My brain hurts... Want a Klein Bottle?: stemerch.com/collections/maths-toys/products/klein-bottle-1 Edit: Some people have pointed out that the dimension in which the sphere surpasses the cubes volume is much higher than 262 and after finding a calculator that could handle these numbers I'm getting the same thing (around dimension 1205 is when it finally happens). And again here is the link for the proof of the 5 platonic solids video: kzfaq.info/get/bejne/ZpqVia-ftdjUdY0.html

The 4 dimensional beings are equally confused when imagining 5d cubes

Now I understand how my dog feels when I'm talking to it

That "knot from hell" got me💀

I've always thought that things people find "supernatural" were just random higher-dimension objects moving through 3D space. It would explain the "appearance" and "disappearance" aspect.

so if 1D cant see 2D as being 2D, and 2D cant see 3D as being 3D and so on - then could we be surrounded by objects of infinite dimension without realising?

The joys of watching a forth dimensional object represented in a three dimensional form on a two dimensional medium.

Let us all acknowledge Maryna Viazovska. She is the ukrainian mathematicia mentioned at 16:33, who found out about the packing in 8 and 24 dimensions. Last year she got a fields medal

60 seconds in, and I'm already far more interested in this than anything taught to me in the 4 years I spent in high school.

In higher dimensions you will find sanitizers that remove 100% bacteria.

I'm going for a job interview tomorrow in a 42-dimensional supermarket, so this video will be very helpful.

This whole time I always thought "dimension" was referring to different realities. So when they say "dimensional beings", it has a whole new meaning to me now. And seems more plausible as well.

6:23 the knot from hell 💀

What if flat-earthers aren’t actually crazy but they are 2d creatures that can’t imagine 3 dimensions?

There are some days when I feel pretty smart. This is not one of them.

Just took elementary topology this semester and this video would’ve been great for understanding euler characteristics and tilings

As someone learning French, I quite enjoyed the seemingly unintentional “loup in disguise” pun 😂

Everyone else: struggling to unknot their “knot from hell” Me: *passes the earphones through 4D space to turn it into an unknot

“Yes, this example is pretty simplistic, but-“ Bro I have no idea what you’re saying

Data sphere packing thing was really interesting, had no idea it was used that way. It's always cool to see how people come up with smart ways to organize information.

This video almost immediately flew directly over my head at the speed of sound

Thanks, I will remember your advice when wrapping oranges in the 42nd dimension.

6:20 "The knot from hell" Unknotting number: *INFINITY*

19:08 The inscribed sphere has radius root(n)-1, and you can use the formula for the volume of the n-sphere V_n to derive n = 262 for the minimal n satisfying V_n > 1.

Oddly enough, I absolutely love this and stuff like this. Fascinating and I actually can follow you. This is how my brain works. You are awesome.

I love to think that in another universe, there’s a 67th dimension KZfaq trying to understand how stuff would work in a 68 dimension world

Wife:I bet he's thinking about other girls Him:how should I wrap those 42nd dimensional oranges

I'm not really into geometry as such, but I certainly found this fascinating! Thanks!

Thanks for sharing this. Much like the parallelogram, every or any quadrilateral can be understood as just an angled view of a perfect square. I hope it makes sense. 😊

When you realise that the Klein bottle on the screen you're watching is a 2 dimentional represenation of a 3 dimentional representation of a 4dth dimentional object.

I’m adding you to my playlist for shrooms-day in quarantine

This intrests me even though I want to be a herpotolgist. Chemistry, Math, and Zoology are amazing things to learn about that I love! I may be in middle school but this is very intriguing! I may be a bit young but I love this and understand this!

I love watching your videos. They're so nerdy and techinal. Love that stuff!

I like how when we talk about higher dimensions it's almost always just about shapes, and nothing about how physics would go nuts

Things are definitely passing through one ear to the other here.

omg i’ve never watched a video where someone actually says the types of math in the end for further research. super awesome and helpful!!!

One application I’ve learned in my CS degree with higher dimensions is using vector spaces for information retrieval. For example, comparing documents with 2 words would have a vector space of 2 dimensions because it would represent a score on which word it was most relevant to. So therefore comparing documents with 10,000 unique words would create 10,000 dimensions and using some math, can calculate a query to a documents relevancy

The ppl who study Higher dimensions are actual mad lads

"And in 262 dimensions..." "Hm... yes, I can totally see that."

Excellent presentation. Tough to wrap one's mind around the fourth dimensional tesseract, no less even the next one!

4th dimensions being: "Yo dude! These guys can't even count ahahah"

If things like these were taught in school I would be way more confused, but also way more interested

14:40 I've seen people refer to the 4th dimension's interior quantity not as volume, but as Bulk.

My first experience with learning dimensions was an excerpt of “flatland”. I was in second grade and I read it in a math and science encyclopedia. I believe it was “Mathemagic” book in the Childcraft encyclopedia set. It’s old but timeless. Children are able to understand way more than we give them credit for. If you’re a parent. I suggest the mathemagic book in this encyclopedia set. It was the key that opened a door to a lifelong love of STEM. I wouldn’t be here without reading that book. One love.

When I was young, I tried to draw a soccer ball with all hexogons and got frustrated when it didn't turn out. Only now do I realize that it is actually impossible....

This is the highest I’ve ever been despite never doing any drugs

My brain cannot comprehend this and cutting a paper mobius strip down the middle had me in hysterics because I didn't actually believe it would just be one long strip. Thanks for sharing this even though I don't understand it at all! 🤣🤣

Excellent video. Very interesting, informative and worthwhile video.

20 minutes ago I felt pretty good about myself thinking that I’m a little smarter than the average person. Then I watched this video. Now I feel like I need adult supervision and should probably wear a helmet at all times!

Imagining a 4th dimension is like asking a blind person(since birth ) to think of red color. And its so frustrating

I need to point out a small error when a 3d object passes through the field of vision. The 2d observer does not see a circle getting bigger then smaller until its trough, because a 2d observer cannot see from above. Therefore, it will observe a point, expanding in a one dimensional line, then the line would shrink again, till it's gone.

I was watching something, I dont remember what, but I fell asleep and woke to a different science video than this, but it seems I left a trail of these video in my sleep. I'm going to continue to follow it now

Goddamn there’s so much about the universe we don’t know. We’re experiencing the tiniest shade of reality, this really seems like magic

My brother: I am strong My sister: I am beautiful Me: I know the cheapest way to wrap 42 dimensional oranges

That was beautiful man. I love that thank you for bringing a smile and laugh into my day.

You taught me so much and I taught my dad and showed him this video Your videos are amazing

6:21 I laughed so hard when it showed "knot from hell, unknotting number infinity" XD

I’m gonna need a drunk history style series where this guy gets plastered and tries to explain these kinds of subjects

This is a well made and engaging video packed with interesting stuff 🙂

It's so much easier and better learning outside of school. Thank you.

Why am I even here, I can't even understand simple maths...

Me: about to go to sleep at 2 AM. KZfaq: wait, higher dimensions!!!

You are fun to watch and hear. Keep going.

I suddenly realized that I'll be stuck in 3D for the rest of my life.

"Draw four dots anywhere on paper", okay (draws 4 dots in a straight line) "Now connect them to form a quadrilateral"......oops

My mind was blown when I saw his face because im so used to seeing his comedy skits 😂.

The last proof in Euclid's Elements gives a proof of why there can only be 5 regular solids. It's mind blowing.

Imagine you are a 100 dimensional student studying geometry, thats gonna take so long edit: just got this notif rn, the reply section is a bit chaotic

Plot twist: The UFOs we see are not actually visiting our earth, they are a part of crossing the 3D version of it from above

I enjoyed the explanation and diagrams depicting how increasing dimensions of the square leads to the decrease in total volume of the sphere. kinda puts our perspective of the universe into perspective in a way that reminds me of the 2d explanation of only seeing an expanding line. we are blind

Zach is funny asf in Zach Star Himself channel and a genius asf in this channel. you're a legend bro 😂😂

it's so amazing that we discovered stuff using mathematical properties of nature without actually being programmed to see it or conceptualize it easily

Damn I wish my brain could comprehend this

The best explanation of higher dimensions, the first person who understands what he is talking about!

13:35 gotta say I know next to nothing but some of the lingo here about this but here talking with chatgpt and pretty asked it the idea of how would it NOT reach a higher volume but still exist as a value in a given dimension in relation to its lower dimensions lol that’s awesome 👏🏼

He's explaining the 4th dimension, i'm watching this in the 3rd dimension, on my 2D pc screen, with my 1D brain With 0D brain juice left And -1D thought process 🥬🥬

I like how all of these shapes are still 3D, since we would have to be in those higher dimensions to fully visualize them.

20:49 Thank you very much. I was just wondering how to wrap my 42D Orange for my friend's birthday. This method greatly reduced the cost of the gift wrap.

Homestly i find all of that simple to understand but it also makes me regret not becoming a mathmatician /physicist, this is fun! Ty

Thank you for this simple and easy to understand tutorial on how to cost-effectively wrap my 42nd dimension oranges during this holiday season. It really helped.

Listen to this as you fall asleep. You may comprehend this better in whatever dimension dreaming occurs.

One of the best videos I have seen youre awesome

One of the most beautiful videos..glad that my friend recommended it...lots of love from a fellow physics student

6:22 “Knot from hell” LMAO

"Kissing numbers" So you're saying numbers get more action than me huh

When you watch the skits, sometimes you forget how smart Zach is

I thought you only made skits but I realized that I have seen your before on this channel when I was learning stuff. Cool.

who tf decided to call it an "unknot" instead of a "not knot"

You're the reason I'm going back to school for computer science and electrical (computer) engineering this fall. Thank you for motivating me and rekindling the curiosity and interest in life that I've been lacking.

I really enjoyed the video and love learning things I couldn’t in school

In Machine Learning high dimensions are very usual and then results for higher dimensions do matter. For example, for doing gradient descent, in lower dimensions local minima are a really problematic. But it turns out in high dimensions, they tend to become quite rare (just if you have 100 or 1000 or 10000 possible dimensions to descent, one of them will descent anyway). That's one of the reasons, why gradient descent outperforms in practice many more sophisticated optimization techniques once you apply it to really big data. There is also this interesting phenomene of finding the best estimator for a parameter given a nr of samples. In 2 or 3 dimensions, it's easy: it's just the average of the data. For higher dimensions, it's usually something like alpha * mean(samples) where alpha is < 1 (the exact number is quite hard to find and IIRC correctly in general an unsolved problem). I'm not much into this mathematics, but while this is highly counterintuitive, it matters a lot again in machine learning (there it's called regularization - and my impression is that it was somehow reinvented there by practicioners, just as it worked much better on real high dimensional big data). IIRC, this is very related to the behaviour of the n-dim sphere (an estimator somehow should be inside the "estimator" sphere spanned up by the samples, but as it has in high dimensions a really low volume as illustrated in this video, you'll have to adapt it by moving the parameter estimation more closer to the origin by multiplicating the samples with a factor < 1) So, like in many parts of interesting, but seemingly unrelevant mathematics (especially for very high dimensions), it indeed has significant counterparts in real world applications (and AI certainly is one).

The world needs more passionate educators such as you. Thank you!

This is the video that led me to discover the existence of pure math. I believe at the end I started googling the branches of math you mentioned. Since then, I’ve learned to write and read proofs, and I’m exploring the realm of proof-based math on my own before I get to undergrad as a math major. It’s thanks to you, Zach Star, that my math journey panned out this way

The most interesting video I've seen in a while

Sorry for commenting on such an old video, but I just discovered it and have thoughts. Many years ago, I played with higher-dimensional simplexes, where "simplex" is the polyhedron in a given dimension with the fewest faces possible. Additionally, for the purposes of this exercise, "face" is defined as a simplex of the next lower dimension. So a 1-D simplex (a line segment) would be made of two 0-D simplexes (points) to define the vertices (endpoints of the line segment). And a 2-D simplex (a triangle) comprises three 1-D simplexes (line segments). And a 3-D simplex (a tetrahedron) comprises four 3-D simplexes (triangles). I managed to draw a 4-D and 5-D simplex on paper (like drawing a hypercube, but with tetrahedrons instead of cubes) and discovered that: A 4-D simplex comprises five tetrahedrons, and a 5-D simplex comprises six 4-D simplexes. Because I was able to draw these, I could count the number of lower-dimensional simplexes in each simplex (e.g., the number of line segments in a 4-D simplex) Doing this, I came up with the following table. The first number followed by a colon represents the dimension under consideration. And the following list of numbers is the number of simplexes of successively lower dimension contained in the simplex of the dimension under consideration. To wit: 0: 1 (A 0-D simplex--a point--contains one 0-D simplex. Yes, that's tautological.) 1: 1, 2 (A 1-D Simplex--a line segment--contains one line segment, and two points/vertices.) 2: 1, 3, 3 (A 2-D Simplex--a triangle--contains one triangle, three line segments, and three points/vertices.) 3: 1, 4, 6, 4 (A 3-D simplex--a tetrahedron--contains one tetrahedron, four triangles, six line segments, and four points/vertices.) 4: 1, 5, 10, 10, 5 (A 4-D simplex--a hypertetrahedron--contains one hypertetrahedron, five tetrahedrons, ten triangles, ten line segments, and five points.) 5: 1, 6, 15, 20, 15, 6 (A 5-D simplex contains one of itself, six hypertetrahedrons, 15 tetrahedrons, 20 triangles, 15 line segments, and six points.) You've probably seen by now that this table is just Pascal's Triangle. I wasn't able to analyze anything beyond 5-D simplexes. I could certainly add another point to the drawing (each increase in dimension adds only one point to the simplex, which is one thing that makes simplexes fun to analyze in this fashion), connect every point to every other point, and count the line segments. But beyond that, I couldn't count simplexes of higher dimensions (triangle or above) than line segments. Does this follow Pascal's Triangle forever? I'm certain that it does, though I don't know how to go about proving or disproving that. But it doesn't involve ratios like between volume and area (or hypervolume and hyperarea), so I'm confident that it does follow Pascal's Triangle to arbitrarily high dimensions. My personal takeaway from this is that using this table, I can know how many simplexes of X-Dimension are contained in a simplex of Y-Dimension, for any dimensions X and Y where X

me: *not understanding so i turn on captions* the captions: this is called Oiler's characteristic me: ah, I see.

socks. the most common thing you'll find in higher dimensions are socks.

man... where were you when I was failing math in college.... I have no idea what you're talking about, but I know you know what you're talking about

I don't know what he's saying for the most part but i like how he says it

12:01 That sound scared me bruh, thought a 4th dimensional being was tryna sabotage my AirPods