Let the higher math begin.

PBS is doing amazing work with their digital content. I'm blown away with the quality of channels like PBS SpaceTime. This is definitely the future of educational content and an invaluable public service.

"If you have a cool way to visualize hypersphere..." you deserve a Nobel

Maryna Viazovska has won the 2022 Fields medal for this work!

I can show you what hyper dimensions looks like but we'll need a ton of shrooms.

Holy cow! An entire channel related to mathematical research? this is beautiful!

Thank god someone finally figured out how to stack 100-dimensional oranges! I was really worried there for a while.

The expanding and contracting sphere was the single most intuitive visualization of a hypersphere I have ever been presented. Kudos.

FIRST (in the 47th dimension) Welcome to the PBSDS fam! Can't wait to see more

It's great to now also have a nice PBS channel dedicated to math :)

A pity this series was discontinued :'-(

I have always sucked at math. Even though I really love trying to learn about all of this stuff, its usually confusing. But I feel like I actually understand this now. Great video!

7:08 this sounds super exotic but after thinking about it, I realized that the math is actually trivial with Pythagoras' theorem. The 2^n balls all have radius 1/4, but the distance from their center to the center of the cube is sqrt(n)/4. So the center ball has radius sqrt(n)/4-1/4. When this becomes > 1/2, the center ball grows out of the cube. And with n=9, this is exactly equal. It might have been nice to show this in the episode, or at least hint at the fact that anyone can do this for themselves with high school math.

This sounds incredibly useful for the next time I'm packing 8d oranges.

Now, this is a math teacher I'll listen to all day.

This is one of my most favorite math paradox! I remember working this problem in Calculus!! :-D It got me to thinking about a hyper-dimensional version of the Gabriela's horn paradox. Where you take the curve y = 1/x from 1 to infinity and revolve that curve about the x axis. If you tried to fill the resulting shape with paint you can do it. But if you tried to paint the outside you would never have enough paint because the surface area diverges. So what happens when you take the 3d horn shape and revolve it about the Y axis? To make a 4-dimensional hyper-horn? Or take the 4-dimensional hyper-horn and revolve it about the Z axis? What happens to the divergent-convergant ratio of surface area to volume at these higher dimensions? Does it remain a paradox or are we left with varying levels of infinities?

This is awesome! -John

I have a way of visualizing n-dimensional spheres where n > 3, it's just that it's illegal in most states...

I've often tried to wrap my head around 4D, but watching this video about up to 24D is just mind-blowing.

Glad I final came over to this channel and excited for the journey ahead. Thank you and horay for knowledge!

A MATH SHOW I'm so excited!!! Subbed, obvi. Welcome to the family Kelsey 👨‍👩‍👧‍👦

now im just frustrated that i can't visualize a hypersphere

I'm discovering this channel only today, but I'll make sure to catch it up! Great first episode, higher-dimensional topology is so cool.

I love this! Just found this channel today and subscribed. Going to enjoy looking at the older videos and looking forward to the new ones.

PBS Digital Studios just keeps getting better and better. Love you guys!

I visualize hyperspheres as regular spheres that have had a lot of coffee

I just love the work of the animators in PBS videos. Pretty neat.

I applaud you at PBS for presenting programming that is not for idiots. Unfortunately, I am too stupid for this. I'll stick to mostly grasping astrophysics over at Space Time.

Wait, so you guys started a math channel over a month ago, and have an actual mathematician hosting it, and I'm only just hearing about it NOW??? This is unacceptable!! Time to go binge watch them all. :) PS: Great job so far Kelsey! Very much looking forward to more episodes.

I'm so glad that people in my country can be that smart! I hope there will be more people like Maryna! (She's from Ukraine)

She could have mentioned why the diagonal distance increases as the dimension of the hyper cube increases.. Pythagorean theorum ! Then how do we calculate the vacant hyper volume of a hyper cube after it is packed with hyper spheres? I hyper wish to see that !

So the big bang might be the effect of a 4 dimensional universe being pushed through 3 d space?

Love it! Math it up.

I like everything about this channel! From the brilliant content to the warm nature of her presentation. Annoyed at myself for not having it found earlier.

This was really fascinating. I had to pause a few times to get the full understanding. Excellent lecture!

I like it! She doesn't speak too fast or too slow, she speaks clearly and is easy to understand. I don't know what the show is called "infinite series" but you've earned my subscription

Yay! I voted for this in the PBSDS Survey! It's finally here! :D Great episode, can't wait for more.

Love this channel. Thanks to everyone making this possible!

I JUST FOUND THIS CHANNEL AND I AM IN LOVE THANK YOU

I have great expectations for this channel! So many good topics to cover... The omega number, analytic continuation, Euler's formula, ... Good luck!!

This is going to be interesting. I'll definitely be keeping an eye on this channel :-)

Excellent, Subscribed! and thank You for the Links!!! Much Love from Sunny South Africa

Several years ago my cousin Patricia was studying math that involved higher dimensions, and geodesic topology. Or something…. A professor told her it was a waste of time, and that her work would only be of interest to Gods and Aliens. Yet Patricia is still working on it despite the criticism. I applaud her efforts and I applaud this show for talking about similar topics.

Its one of the best math explanations for higher dimensions i have ever seen

Woah wait, she was my math TA when I took Linear Algebra two years ago! Go Kelsey! Always thought she was a great speaker and explainer!

My brain just bursted into the fourth dimension. Great video! Totally subbing :-)

First time I came across a video made by PBS ∞ Series. Subscribed !

can I just say that finding this series has forever changed my life.

I am going to echo the sentiment in the comments below, as I was already going to before reading them. My first thought was, "How did I miss this until now?" Now I see that this is new and right up there with Spacetime and Physics Girl. Space Time covers A lot of Astrophysics and science misc. Physics Girl covers the experimentalist side of thing. And now we have this. All of this work is immensely appreciated. These three and Numberphile are my favorite channels. If I count Numberphile as a special elective, can I get a "PBS in Math and Physics"? Hmm... that was a joke while I was writing it.

This is the first math vid i came. Thanks for that!

I have been struggling with ways to help my friends visualize higher dimensional geometry. Take it as a high compliment that I'm stealing your way! Brilliant, brilliant stuff young lady! 😎

Great series and well articulated host. Liked and subscribed 👍

First!

thank you for making me realize that I have to explore more in mathematics.

Thanks for sharing this. It's simple and understandable.

Co-ordinates, NOT co-ordinants!

A great addition to the excellent maths-communication videos that have been accumulating on youtube..!

Can a fifth dimensional pencil draw 4D images? Can a 6D pencil draw 5D images? How would you shade it? What shape would the paper be? My head hurts.

I love this channel and PBS Space Time. Most other channels shy away from maths but with these channels a basic understanding is a prerequisite. Even Numberphile and Sixty Symbols keep the maths pretty basic. They're still great channels though!

If you have a cool way of visualising hyper spheres, don't put it in the comments, write your own paper first.

Cool!! The way a hyper sphere as described here kinda makes me feel like our universe is one and we are stuck watching the 3D while the dimensional order of the universe goes up. Need to learn more! thank you for making all these videos!

The hyper sphere bursting through the 10th dimension and going all sea urchin looking sort of mimics my understanding and mental picture of how stars morph from one state to another, and the Big Bang for that matter. If it’s a helix or perfect wave graph of demolition and creation over the LONG course of time these dimensional rules make sense in a perfect order kind of way. The same way quantum stuff is intuitive on the surface you know

This is amazing! Amazing narrator!

YAY a new math channel!

Came for the content. Subscribed for the host. I'll show myself out.

It's relatively easy to understand that higher dimensions are degrees of freedom. Thinking of dimensions as right angles could be a limitations. A higher dimension just encapsulates lower dimension, like sphere encapsulates circle. I am not sure why we focus on spheres, hexagonal shape has all neighbours clearly defined without gaps. Higher dimensions and their shapes could be defined well at 0 to 1 fraction level. At fraction level, behaviour of multiplication and division changes. Using slope calculation to find points on a line also may not work correctly and line may start to bend as we explore deeper in fraction level. Thus at fraction level higher dimensions could be more visible.

Do i have a cool way of visualizing hyper spheres? Yeah, it's called, N-Dimethyltryptamine. ;-)

I don't understand it, but i subscribed..

Imagine a baseball being thrown at your face, the ball gets bigger and bigger as time passes, you could think of the baseball as a 4 dimensional object where time is the 4th dimension. Think about this until it hits you :)

I've never visualized hyper spheres, but I have tried to do the same for hyper cubes. I have made a 3^n model of a hyper cube where each dimension adds another •3 to the coordinate to find its hypervolume(?). I can only model it however as the closest I can think to see it is the perspective model and one other I don't remember the name of. All I can do is conceptualize lines within the polytope.

Kelsey said "We only know the best arrangements [to pack spheres] in dimensions 2, 3, 8, and 24." Well, though it be trivial, we also know the best way to pack spheres in dimensions 0 and 1. A 0D sphere is a point and a 1D sphere marks a line segment.

tl;dr: It's great to talk "about" the math, but it would be *far* better to actually *do* some math! While the details of higher-dimensional optimal sphere packing are understood by few without a math PhD (and, thus, 99.999% of KZfaq viewers, including me), it would be great if an accessible proof (or at least demonstration) for the 2D case could be presented, along with mentioning the math tools and techniques needed to work in higher dimensions. One missed opportunity may have been to directly calculate the space-filling percentage for a given (simple) packing (or, better yet, comparing a pair of simple n-D packings/arrangements). This should be simpler than finding or proving an optimal packing, yet still give a hands-on feel for working in higher dimensions. Just calculating the volume of an n-D sphere would probably have been worth doing (en.wikipedia.org/wiki/Volume_of_an_n-ball). If you connect the centers of a minimal (tile-able) example of an optimal sphere packing, what kind of geometrical object results? In 2D it is a triangle (2D -> 3 faces), in 3D a tetrahedron (3D -> 4 faces). What about in 8D and 24D? The "packing" *IS* the geometrical object described by the arrangement of the hypersphere centers, right? Why not make this explicit? Don't fear the math! While many of us may have forgotten most of the math we learned since turning 12, it would be great to have motivations (and pathways) to reclaim lost math skills and develop new ones. I suppose this really involves identifying your target audience. But don't aim low! I'd recommend a multi-level approach that "informs without alienating" so that the math-less and the math-ful can both enjoy the video. And all viewers can be inspired to become more math-ful. At the very least, we all should learn some new math vocabulary, even if we don't learn the math itself! Becoming "buzz-word compliant" is important: Even if I know nothing about, say, "SU(3)", if I see that text used in two different contexts, I should be able to infer that those contexts may be related (though perhaps only by using common math/geometry). That's useful! Vocabulary matters. Take a look at PBS Digital's Space Time for examples of how to handle "real" math at various levels of abstraction. Matt really has a knack for showing the "whole shebang", then taking it apart and showing how some parts have useful (and more accessible) approximations. But then, he gets to leave the math domain (for physics), and you don't! Numberphile is simply awesome in this area. Perhaps definitive. How can/should Infinite Series compare to and/or be different from Numberphile? Is Infinite Series a "math news" channel, or should it be more? Should it be limited to describing the exterior appearance of recent math developments, or should it try to bite into the sweet, juicy math interior? I'm not suggesting that Infinite Series have anything like formal, pedagogical math tutorials, but instead provide more along the lines of mathematical sketches, recipes and approximations that could be explored by those with sufficient curiosity and determination. Once an Infinite Series video reaches its intended run time, an ideal addition would be to include FMI links to additional learning resources. There's no need to reinvent any wheels. But there is a need to curate online math resources applicable to Infinite Series topics. PLEASE be sure to *always* include direct links to the original and fundamental papers involved (or full citations when the paper isn't freely available online). Some of us like to admire the pretty LaTeX math squiggles while reading the abstract, introduction and conclusions. Regarding the specific content of this video, there are MANY videos that provide very accessible visual explorations of n-dimensional geometry. Was there a need to repeat this well-worn path? Why talk about slicing/viewing higher-dimensional solids through lower dimensions? This felt like a fruitless detour to wind up at "hyperspheres are weird". I'd recommend focusing on issues directly related to getting a handle on hypersphere packing. I mean, if you want to do "An Intro to Hypergeometry" video, please do so! But that wasn't your stated topic for this video. There may have been an actual fail in this video: I believe it was a mistake to say that a higher-dimensional n-sphere "bursts through" the enclosing n-cube. It may have been much more useful and far more illustrative to say that the "completely contained" sphere gets smaller in higher dimensions! This would *directly* lead to a great illustration of *why* packings in ever higher dimensions fill a lower percentage of the available volume. While it was correct to say that optimal packings are known for only 2, 3, 8 and 24D, it is also important to generalize beyond this to say that very efficient packings have been identified in hundreds (?) of dimensions, but that proofs of optimality have been developed for only these 4 specific ones. The best-known packings for other dimensions can often state how closely they approach the theoretical optimum when the optimal packing and proof are not yet known. Infinite Series has a great mission statement: "Infinite Series [is] a show that tackles the mysteries and the joy of mathematics. From Logic to Calculus, from Probability to Projective Geometry, Infinite Series both entertains and challenges its viewers to take their math game to the next level." Please provide concrete steps along the path to that "next level". Looking forward to the next episode!

The volume of the box increases faster than the volume of the sphere. You can see this if you think of the empty space as a solid. It goes from nothing in 1D, to a gasket in 2D, to swiss cheese in 3D. Intuitively the volume grows around the relative sphere. With each dimension added, the number of "corners" around the sphere doubles. There's 2^n "corners". The box grows corners, the sphere doesn't.

Kelsey is going to be the next math/ science superstar.

Astonishingly beautiful!! (I mean the math as well as the host)

Very much looking forward to forcing my family and friends to learn some math!

Awesome video! I used circle and sphere packing with variable radii and Poisson jittering to try and optimally sample the surface of a spherical planet when distributing objects like trees, mountains, rocks, etc. Since I started with the larger objects and proceeded to fill the gaps with the smaller objects, I managed an average 72% efficiency. The Poisson distribution resulted in a nice, 'natural' distribution, free of the grid-like tendencies of our previous system.

Learning just took place! Well done!

This is lovely gift for me, this channel that is, do more number theory vids.

Thank you to Physics Girl that made me discover this channel. I'm already fan :) From France with love,

DMT

Understanding the 4th dimension is basically looking at yourself. You look in the mirror and you see your physical self, but you also have thoughts(which exist electromagneticly) and you have a electro magnetic field that surrounds your body, produced by your heart so we are 4th dimensional beings. Our body's being a physical shape but only another shape which is within more bigger layers of the same shape which are just energy. BUT IS VERY MUCH THERE JUST AS YOUR BODY IS. and learning to see these energetic aspects of yourself being your "aura" and beginning to identify back with your aura as we're supposed to, is entering 4d than 5d than higher dimensions. Basically looking at a person and seeing them within another energy them within another energy them within another energetic them. And this is where ghost come from. The energetic aspects of a human just no physical body and that explains us. We just have a body right now

You can easily talk about, for example, 7-dimensional space by saying each point can be in 3-dimensional space, with a symbol, size, color, and rotation. It's an easy way to disentangle mathematical dimensions from physical dimensions. So, you could describe a 7-dimensional point as (10, 4, -3, "A", 20mm, red, 30°). And you can visually put it on a chart along with other symbols to reinforce the idea. You don't have to get abstract so soon; it's a tangible way to get someone's head around higher dimensions before launching into the harder math.

THIS VIDEO IS AWESOME!

I visualize a hypersphere like I visualize the afterlife: I know it's probably there, I have no idea what it actually looks like, and it is absolutely terrifying to comprehend.

Here's my cool way of visualising hyperspheres: Ingest LSD

as a person who is enthused by body language, i must ask you to put your hands down! Your hand gestures are the equivalent of saying "yeah, um, like so... yeah". That being said, this is great work. I subscribed ;)

I know one cool way of "visualising" the three sphere. Namely, take two solid tori and glue them together by taking meridians of the first one and glueing them to parallels on the second one :) That is to say, please make a video about the Hopf bundle.

I'm sorry, but, wot? Perhaps there exist "hyper-dimensional" beings that are laughing their ass off, because we are going "WTF" with something that they perceive as natural as we perceive three-dimensional objects as natural. Fascinating stuff.

I am loving this video ! Congratulations for the great work. If possible, in the future, could you make a video about Poincaré conjecture and the solution of it. . Thank you.

Fascinating stuff! My interest in DMT brought me here. I have an interest in DMT because of a near death experience I had as a teenager, which closely resembles the recollections of people using DMT. Trying to describe the kaleidoscopic machinery of a massive sphere is almost impossible. The more I have looked into higher dimensional theory, the more convinced I am that there is a connection.

"coordinants"

ok I just confirmed that my suspicions were right, I'm a tuna

AWESOME! thank you!🔴🟣🔵🟢🟡🟠🟤⚫️

Just discovered this show. Love it. Stage tip: wondering what to do with your hands? Best thing to do is....nothing. Let those hands drop to your sides.

The worms that live in line-land can only see 0-dimensional points as flat-landers pass through their realm. Flat-landers on a plane can only see points, and lines as solids pass through their domain. So our hyper-buddies are like X-ray machines when staring down at us. I like explaining higher dimensions through drawing perpendicular axes. We can barely see the 3rd z-axis in our minds on a 2-d chalkboard because the z-axis is written as a perpendicular slant to the x and y axes. Plotting the points of higher objects is tough or impossible because of our limited 2-D vision. Some imagine time as the 4th dimension. Like, a series of 3-D spaces moving through a fourth degree of freedom. maybe, I don't know but it is cool. Then 5-D are sections of 4-D spaces moving in another degree of freedom. Some would say alternate 4-D timelines? Some neat stuff!

Dont understand it but this was dope as hell

Great show! I also gotta give a shout out to the fact that a woman is presenting this maths series, which is a great way to fight the cliches in a world where mathematicians are (unfortunately) often visualised as being men (which is so wrong). Way to go and keep the cool content flowing!

Watching this again, I miss you Infinite Series

Anyone else watch the movie Cube 2: Hypercube?