well.
@@freddielo4330exactly
I was hooked the moment you said ‘let’s rotate it into the 4th dimension’
It's amazing how you make something that seems impossible, so intuitive.
You again have represented a concept I've never seen anyone else visualize before; the moment you rotated the "flat" cube to reveal the W axis to show the perpendicular angle of the 4th dimension. That's always been one of the hardest things to wrap my head around, but I thought you worked up to it nicely! Great videos.
Yooo part 2 dropped
Edit after watching: This really helps explain how the 4th dimension can be visualized! Very cool.
This is insane and I adore it. Flattening out the 3 dimensions and stacking them into a fourth one is what completely blew my mind. I can almost *see* it happening and that means we need a 5d sequel 😂 thank you so much, brilliant visuals!
If you kind of think about it the 5th dimension could be the vector instead you just call it the 5th dimension
let’s go 5d
waiting for the 11d update
I really appreciated how, at around 2:30, you clearly explained your graph-a clarity that's often missing in other presentations. It was great to see that you identified your axes as vectors and then went the extra step to convert them into distances. That level of attention to detail is refreshing! I haven't finished the video yet, but I’m looking forward to seeing how you maintain that consistency throughout. Based on what I’ve seen so far, I’m confident you will!
Having the gauges/disks represent the _squares_ of the vectors in those directions rather than the actual length also makes them much easier to work with since they'd have a constant sum. Why that's not used more often is beyond me.
This feels like some kind of forbidden ritual meeting. Like there is a city of math and you are this math dealer who lives inside of a hidden alley sharing secret info that no one else knows about.
Edit: Holy $&@/! This blew up so much! Thank you all!
Y’all, I’ve watched this 3 times now. I think I’m getting it.
@@yellowbacon69 looks like you didn't take the oath
Yeah it gives some conspiracy vibes 😭🙏
@@alert2 nah I did, but it was midnight.
He’s hiding in the W dimension (pun fully intended)
So far the models have been with straight-edge, cubic figures, which makes sense. But when the 4th dimension really gets interesting in my opinion is with non-uniformity (or a "curved line") in the fourth dimension. For example, a 4D object which on one end of the 4D length is a cube, but morphs into something else (like say, a sphere) as you move along the 4th dimension. Or a figure which only exists along one segment of the 4th dimension (i.e. a sphere that appears and gets larger, then shrinks back into nothing as you scroll along). I would love to see that kind of thing covered at some point in this series.
We definitely will - especially when we get into cubinders and hyperspheres and the like.
@@HyperCubist
Challenge:
Depict a Klein bottle in its full 4D glory - no self-interactions!
@@cezarcatalin1406 Here's a twist: it's impossible to depict a Möbius strip in 2D without self-intersections. Similarly, the "3D cubes" being shown in this video are absolutely _full_ of self-intersections, because they're actually 2D images, and the only times they _don't_ have self-intersections are when they're seen directly face-on, or edge-on in 4D, at which point they're unlikely to be recognized as cubes.
Wait things can “Morph!” In the “4th?” 🤯 can’t wait to see part 4 !
@@cezarcatalin1406 I've been planning that exact video for a few years now. I'll meet the challenge - provided you accept a 4D visual framework.
I have watched so many videos on this over the years and this is the only one that makes sense. I can’t wait until the next one!
Same. They never get to the point.
21:53 You don't need to tell us to like and subscribe. Anyone with a SOUL is slamming that bell as hard as they can to learn more.
This series is incredible so far, and I have deep respect for you refraining from making the "whole new dimension of possibilities" pun.
The human brains ability to perceive higher spacial dimensions is like a muscle that needs to be strengthened and worked on everyday. It is 100% possible that one day this will be second nature to us. Thanks to resources like this, produced in such a concise and immaculate way, more and more people will learn. Great video 👍
The problem is that we have no reference for it in the physical world and therefore have no experience to draw from. But interactive 4D computer models can give us that experience and teach our brains to deal with 4D information. It's a weird sensation, you feel your brain rewiring the more you play. Kind of like when you study a foreign language and you start to actually think in that language.
@@HyperCubist what about being able to actually see more than 2 stacks by changing the properties of light? can we do that next?
@@Sesquipedalia Now _that's_ a tall order.
@@Sesquipedalia blud really thinks we're light benders😭😭
We have 2 eyes, bro
@ob3ythee.t.128 Yes but since we see in 2D (a 2D version of our 3D world), our brains can really only visualise one dimension above that, therefore visualising in 4D which is two dimensions past our 2D sight will never be second nature. It’s like a flat organism trying to picture 3D, when his vision only consists of a 1D version of his 2D world which is just a line. However I do agree that with practice you’ll be able to better understand the logistics of the 4th dimension, but you would never be able to truly perceive/visualise it.
I genuinely have never seen any videos that have changed my world perspective this much. Thank you for this series
Watching these two videos has changed my whole perception on dimensions (dare I say reality). I love the way you explain everything, really easy to understand and has led me down some wonderfully deep thought experiment rabbit holes.
Waiting for the next episode like it’s my next binge worthy series.
You first video on 4D was the most interesting video I’ve seen on the subject. Looking forward to future projects
Part 1 felt like a great intuitive way to visualise what I already knew and understood about 4D, but this one just went above and beyond in terms of showing some of the fundamental logic and tricks you can do once you accept the framework, and all the consequences that come with it
This is single-handedly the best explanation I’ve ever seen for how the W axis can be perpendicular to parallel 3D planes. Your analogies are genius and I look forward to the third video in this series
I'm not young to lie, but this is opening up a completely new way of understanding life
I'm not exactly sure yet but I can feel some sort of profound philosophical thing coming to me, something about people hiding their true selves
Your chips stacks shenanigans melted my brain cells 😂
This video series is one of my favorites on YT! Thank you for publishing these
The effect of rotation with the emergence of the fourth axis is truly mind-bending. It feels like something in my brain has been reorganized. I wonder how the same rotation would look like in VR headset, where the depth is perceived much better because of binocularity.
Looking forward to the 3rd part.
yeah I definitely want to see these concepts using perspective projection in a 3d vr environment
well thats rather simple to try, isnt it.
Thank you! I never thought I could wrap my head around 4d but here I am! These videos are extremely well made!
Thank you! Much more to come.
Ok , the only reason that you don't have that much subscribers is that you don't upload that much. And I dont mean that in a bad way. You put sooo much effort in all of your videos! Keep up the good work!
Yeah I thought I"d upload once a week lol.
@@HyperCubistthen I am waiting with bated breath for that part 3 next week PLEASE
@@HyperCubist maybe, one day, you'll release another video about spinors🥴😊
Also too tough subject as for my brains😵😬
Great videos! You're doing great job! Waiting for the next series 💪🏼
@@HyperCubist
Please take your time, though. If you're not finished in one week then it's fine. No need to rush, your work is awesome!
No rush!
Quality over quantity! Viewership will come at is due time
Great series, I've been working on visualizing 4D for a while now. Honestly I get that the stack game was an analogy for the vectors, but it came across very muddled and offered no insight (for me at least). The partial rotations into the w-axis are much clearer and actually improved the 4D conceptual space in my brain. Thanks, and keep up the excellent work!
To me it’s flattening a picture of a cube but with frames, like the film on a movie, and stretching it to see it different times but being able to still see them in 3D when you turn them around 360 degrees. This is awesome!
gosh yout videos make my brain tingle in a way i hadn’t have in a while. thank you so much! i hope i manage to grasp this entirely new and exiting concept of understanding a reality i can only see in my imagination.
Babe wake up part 2 just dropped!
Brilliant way to help us visualize 4D. Instead of adding a fourth dimension to our 3D reality, you converted our 3D reality to 2D, thus making the fourth dimension a much more familiar axis for our understanding. Absolute genius! Thanks for sharing!
Now to take it to the next step. If you set your total chip stack equal to “c” and make your mechanism of rotation total 4 velocity you have a visual representation of Special Relativity where “length contraction” an “time dilation” are simply the amount of chips in the hidden stacks relative to the observation point. Of course our experienced observation point is from within the 4D object like the LEGO man instead of from the outside perspective (god view) we saw in the video.
I have been working on a grand unifying hypothesis for a few years now and I always struggled to explain what you have so eloquently shown in this video. Someday when I am finished with what I call the Time-Force Hypothesis I will have to find a way to make a UA-cam explanation video like yours. Excellent work 10 out of 10! Keep it up!
commenting here to see if anything comes up of this
I'm having a Hypothesis about a new form of a universal cycle and therefore explored the world of 4D among other things like black holes and bubbles. I would really love to see and understand a visual representation of Special Relativity. Unfortunately what you just said did not connect with me. Can you point me in the right direction where i can find more info on it?
( @HyperCubist , will you touch on this somewhere in the future series? )
edit: oh my bad, i'm actually looking to visualize general relativity from god view. 😶🌫 not special.
Time cube fans just love the joke.
If I do touch on Special Relativity (let alone GR), it will likely be in a separate series. I'm trying to keep this as purely math/geometry as possible, without any physics baggage. Partly because I don't have quite the depth I'd like to have in physics, and partly because you don't need any physics to understand four spatial dimensions, just geometric intuition and maybe some basic math. That said, I do use time as a crutch now and then for conceptual purposes, and may talk about light cones and the potential hyperspherical shape of our universe as physical examples.
@HyperCubist honestly you should skip that. Its 2024 and generalist content is a waste of time. You should focus on your thing. Its the audience job to be motivated or not.
The stack model is just perfect! Suddenly everything starts to fit. When you asked about difficulty to perceive it, I was not at 25/10 but may be at 3. It's easy and very logical this way. Thank you so much!
This is absolutely amazing ! Thanks ! Cant wait for part 3 !!
This is the best video for explaining the fourth dimension! Thank you for making this!!😊
It amazes me how much of this actually clicked in my brain and makes sense for me.
I love how you slowly build up the visualization technique, it's very crazy to think you were able to make visualizing 4D space take close to no effort. I specially liked the part at 18:18 where it all made sense. Thanks for the videos!
This feels like the closest understanding i have towards the 4th dimension!! BRAVO
I understood every single thing at first time. Not only I am proud of myslef but also i am intimidated by how fast can process things. You're doing awesome job. Your way of explaining is great. English is not even my first language but I just needed to watch two of your videos about 4D one time with no repeats to understand. Again, awesome job. Please keep doing what you're doing!
People have already tried to make videogames in 4D.
This video series is opening that door for more game developers!
I would love to check out those games. Would you mind sharing their names?
I've never been very good, or interested in maths, but always loved the idea of the 4th dimension. Thanks for this series and making it quite understandable!!!
Can't wait for part 3.
YES i have been refreshing you chanel every week
I really have to thank you for this video. You not only explained to me how I can imagine 4 dimensions but also that complex concepts are often taught worse than they could be taught. In the past, I have watched several videos on the concept of the fourth dimension but they never actually explained how I can imagine it. They always made a comparison to how the second and third dimension work but never explained how that can work between the third and fourth dimension as you did.
Now for context: I live in Germany and have Maths "Leistungskurs (LK)" and Physics LK. These are special subjects were you dive deeper into the topics.
Especially in Physics LK we never actually learn the concepts that are taught. We're always taught about the formulas and all the mathematical stuff but to this day I have not understood a significant part of the given topics. For example the theory of relativity. We were taught the concepts but I have never fully wrapped my head around it. The teacher really tried to do so but due to the heavy focus on actually all the mathematical stuff it's never been possible in the way as you explained the fourth dimension.
If you could do another video series on the theory of relativity that would be really cool! 🙏
I already know shii gonna be fire after a blinker
Oh absolutely
The moment you turned the cube into the 4th dimension and caused the other dimensions to flaten out i got this weird detached from reality feeling for a split second. Thanks for that!
Been waiting for this.
Bookmarked & Saved the playlist (just in case the algorithm fails to show new released video.)
Just came back to look through the channel to make sure there wasn't a video I missed. I don't trust YT's Algorithm. I see that Part 3 still isn't up. Will wait 👌.
Wow, this is my far the the best video on the 4th dimension I've seen to date!! Thank you for the time you have put into this!
During episode 1 I was a bit dubious on whether or not this would actually allow me to visualise the 4th dimension, but this part helped me make such huge strides that I'm pretty sure this series will deliver on its promise.
It's a gradual process. Especially once you get to play with models, you just start to get accustomed to certain things and grow to accept them as normal.
I'm a math teacher and have been fascinated with 4D visualizations since I was 12 or so, but never saw a visualization so understandable and tangible as this one. To realize that our perception of a cube on a screen is analogous to looking down from further up in 4D space is fantastic. This is amazing! Thanks!
Yooo I was just rewatching the first one yesterday. Been looking forward to this.
This just hit me and now I can take objects around my room and just imagine the different shapes they make because of how they sidestep into 4D space! Thanks so much!!
Great video. Just one note: Something I've missed about the "Adding The Fourth Dimension"-Part of the video, is the discussion of lighting, which is important, because the way light travels to the camera determines the geometric 2D-projection that reaches our screens and helps us understand the 4th dimension.
In the previous 3D case, the lighting was quite straight-forward, behaving in more or less the same way as in real 3D space.
In your 4D demonstration however, it appears that the light rays can actually travel through the 4th dimension, which is fair enough, but not trivial, and really important to mention, because now the appearing "rectangles" or "cuboids" with shortened edges (3D-projections of objects being tilted into 4D space) are NOT the objects that ARE actually in the "3D slice" viewers are accustomed to.
If we just want to see what ACTUALLY IS LOCATED in our 3D slice of the 4D world, than we would only have lines instead of squished squares and areas instead of squished cubes.
Also a demonstration of a hypercube (4 pairwise perpendicular edges per vertice) in this visualisation would've been fitting.
I think I understand what you're saying. Is that like, if you rotate a 3d cube along the WX-plane, then a human who sees only 3d will just see the entire cube disappear except the middle of it as a flat YZ plane?
@@anipodat394 Short answer: Yes, at least I think so.
If you are rotating a 3D cube along the WX plane, it should "disappear" in the sense that the observer limited to light rays traveling exclusively in his 3D slice would only see a 2D area. Note, that this area doesn't have to be a rectangle, if you would rotate along something nontrivial, like along the x=y=z=w line e.g. Also note, that the points you can see by rotating in the 4th dimension don't have to be limited to the boundary of your 3D cube in 3D space, so the area could also display points from inside the cube. One rather morbid demonstration of that is the rotation of a human body in 4D, which would yeald 2D cross sections displaying slices of his inner organs.
@@padgames8179 Yes, it's fascinating to me how you could see inside things this way. Check out the UA-cam video on how player skins look in the 4D game "4D Miner"!
As I understand it, a 4D being who sees in 3D (as we 3D beings see in 2D) would see all our 3D "surfaces" at the same time. A human would be "flat" to them. They'd see the whole body; nose and back of the head and I think soles too (sounds quite strange), at once. Same for everything inside the body.
@@padgames8179 For a demonstration of this "looking inside", check out how player skins look in the 4D game, 4D Miner.
I guess a 4D being, actually seeing in 3D, would see us as "flat". They'd see inside us, every single level of depth. Also, they'd see all our sides at once. Like face, back of head, and even soles? all visible at the same time (hard to imagine)
I understand the concept of this kind of, but it's blown my mind. I think this is the absolute best way to understand and visualise the fourth dimention. Absolutely outstanding video, look forward to the next one.
aw dang no part 3 yet
The stack game really helped me accept that a 3D object rotating into the 4th dimension and blinking out of existence from the 3rd can make logical sense. That was always something that I felt I had to suspend my disbelief going into videos like these.
Hello! Independently discovered my roomate had seen your last vid too. We at Ilum dorm 19 are avid fans. Unironically clapped when this showed up in his recs. Thanks for the great new vid!
Very cool thanks! What college?
@@HyperCubist yes very good video Ty but be careful the comments are watched by bots and it’s best for them not to know where he lives
@@maxcrocky9426 true that.
I just feel like that is on the cusp of opening me up to something.
Theres points everynow and then when I just feel like Im about to push through into something, or Im just im sensing somethint that my senses cant register but im absolutely feeling.
These videos feel like theyre about to provide me with some way of understanding this feeling.
Really greatful for these being made
Let’s goooo really enjoyed the first one.
Your visual expressions of this complex subject open a, dare I say, new dimension toward explaining so much of what we know is real, but are baffled with how to explain. Your two-part series are an absolute masterpiece, and should be mandatory viewing for anyone entering a class on geometry - or anyone with even the slightest curiosity about anything at all. I'm left with the sensation that, "Something just changed."
Thank you! Much more to come!
BRO PERFECT TIMING!
I don't see it as complicated. 4D emerged quite naturally when hiding the other "3D" vector. The stacks concept is super easy to grasp. Kudos. Crazy how a cube now looks like a triangle. It has always been there, right in front of us... Kudos and thanks.
How is this free? 😭
Welcome to the Internet, brother
Where you can learn calculus, quantum mechanics, and four-dimensional geometric visualization, all for free 🎉🎉
But you will need to pay if you don’t want all of your personal information stolen 🙂
*insert Nord VPN ad
I've waited so long for this. I always knew ppl were wrong to say "our brains cant do it" like neuroplasticity doesn't exist lmao. The bit you said about vr was extra special to me, no matter what we only have 2 eyes EVERYTHING ELSE IS JUST THE BRAIN.
Imagine figuring out all the freaking linear algebra to do this... It's already freaking hard with 3 dimensions, lust image the rotation matrix in 4D🤯🤯🤯🤯
Honestly deriving new formulas in maths is so long. Some of the simplest things have such long proofs and steps beforehand.
@@Prashant-pm7iz absolutely agree it's madness
4d rotation matrix is actually very similar to the 3d one. rotations always occur on a single plane
@@smallkloon They get trickier when you add in perspective (which I haven't done in this vid).
@@HyperCubist of course, excited to see you go into it for future videos!
Oh my god. I can see it!! I'm starting to combine my knowledge of the rank-nullity theorem with this video and I'm starting to be able to picture objects rotating "away" from us into the w dimension, causing comprehension. This is insane!!! I can't wait for more videos. You've made this mathematician really happy. The stack game as a visualization for the effects of a view point (so an orthogonal "perspective") was such a smart choice.
EDIT: I think I'm starting to get it because I know what linear transformations look like so well, so when I see them violated here, such as parallel lines no longer looking parallel, I realize that some dimension must be lost into another dimension, which is the stack game!!!
EDIT2: 20:00 And this is why 3D would look flat to 4D creatures!!!!
Yes a lot of Linear Algebra concepts come alive when you start to see this way. Later we'll see a visualization of how 2D planes can intersect at a point that would make Gilbert Strang's jaw drop ;)
Babe! Wake Up! Visualizing 4D pt 2 just dropped!
thank you for making these videos, maybe my favorite videos published by any youtuber this year.
waiting for pt. 3
Oh thank God you posted another one. I just saw part one of this yesterday for the first time and I have not stopped thinking about it all day. 🎉
13:55 i don’t understand the stack game at all, but i completely understand how these rules apply to the model
All the stack game does is track rotation by visualizing the various components of an object's dimensions. The idea is to take the context out and strip rotation down to it's fundamentals, making it easier to accept a fourth dimension as just another place to store information.
@HyperCubist you explain everything so well. Thank you for making these videos.
Finally! I've been day dreaming about this for ages but my computer visual skills were just not up to par to make it visually happen! Tysm! Finally seeing it as imagined is so gratifying!!
Hell yeah. Hypercube hype!
This stack model is so good. I must say I'm not disappointed by episode 2 and now I can't wait for the rest of your course.
Thanks a lot and keep up the great content
Woo! more of this! Ive been waiting for this! Also have you played 4d golf? it's a great game for intuiting 4d
Yo I think you're a genius. Like a literal genius. So many people have tried to explain this and I've watched so many videos.
This is amazing and so clear! I totally get it.
Also I think I understand how we could achieve time travel.
Idc that it took so long to release I'm just happy it did at all and not 20 years into the future
This is like the best description of 4d EVER! I LOVE IT! KEEP UP THE HEART WORK!
i love you for making these videos
as soon as you were explaining the 4th stack everything clicked. nice.
Oh so THATS what happens to things when I drop them and can’t find them
Please extrude yourself
I feel enlightened after watching this, just like after the first part. I love mathematics content on youtube, but this is the first time Ive ever learned something to do with math on youtube that will actually stick with me. One of the only things out there that I'd actually support on patreon, if only I wasnt a broke university student
Yeah very long awaited video, next time plz be quick 😅
I cannot express how much this has helped me
At least, not with a 3-dimensional explanation :D
(Thank you so much, each and every demonstration led up to a click in my brain, and more wows as each elaboration further enhanced how I could view the 4th dimension, the peak being at 21:15 when I completed the sentence with HyperCubist out of pure eureka-style excitement, simply amazing!)
I'd like to live in a Hyper cube
Re-read the 3 Body Trilogy recently and felt like an absolute genius when the 4th dimension shenanigans were happening. Thank you for these 🙏🏻
Wake up babe, new dimension just dropped
I've tried to bring this concept of "squashing axis" into VR and oh boy, that looks way more amazing (and maybe even terrifying) than on a flat 2D screen. 😮😮😮
Actually SEE 3D things from every side of a 3D plane is breathtaking.
Can you explain a bit more? You really saw 3d? How did it feel? Did it make you better understand 4d?
The dislikes are from 2D creatures
I am fully hooked and really feeling my brain grow. Can't wait for the next part!
OMG I've always tried to wrap my head around this but after watching the two videos, it's much easier to visualize 4D now. Thanks a lot! Great visuals
There is so much love in this explanation I was able to grab the concept and even made it to a 4D imagination room in my mind. It feels like starting to see things from more than one perspective like use more imagination fuel every minute and simulate other perspectives simultaneously. We can also relate to the infinity mirror to help the mind gap. It would mean a 3rd point of view when talking about a aspect that has maybe many different point of views on it helping to shift the mind more easy around and avoid sticking to only one idea. :D
i love how understanding the narration is with how difficult it is to understand the topic
Possibly the best educational presentation of anything that I've ever seen
These videos give me a new understating of the phrase "expand your mind."
No way, I just subscribed yesterday and I got recommended this video today, a month after your last video! What a gift!!
This is maybe the most unique and intuitive model of flattening the axis that I have seen. Amazing video.
omg! i'm struggling to conceptualize 4d space for few years already and have watched few dozens of videos on this subject but this is by far the best series to explain and visualize 4d. but i'm still not there yet. i feel like i'm 95% away from the top of the mountain but not 100% yet and remaining 5% are the steepest and the hardest to climb... thank you for your work and i'm waiting for the next video in this series. liked and subscribed!