Kissing Numbers - Numberphile 

"Imagine an eight-dimensional sphere." Okay, no problem 🙃

8:26 - "So imagine eight-dimensional spheres..." Brain: "Nope." [packs suitcase and leaves]

My kissing number is zero

Does Dr. Grime have a framed piece of Brown paper from Numberphile on his wall? That is adorable 😊

Part 2 of a trilogy always leaves you wanting more--looking forward to part 3. Glad to hear Dr. Grime say that it's hard to imagine how things look in extra dimensions--I've always struggled with that myself. But I also want to know which of those billiards tricks Dr. Grime can pull off.

5:03 - That's not a theta, that's a small aubergine.

Isaac Newton? Never heard of him...

I took my pants off for nothing!

Dr. James Grime puckering up in that thumbnail meant I had to watch this video hahaha

This was the topic of my bachelor thesis. Had much fun with this. And some sleepless nights in the End :X. Cool to see it on the channel! :)

Oooh, the 24-cell represents the 4-dimensional kissing number! Another reason to love that figure!

Me: Do you know that there's a cool thing about how many kissing points can a sphere have and it's bla bla bla Friend: So it's 12, but why? Why it's not 13? M: Yeah there's a reason for that. Some mathematical proof. F: What is it? M:.... The- There's a reason. Uum, I couldn't quite understand you know but there's a proof.

the 24 dimensional one should be 196560 not 196500

Loved meeting you in Denmark at my school James, and thanks for the pictures. Cheers!

soooooo amazing, you guys always blow my mind. Thanks!

I have been waiting for this new video!

Error - 196,560 is the 24th Kissing Number It seems that regardless of the answer for the others, the number is probably going to be divisible by the number of dimensions.

A four dimensional sphere could be imagined as a 3D sphere changing with time.

"We're gonna talk about kissing numbers!" Continues walking up the to screen. Everyone: OH NO HE'S GONNA- Me: He's a number?

LOL at "Nobody Knows" 8:15

My favorite part of this channel is when James Grime says 'NOOM-BAH' in the iconic way

In the cosine formula you have to divide by the product of the lengths. In this case it doesn't matter because it's 1, but it might have been worth mentioning it.

"Surely, using these formulas this is enough information to work out how many kissing points we have on a sphere" Well yeah, obviously....

I'm a mathematician. I work on my "The kissing problem in three dimensions" paper.

I was just watching another Numberphile video when this came up. Wow!

The kissing number in one dimension is two.

James “a bit of Pythagoras” Grime is my favorite mathematician.

Old but gold.

7:10 A 4-dimensional sphere looks like our Universe, according to some physicists. So, at least, we could see the local structure, in some part of one.

I've had similar conversations to this, but not about spheres. more like organizating books into a cube. or how many pens i can fit in a can

"Its less exciting than it sounds" is a rule of thumb when it comes to math. But most of the time it's still interesting.

I love this guy

Sphere trilogy? Starring Sigourney Weaver? Yes pls

5:00 Wouldn't drawing 3 circles/spheres touching each other be a much easier explanation? You would then get an equilateral triangle, formed by the 3 centers, and since you cannot get any of the spheres any closer to each other (only farther away) - the angle is always 60 or more degrees. Much easier to understand and visualize than a cosine theorem, I think

Fun fact: You can change the position of any two kissing spheres in the 3D configuration without all 12 spheres losing their contact with the inner sphere (meaning you don't need to lift any of them)

Correct value for кissing number in 24-dimensional space is 196560

I'm suddenly gripped by a desire to find the kissing number of dimension 5...

so dimension 8 and dimension 24 were solved you say? I see a pattern there but what happened to dimension 16?

Shouldn’t this come with the caveat “for all Euclidean spaces”? 😉

I knew the answer was 12 due to the packing problem and knowing that the hexagonal packing is (along with a few others) the most efficient packing.

Why are those picture frames in the background never put on a wall?

6:27 legend says this was the biggest expression of happiness he has ever shown..

Wikipedia gives the kissing number for 24 as 196,560, not 196,500.

My first numberphile video where I fully understood the maths before it began!

I knew Newton was probably right because I already knew the regular packing density is the greatest packing density for 3D spheres.

I love your channel very much

“a guy called Isaac Newton, I don’t know if you’ve heard of him” 😅

I like it that james surname (grime) rhymes with prime

Now I want to know about the packing of unit spheres around a central sphere of arbitrary radius. With central sphere radius you can get 2 spheres around it. How big does it have to be to get 3? 4? 10? I guess it approaches just the function for surface area at large R. I wonder if this has implications about quantized packing of quantum modes around small things.

my brain imploded after 3 dimensions

All hail the Singing Banana!

Will there be an exploration / explanation of the Leech Lattice?

Is there a video that explains the multidimensional graphic you use in the videos?

Dr. Grimes: So imagine 8 dimensional spheres...

Numberphile. Nobody, nobody, nobody does it better.

Nooom-BAH...Love it

I glad to see you, maniac )))

Do you think you could do a video explaining math(s) found in music?

When he mentioned the Kissing Numbers for dimensions 2, 3, and 4, I was so excited that there was a pattern! (For dimension d, it's K[d] = 3 * 2^(d-1)). And then he mentioned the numbers for dimensions 8 and 24, and the whole thing broke down :(

It's interesting that they all divide by 6 (except k(1) = 2)

Better mention my man Kenneth Rosen

Circles/Spheres and their related shapes in higher dimensions are interesting in a number of ways, but one of the ways they are interesting is you can use the exact same definition for them in any number of dimensions without any funny work. That is to say, say we are in some dimension with N directions you can travel and we are at that dimension's origin point of 0, and we want to describe some sort of shape to people in this dimension a Circle/Sphere/Ect, we can say 'The surface of this shape is the made up of all the points that are an equal distance from me.' (Or perhaps some better worded definition but even this works) In 2D space that makes a Circle, in 3D space that makes a Sphere, in 4+D space that gets you Hyperspheres. We can even go the other direction. In 1D space (a line), we get two points X and -X. Other shapes don't have this easy ability. We can for instance look at a square and a cube and see that they are very similar, but it's not so easy to come up with a single definition that when put into different dimensions results in the correct shape for that dimension. If we use "a plane figure with four equal straight sides and four right angles" from doing a quick google search on define square, we can easily imagine what that shape might look like, but with that definition you'll always get the square version regardless of how many dimensions you have. Want a cube? Need something else. Want a hypercube? Again need something else.

if there's enough room to fit 12 spheres of the same size and more but not enough for 13th then what is the largest sphere of a smaller size that could be placed instead?

Now, circle packing is a common problem. What about kiss packing? For some ratio, probably an integer, between sojere sizes, what is the maximum density upon the surface of a subject sphere? For 1, it seems, this is 12.

This is going to help my 8 dimensional billiards game no end

Haha, shoulda kept that one for February the 14th

the real *numberphile*

I had discovered that by my own with a coin ..that around a circle we can fix 6 circles of same size and you guys somehow stole it I dont know but everytime I discover something a majority of time someone comes and tells me that it has already been discovered

8:26 I can’t imagine 8 Dimensions, you have said before that being 3 dimensional beings we cannot comprehend hyper-dimensions... And of course, I am the only one who would have picked that up...

When you have a perfect sphere that doesn't smash how big is the area that actually touches

Damn that's a cool proof

Hey in your next video can you talk about my Number S: An integer which you can multiply by another integer and get a decimal result. But if you multiply it by a decimal, you get a integer.

6:53 the fourth is time! It (or any other) does not change the other three!!! Unless you define them differently then i should ask HOW do you definition for them...

The Number Phile Pod Cast

How many spheres could you fit if they went to second base?

Can you please do a video spherical and cylindrical coordinates?

"A guy called Issac Newton, I don't know if you've heard of him..." Yea, never heard of the guy who co-founded calculus and set up the foundations for classical physics...

Can't we find a global formula for all dimensions ? Are they hypothetic ideas with the results we have for 2,3,4,8,24 ?

Hey, funny thought. Could kissing numbers have anything to do with Highly Composite Numbers? Anyone care to investigate. It appears to work for D 2,3,4 and 8.

Another james video without prime numers

*James Grime kissing numbers* Now there's a video I want to watch

The squareroot of -1 is 'I'm but anything x1 is 1 so does that not mean that the sqareroot of -1 is -1

Circular argument!

Save this video for Valentines Day!

Is there any significance to the kissing numbers all being multiples of 12? (Well, or 6...)

Flash backs to high school. I feel sick.

Ok.... so what would be the smallest diameter of an inner sphere that did have 13 spheres kissing? each with diameter one ?

Please do a video on octonions. I'm having trouble wrapping my head around them.

Hey James, how many sides would a Rubik's cube have, if we can scramble it so on each side all squares are different colors? I was talking with my dad about it last night, but neither of us are not mathematicians.

what if we used different sized spheres. why isn't that addressed?

I have the notifications on so why didn’t I get this notification? 🤬

If it is 6, 12, 24 the kissing numbers look like are just doubling. Until it doesn’t.

Surely the coordinates would be x1, y1 and z1 ;)

I was told 24D kissing number was 196,560 not like he said 196,500

why is the apature so low? Just makes things out of focus all the time.

I think that is the reason that IRON (atomic number 26 = 2 *13) is the minimum energy nucleous. I don't know - because nobody is paying me to find out - but I´m pretty sure. I think all the models of the nucleous of the atom is wrong.

Welp, it is numberphile after all.

It's a sphere trilogy because spheres exist in three dimensions, but there are only 2 videos because spheres only possess two dimensions.

What's the greatest r such that a sphere of radius 1 "kisses" 13 spheres of radius r?

He sometimes gives meSheldon vibes but more freindlier version