# A Breakthrough in Higher Dimensional Spheres | Infinite Series | PBS Digital Studios

We all know how

three-dimensional oranges are stacked in grocery stores, but

what do you think the best way to stack 100-dimensional

oranges is? I’m Kelsey

Houston-Edwards, and this is “Infinite Series,” a new PBS

digital show about mathematics. The study of math

is old, really old. But right now it’s

expanding faster than ever. Each week, we’ll

explore a new corner of this puzzling universe

that’s being carved out by mathematicians right now. For today, it’s hyperspheres. What the heck are those? Well, it’s a sphere

in dimensions that are bigger than three. And while stacking

hyperspheres has surprising practical

applications in error-correcting codes

which help ensure accuracy when transmitting

data, like through cell phones, satellites,

and the internet, I want to talk

about them mainly. Because they’re cool

their properties are counter-intuitive, and

even seemingly simple questions about hyperspheres

are hard to answer. But before we get there,

let’s start with spheres. It’s easy to define a

sphere in any dimension. It’s all the points that

are a fixed distance away from a central point. On a two-dimensional

plane, you can imagine holding a

rod fixed at one end and spinning the other

end to trace out a circle. In three dimensions, the same

trick works, but now the rod has more freedom to move

and traces out a sphere. In higher dimensions, this

traces our a hypersphere. So what does that look like? Well, before we go there, let’s

talk about what mathematicians mean when they say

higher dimensions. We can visualize one

dimension as a line. But another way

to think about it is all the points described

by one coordinate, which we usually call x. Two dimensions is

just all the points described by two coordinates,

usually called x and y. Three dimensions is

just three coordinates. That means it

takes three numbers to describe a point in

three-dimensional space. Two numbers alone isn’t

enough to give directions to a specific point in

three-dimensional space. Even though you can’t

really visualize it, four-dimensional space

is just the point specified by four coordinate. Eight-dimensional space is just

eight coordinates and so on. Mathematicians have wondered for

centuries about the optimal way to fill space with equal-sized

non-overlapping spheres, what’s known as sphere packing. You can try the two-dimensional

version at home. How should you arrange

pennies on a table so that you have the

least amount of table showing between the pennies? At best, you can cover

about 91% of the table. What about in three dimensions? Way back in 1611,

famed astronomer Johannes Kepler

guessed that the best way to pack three-dimensional

spheres are this and this. These 3D sphere

arrangements are intuitive. It’s how tennis balls or oranges

are often stacked in stores. They waste the least amount

of space between balls, filling about 74%

of total space. Kepler was right, but

it took nearly 400 years for mathematician Thomas

Hales to actually prove Kepler’s conjecture. But what about sphere packing

in higher-dimensional space where spheres are

less intuitive? Mathematicians have

struggled to prove anything about sphere packing

in dimensions bigger than three. However, just a few

months ago, in March 2016, mathematician Maryna

Viazovska proved the best way to pack spheres in

eight and 24 dimensions. The standard way to pack

spheres in three dimensions has an analog in

higher dimensions, but the spheres

move further away from each other as the

dimension increases. Then something special

happens in eight dimensions. In eight dimensions,

there’s exactly enough room between the

spheres to squeeze in new ones. For a detailed explanation

of Viazovska’s ideas, check out the “Quanta”

article in the description. Now, here’s the

totally wild part. Mathematicians don’t

know the best strategy for packing spheres in

any other dimensions. We only know the

best arrangements in dimensions 2, 3, 8, and 24. That’s it. All the other dimensions

are pretty mysterious. Part of why sphere packing

in higher dimensions is so difficult is that

hyperspheres are basically impossible to visualize. People have used different

visual representations of higher-dimensional

spheres, particularly in four dimensions. One method mathematicians find

useful is to look at slices. Think about it this way. If you pushed a

three-dimensional ball through a two-dimensional

surface, like a table, the portion of the ball

intersecting the surface would always look like a disk– first a tiny disk,

getting bigger, and then getting smaller again. The disk is a 2D

slice of the ball. Similarly, if a crazy

four-dimensional creature took a four-dimensional ball

and pushed it through our three-dimensional

world, we’d see it as a three-dimensional ball– first a tiny ball, then

bigger, and then smaller. These are the 3D slices of

a four-dimensional ball. After four dimensions

our visual imaginations are pretty much useless. Here’s two thought

experiments to show how weird hyperspheres are. First, what happens if you

confine a sphere to a cube? Start with a 2D square,

and draw a circle that just barely touches the four edges. Analogously, in three

dimensions, draw a cube and inside it a sphere

that touches all six sides. The circle takes up

79% of the square, but the sphere only

takes up 52% of the cube. The pattern continues

in higher dimensions. Inside an n-dimensional

cube, nestle a sphere that just barely

touches all 2 times n sides. As the dimension n

goes up, the percentage of the cube that’s

occupied by the sphere gets smaller and

smaller, approaching 0. In 30 dimensions, the sphere is

about 10 to the negative 13th the size of the cube inside it. This is about how

big a grain of sand is relative to a sports arena. Except that in this

analogy, the sand grain is touching each wall

of the sports arena and is still round. Yeah, I know that’s weird,

but just stay with me, and let’s try one

more experiment with cubes and spheres to

see if we can understand what a hypersphere looks like. Take a two-dimensional box,

and cut it into quarters. Draw a circle filling each

one of these four boxes. Now draw another

circle in the center of the box that just barely

touches the other four circles. Notice that the interior black

circle is far inside the box. Now let’s try the

whole thing in 3D. Start with a cube, and split it

into eight parts, or octants. Place a sphere so

that it perfectly nestles inside each octant. In the center of the

whole cube, draw a sphere so that it touches each of

the other eight spheres. Again, this sphere is

way inside the cube. We can repeat the same

thing in higher dimensions, nestling 2 to the n

equal-size spheres into an n-dimensional cube and

constructing a central sphere that touches the others. The central sphere gets bigger

as the dimension goes up. But here’s the totally

mind-blowing part. At nine dimensions,

the central sphere touches the sides of the cube. After nine

dimensions, the sphere actually bursts through

the sides of the box. Why is this happening? Roughly, because as

the dimension goes up, the distance between

opposing faces of the cube stays the same while

the diagonal distance between opposite corners

gets longer and longer. . See. Hyperspheres are totally

counter-intuitive. But that’s part of what

makes them so awesome. If you have a cool way of

visualizing hyperspheres, let us know in the comments. I’ll see you next week

on “Infinite Series.” [MUSIC PLAYING]

a lot of bullshit information,that is not proved and stolen from others ideas……and finally you didnt give an answer about 4th dimensiion and how a sphere would look like…….and i dontmind how people watch this and beleive all your lies…this chanel is for dumbs

I think after the part where the cubes "multiplied" in dimension, I could understand how a smaller sphere (which is no longer just a sphere ofc) could touch all sides of the hyper cube…

Remember that Ball that change on your computer desk top. THAT !

Her hair looks like a helmet.

If we were able to see any object in 4D it would look like a sphere of a colossus of colors. In order for us to understand what we are looking at, we would have to shine a light on it so that it projects a 3D shadow. If a 4D cube projects the shadow of a tesseract, would it be safe to say that a tesseract is a 4D square?

Is that a pocket illusion on her shirt ?

The reason why sphere packing doesn't work all of the time is because, if you watch the "8e Dimension", you'll see and understand that it is the packing of Tetrahedrons that takes up all of space.

IF it is counter intuitive then it just ain't true!

So what happens when lots forced to overlap? Do they ever end up coexisting ? (2 diff dimensions both able to pick up on each frequency? Then we might see the Greys not all looking like bland non texture loaded avatars that we are not able to “register”?

Assuming there are 100d oranges.

According to this explanation, it would be untrue to say that every place is equally at the center of the universe. That only holds true for a 3d sphere.

Nize ,you have tried to make this topic easier to understand…

4 :19 Is there any example that we have done. Such as passing 3d object to 2d world.. same way any 4d experiences in our 3d world.. if you research just put a video on this buddy..

It would be better if you had an experiment.

The golden ratio has turned out to express a multidimensional multifractal equation. Current physical theories imply it is a notably classical appearing Fractal Dragon blending into a humble Mandelbrot, while classical mathematics have proven to be wildly inefficient and too limited. Efficiency in even mathematics has some sort of real meaning.

I didn't get past crystals lol. I'm to sober!

absolutely useless knowledge

100 dimension is deferents between deepness or highness 🙂

Sounds like bullshit

Ok i think i get it

Lol the dislikes

666

Was that creature is the godzilla

To visualize higher dimensional shapes, let's start in 2d picture a circle, then picture the circle oscillating through itself, this forms a sphere repeat the process with a sphere oscillating through itself and you get a hypersphere, repeat process unto any number of dimensions

Smoke DMT, see 11 dimensions.

They got it wrong. Once the circle goes beyond the square you need to reverse the process. A 8 d sphere is pushed through our 3D reality example. It starts small then grows to a point before it starts to get smaller. You need to adjust your video. The circle does not outgrow the box.

What are these cordinants you speak of.

4 dimension maybe is time , seeing sth growing old is seeing an 4d object moving through out 3d world

She has nice dimensions.

Cheating.. Still 3dimention

People figured out how to make money without doing any real work and bullshiting the hell out of everyone. AWESOME!!! just like politicians.

Who shouted "FOUR!" within the first 45 seconds, then switched off?

first you have to convince the spheres to do whatever it takes to fill in the space . A little cooperation goes a long way

When I was in 9th grade, taking Algebra I, my teacher noticed my interest in mathematics and programming and encouraged me to explore hyper objects. Being a programmer, I chose to use triangles because I could visualize the dimensions with midpoint up to one configuration of the hyper-tetrahedron (or whatever it is actually called).

The idea is simple. Take a point and place a point on it. Then lift it up into a line segment and lay it down. Now place a point in the middle, connect it to the other two points. Then lift it up and lay it flat as a triangle. Now place a point in the middle and connect to the other points and pull it up into a tetrahedron. Then place a point into the middle and connect it to the other points. Lifting up in 4D is not something we can do but this is where the programming comes in.

My father had gotten me the first edition of computer graphics gurus which was written in C. (C++ had not quite been invented yet) I used the basic transforms to rotate this 4D object and project to a 3d object which was displayed as a wire drawing on the 2D screen. I had eight keys I used to rotate in all four dimensions using matrics transforms and it was really cool. I verified all of my guesses about what other shapes should appear and found other very strange shapes.

I got lucky with my ignorance as I am only now ready to find a way to repeat this program from the perspective of actually understanding what I am doing. Meaning, I have yet to successfully repeat that program. however, I hope my description can inspire others to try this out for themselves.

My hypothesis suggests that this girl is tight, iykwim.

This just in,…Physics Girl finds a Man with 2 hyper dimensional balls.

Would you imagine if we apply this to stack 6packs of beers ?

Reallh, i think you are full of crap. You cant prove any of this, they are just thought experiments. Keep daydreaming.

This legit makes my head hurt…

What she said prompted a thought:

The definition of a circle and sphere are essentially the same: An infinite number of points equidistant from a single point.

The difference being that a circle's points are defined by 2×1 matrices, and a spheres points by 3×1 matrices.

Since each higher dimension includes each lower dimension (a line is in infinite number of points, a square an infinite number of lines, space an infinite collection of planes…), it would follow that the definition of spheres in higher dimensions would also be the same: an infinite number of points equidistant from a single point…except with points defined by larger and larger matrices.

So a 4D sphere's points would be defined by 4×1 matrices, 5D by 5×1's, etc etc etc.

Easy logic to follow and easy numbers to write down…but attempting to visualize what that looks like is giving me a headache.

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.

There are only 3 dimensions.

I like your spheres, they make me hyper ?

Read F. Buckminster Fuller – Synergetics… A must read. Everything is based on closest packing of spheres, namely octaves and fibronacci numbers.

What's up with higher dimensional objects which are not regular?

i just dont get anything , only enjoy watching that cute girl , i have to re watch

Modern physics is starting to sound like a clown show. The best

addition to the standard model on light is in Lesseirg papers https://www.youtube.com/watch?v=3_73gw3Yf_Q&t=313s

and the mischief of black holes https://www.youtube.com/watch?v=po1CwKygzyA&t=33s

I think a good place to start might be to assume that each sphere has D+1 point contacts with nearest neighbors and that these contacts are equally distributed about the solid angle. Disks meet at the vertices of equilateral triangles. Spheres meet at the vertices of regular tetrahedra. One might project that an equivalent statement applies to each higher dimension. Once this construct is regularized, we could initiate proofs that it is or is not optimum. We probably also should be aware that solutions might not be unique in some higher dimensions.

Isn't it weird explaining multi-dimensions in a 3D space? Here we are visualising multiple dimensions on a device in two dimensions emulating three dimensions. It's too out there for me.

If they referred to them as matrices or arrays it might make more sense. But dimensions imply a physicality a measurable, perceivable entity. Whereas matrices and array are logical collections of data in multiple arrays. So for me this is a problem with definition not mathematics.

I'd love to know what these practical uses are? It may help understand the value of these theoretical mathematical excursions. Without a practical purpose you have to ask the value of these leaps in theory?

Im lost lol ……my iq dosnt compute this lol

God, I’m so smart now..

I use lasers 😉

Visualize? Sperm and Egg making a Human.

The good part about 4D space is that it is physically impossible for your earphones to get tangled.

The bad part about 4D space is that the same can't be said about paper.

But who needs paper when you can draw inside 3D cubes.

Ye they thought that the planets were held up by celestial spheres too.

"How do you stack 100 dimensional oranges?"

No need, they've probably stacked themselves quintillions of times over.

false logic

There is only 3D here, in this realm we live in daily.

Outside of it there is an infinite number of dimentions.

Fight me!

Not hyper spheres but proof of dimension separation on this paper.

Latest discovery on time dilation, fixed light speed and

understand Algebra, then download this open access, no login,

no ads, 2018 Cern archive journal paper.

https://www.zenodo.org/record/1447218

Not hyper spheres but proof of dimension separation on this paper.

Latest discovery on time dilation, fixed light speed and

understand Algebra, then download this open access, no login,

no ads, 2018 Cern archive journal paper.

https://www.zenodo.org/record/1447218

hyperspheres are best explained as losing track of what’s being said and losing interest quickly

This is total crap

I think my car keys ended up in that dimension. …..

So esentually your saying we live in a single demensional space…and the 90% angles of the square are representitive of the the temperal offset due to expansion being closer to the event horizons unpredictability but while still trying to catch its mirrored self in the singularity that it has no choice but to chase due to gravity mass volume etc…

I'd smash

There are a few things limited to number 3: The white light consists of only 3 colors: red, yellow and blue dot. The other colors are a combination of the three. Fire (flame) is formed with 3 elements: fuel, oxygen and heat point. The tripod does not balance at all times. A triangular figure can not change its shape, that is, it is rigid, or a figure like many sides of 3 can be deformed. It is also 3 dimensions, there are only 3 and can not be anymore. The rest is an imagination that can never be solved.

I have watched quite a few videos and this is the best explanation I have found. Thank you.

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

Mirrors and holograms

Her voice was hard for me to listen. Sensitive ears.

I been to the 5th dimension of a dmt breakthrough..it is REAL!!

Okay … "cool" is not a valid reason to understand something – and theories (mathematically correct or otherwise) are just theories.

She's cute, but she has the charisma of a dish rag. Fix this.

This ain't true so much ! Any more dimensiones than 3 are explicable by 1xyz …in space

Can’t you just imagine the 4D world as a line made up of infinite 3D points? And then the 5th dimension as a plane made up of infinite 4D lines and just keep on going that way?

I'd like to see all dimensions of the good looking presenter.

I was on complete 4D an it scared my shit off.

Why does she speak with her hands so damn much?!

yes I've much more cooler way to visualize a hyper sphere, see my balls

I saw this somewhere, but basically you can draw any dimension of a cube onto a 2D sheet of paper.

You won't know how it behaves or moves, but you will begin to understand at least 1 of its possible shapes.

It works like this:

1D:Draw a straight line(Line)2D:Draw another parallel straight line and connect the ends(Square)3D:Draw another square but offset towards one corner, then connect the corners of the 2 squares(Cube)4D:Draw another cube offset to one corner of the original and connect the corners of the two cubes(Hypercube)5D:Draw another hypercube and connect the corners of the two hypercubesI think you get it by now. Basically, you just draw the same thing and connect the corners for as long as you want to get the next dimension.

As said before, you wont know how the shape moves or behaves, but you will know at least one of its possible shapes.

It's very difficult if not impossible to try to visualize objects of dimensions n>3. Because our brains are not wired such way….Our brains are stuck in "Survival Mode" wich make 4d perception not a high priority…the only possible way left is mathematical modelisation…the book "Flatland" written by Edwing Abbott is very inspiring: Unless you manage to leave our 3D world, you never will have a grasp of what 4d objects look like.

Please don't belittle me because maybe this is a very stupid question but here it goes… Why are we limited to believe that a circle must fit into a square? Why not, for example, allow a dimension to curve?

A line is not one dimensional, it's 2 d

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.

"In Math you come up with infinite numbers, in Physics that usually means you are wrong."

After 8 minutes, I still don't know what sphere packing really is.

I'm struggling understanding the 4th dimension and she is talking about the 24th…. Now I don't even know my name…

bawxes

Where does the dimension of time fit in? Is this the fourth co-ordinate? Einstein saw space (measured in 3D cordinates) as co-relevant to time…

and i though 3rd dimention maths was hard

Her hands are distracting

I really really really want to know what shapes in hyper dimensions look like

Ничё так баба…

"And while stacking hyperspheres has surprising applications in error-correcting codes, which help ensure accuracy when transmitting data, like through cellphones, satellites, and the internet, I want to talk about them mainly because they're cool."

She seems very anxious to prove the importance of her topic for technology as the standard of relevance, but…regrettably there is no reason to think "the stacking of hyperspheres" is critical for any of those things.

PBS's intellectual lying is one of its longstanding fortes lol.

It's like exercising at your max weight in your brain for every set in every exercise.

I’d bang

quantum mechanics, quarks packed, but not touching, into nuetrons and protons, electrons orbiting at a distance, emitting photons when they droping an energy level, sounds like mathematical n=dimensional sphere packing to me

Can i 4D print my dick in higher dimensionnal spheres? ._.

A Rachael hairstyle that I actually like bcos she looks fantastic with it.

3:33

It tells me that there's a point where it proves there is a limit to the number of dimensions that actually exist.

That hand thing is annoying