Kissing Numbers - Numberphile

[@imveryangryitsnotbutter]

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

[@666Tomato666]

could you say that it "screamed geometrically"?

[@sashimanu]

*packs the suitcase with spheres

[@f.f.s.d.o.a.7294]

Packs with eight-dimensional sphere. So many; so heavy.

[@Triantalex]

false.

[@fatmn]

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

[@loganm2924]

I just commented this, before reading this comment... 🙁

[@philosophiamourningstar9424]

Eight dimension....no problem....🤔but what's a sphere?

[@pegy6384]

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.

[@Jesse__H]

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

[@SchutzmarkeGMBH]

It's the one from the Grahams number video, signed by Ron Graham himself.

[@BlackWhiteCloud]

I think this is Brady's place, he shoots a lot of videos here, not only Dr Grime's.

[@koolguy728]

Thats Brady's house

[@ulture]

you can tell it's Brady's because he has all his precious metal play buttons on the floor

[@saetainlatin]

That brown paper contains "the how" Fermat proved his last theorem

[@pakan357]

Isaac Newton? Never heard of him...

[@flyingskyward2153]

He invented apples

[@HomeofLawboy]

Oh! He's Bill Gates' brother!

[@sergiokorochinsky49]

pakan357 is that a quote from Leibnitz?

[@1224chrisng]

any relation to Fig Newtons?

[@eideticex]

He's just this guy, ya know.

[@RFC3514]

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

[@MrMineHeads.]

Why do people call eggplants aubergine? Are you french? Say eggplant you posh!

[@RFC3514]

+Hassan Tahan - To answer your first question, the reason why most people on the planet don't call it "eggplant" is that it's not a plant (it's a fruit), and normal cultivars are purple (not exactly a normal colour for eggs). The term "eggplant" is a reference to a specific cultivar of the plant, that produced _white_ oval aubergines. The purple elongated variety is only called "eggplant" in North America and Australia, where presumably eggs look very different from the rest of the world. Also, the line is a reference to (Nina Wadia's "mother" character in) "Goodness Gracious Me".

[@MrMineHeads.]

@@RFC3514 i'm playing joke btw

[@nowonmetube]

@@RFC3514 thx 👍🏼🙏🏼🍆🥚

[@Phobero]

My kissing number is zero

[@Boooo]

Maybe one day

[@Nordzumu]

I'm dying m8

[@DerGully]

Well, the kissing number for zero dimensions is zero. You must be a zero-dimensional sphere.

[@h0verman]

that comment is made 12 times funnier by the emotion expressed through that profile pictuere

[@willsuttie3683]

r/suicidebywords

[@KingOfTheUnderdogs]

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.

[@HaileISela]

the reason is that the dozen around one are the vector equilibrium. they are really four intersecting hexagonal planes. and just as six around one is the maximum in a plane, the four planes of spacetime manifest as four hexagons. the irony of all these weird and complex problems presented in this trilogy is that it is completely unaware of the lifework of Richard Buckminster Fuller who developed a non-axiomatic, completely experiential geometry called synergetics. among the many things he figured out in it was the vector equilibrium as the nuclear geometry, the point of origin of his actual, dynamic, four dimensional coordinate system based on the behavior of spheres, with the unit volume of the "tetrahedron" as whole rational base unit. he also proved the commonsense of static "3D" space held since ancient greek times to be false. the basis for all science, the axiomatic "pure" mathematics, does not really hold up to the standards of an operational, experimentally verified analysis. quite the opposite. if you want to know more about this, check my playlist on synergetics.

[@Triantalex]

??

[@bradleyparrett4483]

I took my pants off for nothing!

[@remanjecarter2787]

Pants? You mean wearable Klein-bottle?

[@paulgoogol2652]

just keep looking, you will find it eventually 8)

[@TrveNyshya]

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! :)

[@kinyutaka]

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.

[@loveforsberg530]

What leads you to that claim? To me it sounds unnatural.

[@kinyutaka]

It's a simple observation of the known and suspected kissing numbers. 2 (1st Kissing Number) is divisible by 1, naturally. 6 (2nd Kissing Number) is divisible by 2. 12 (3rd) is divisible by 3. 24 (4th) is divisible by 4. 240 (8th) is divisible by 8. And 196,560 (24th) is divisible by 24. The unknown numbers lower bounds are also divisible by their number of dimensions, implying a link between the regular distribution of spheres in Nth Dimensional Space and the number of spheres that surround a central sphere within that space.

[@eduardopupucon]

+Red X that claim sounds very pareidolic

[@kinyutaka]

But the error correction is correct.

[@shanathered5910]

@@loveforsberg530he's actually right about it being an error. just read up on a mathematical structure called the "Leech lattice" it's very interesting

[@WildStar2002]

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

[@Archanfel]

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

[@ben1996123]

the 24 dimensional one should be 196560 not 196500

[@vexphoenix]

Why exactly??

[@Czeckie]

check out Leech lattice, that's the regular object behind this result. It just gives rise to 196560 spheres.

[@nowonmetube]

And... Why exactly?

[@dylanslingsby7643]

this is what i said and i was scrolling to find another person who also said this. i cannot find anywhere else on the internet that says it is 196500 they all say 196560

[@saurabhratnalikar8663]

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

[@danielemessina1979]

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

[@epkoda]

that's what I always do too! sadly this method cannot be used to imagine a 4d sphere moving in 4d space, so it only works for static objects. I still think it's pretty cool, it actually helped me intuitively understanding why unit hyperspheres occupy a lesser and lesser proportion of the unit hypercube as dimensions increase!

[@nebelung1]

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

[@nebelung1]

that would be even better

[@TheGanamaster]

Probably the dimension 96 would be the next...or am I wrong and the correct guess is the dimension 72 will be the next...?

[@SauravKumar-mz1bs]

How many kissing points will be there in 1 dimensions

[@darreljones8645]

SKR, 1 dimension is trivial. There are only two points next to any given point on a line, so the 1-dimensional kissing number is 2.

[@HL-iw1du]

TheGanamaster there are an infinite number of patterns that fit a finite set of numbers

[@robinvik1]

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

[@Gismo359]

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

[@HL-iw1du]

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

[@MaeLSTRoM1997]

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

[@PC_Simo]

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.

[@alexpotts6520]

The kissing number in one dimension is two.

[@PhilBoswell]

Erm…what form would a one-dimensional sphere take?

[@Ontonator]

Two points (known as a 0-sphere). An n-sphere is defined as the set of points r units from the centre of the n-sphere in (n + 1)-dimensional space (a circle is a 1-sphere and a sphere is a 2-sphere). Note that this does not include the inside of the sphere, only the surface, hence the two points instead of a line. (The inside of a sphere is a ball, the inside of a circle is a disc and the inside of a 0-sphere is a line segment.)

[@swordfishxd-]

3

[@magnusbreinholt350]

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

[@macronencer]

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.

[@00bean00]

Sphere trilogy? Starring Sigourney Weaver? Yes pls

[@frederf3227]

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.

[@stevethecatcouch6532]

Brian Clark, you left a key number out of that post. Was that first central sphere of radius 0?

[@stevethecatcouch6532]

For 3, the radius is 2*(sin(30)/sin(120)) = 1/sin(120), about 1.155.

[@Gvozd111]

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

[@TheGanamaster]

Yeah, a lot of shy people would read it...

[@paulgoogol2652]

it is hard to land a kiss in a 3d environment. the vector calculations involved are way over my head.

[@sundaranarasimhan58]

Nice....

[@sundaranarasimhan58]

Nice....

[@alexakalennon]

Thats gonna be a classic

[@vanyasketches5154]

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

[@dylanslingsby7643]

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

[@shanathered5910]

Isn't the kissing number for 24 dimensions 196560?

[@John-pn4rt]

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

[@MrBrain4]

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

[@kylecronin3212]

LOL at "Nobody Knows" 8:15

[@peachu7]

What, do you have some information we don't?

[@U014B]

*_[Theremin music intensifies]_*

[@MahraiZiller]

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

[@rq4740]

There's always this guy haha

[@angelmendez-rivera351]

Talking about spheres only makes sense in Euclidean space, so the caveat is completely redundant.

[@Sam_on_YouTube]

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

[@ZipplyZane]

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.

[@kingxdedede7327]

But that's for global packing, if you were going to tile all of space with an arrangement what would the concentration be, whereas because this problem doesn't require tiling space, irregular structures that have higher local densities can be used instead. It just so happens that the regular third-dimensional structure is the best one both for packing and for kissing numbers.

[@michalbotor]

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

[@DavidFugl]

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

[@MhDaMaster]

"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.

[@naomiperez7482]

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

[@insan3d0wn3r]

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

[@eve36368]

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

[@djsyntic]

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.

[@philipphoehn3883]

Hypercubes can simply be defined by their vertices in Cartesian coordinates (±1,±1,±1,...)[xN] N-Spheres aren't the only shapes with dimension general definitions.

[@alexpotts6520]

A square is a 2-dimensional figure with 4 equal sides and 4 right-angled corners A cube is a 3-dimensional figure with 6 equal faces and 8 right-angled vertices A hypercube is a 4-dimensional figure with 8 equal hyperfaces and 16 right-angled hypervertices An n-cube is an n-dimensional figure with 2n equal n-faces and 2^n right-angled n-vertices

[@rabbit2840]

not true, one can simply take your definition (every point that has euclidian norm equal or less than R) and exchange the euclidian norm for the maximum or one norm to get a cube.

[@danielbenyair300]

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...

[@pedrogonzalezgil]

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

[@EulyDerg]

"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...

[@manueldelrio7147]

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

[@ilyrm89]

I have been waiting for this new video!

[@alexisdc91]

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 ?

[@ultimateo621]

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

[@maciej1276]

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

[@CalvinWiersum]

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

[@sjdjsfjsjf8446]

Old but gold.

[@TakeWalker]

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

[@MagicGonads]

They are all multiples of 2 so far, not 6 as k(1) = 2

[@deadgavin4218]

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?

[@parsuli.]

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.

[@dylanslingsby7643]

in my "opinion" NO I thought a similar thing about factorial numbers but it was wrong, with such a small number of known kissing numbers its easy to see them in all kinds of other mathematical fields try to find a pattern by looking ONLY at the kissing numbers (if you're still interested)

[@Phalc0n1337]

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

[@TheHardwareDeveloper]

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

[@dylanslingsby7643]

teach your self what has NOT been discovered then try discovering something a new pattern a new invention

[@jfb-]

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

[@dylanslingsby7643]

there is a pattern in how they divide but its more complex than them all just dividing by one number

[@loganm2924]

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...

[@MrRyanroberson1]

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.

[@flightwithtools]

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

[@WarzSchoolchild]

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

[@MrInitialMan]

So all the known highest kissing numbers are divisible by 6. Interesting. (2-D is 6 itself; 3-D is 12, 12/6 = 2; 4-D is 24, 24/6 = 4, 8-D is 240, 240/6 = 40; 9-D is 306, 306/6=51; and 24-D is 196500, 196500/6 = 32750)

[@shanathered5910]

196560, not 196500. look up the leech lattice

[@moroccangeographer8993]

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

[@MasterStroke.]

Numberphile. Nobody, nobody, nobody does it better.

[@RibusPQR]

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

[@SilentBudgie]

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

[@rioga98]

Haha, shoulda kept that one for February the 14th

[@JohnLeePettimoreIII]

All hail the Singing Banana!

[@Superman37891]

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

[@hectorryansmith8440]

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.

[@hectorryansmith8440]

Also S x 1 is not equal to S and S x 0 is not equal to 0

[@ragnkja]

An integer multiplied by an integer is always an integer.

[@suncu91]

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.

[@peglor]

Each side has 9 squares, but there are only 6 colours on a standard Rubik's Cube. It could be done on a 2x2 one though... Beyond that you either end up with something other than a cube or just start dividing the faces into arbitrary numbers of squares. Sticking to square numbers again gives 1x1 (Not much fun as it's always solved) and 2x2 faced cubes as the only possible answers.

[@eve36368]

240 & 196500 & 306 are all divisible by 6. is that just a property of regular shapes or is it merely for every kissing number?

[@dylanslingsby7643]

The 1D kissing number is 2

[@collintmay]

Something not covered in this video: Are the n-dimensional spheres discussed towards the end surrounded by n-1-dimensional spheres or just 3-spheres?

[@MrAlRats]

The discussion is about n-dimensional spheres surrounded by n-dimensional spheres in Euclidean space.

[@phoggee]

Why do you waste so much paper?? Can't you use a blackboard or something like that?

[@amathystt3354]

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

[@IceMetalPunk]

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 :(

[@yomommamadthicccuh]

keep working!!

[@Jupiterninja95]

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

[@zerebusgarago]

This one sort of confuses me. Why isn't the maximum amount of spheres that you can fit around a unit sphere the surface area of a 1.5 unit sphere divided by are of a unit circle or PI?

[@morethejamesx39]

Zerebus Garago because you’re not accounting for the gaps in between

[@Cammymoop]

I wonder if the rest are multiples of 6

[@sergiokorochinsky49]

FelixNemis I would rather suggest multiples of 8 and/or 24, but as James said, the trick of simmetries stops working at higher dimensions...

[@ratanbharadwaj7564]

X , Y, Square, Sin , Cos, Tan, Theta these were my worst enemies at school

[@connormccann1999]

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

[@adlsfreund]

Two questions: 1. Why can't we work out the kissing numbers for dimensions 5, 6, etc by simply extending the formula like he did for dimension 4? 2. What does he mean by irregular shapes? It seems obvious to me that if you take an arbitrary shape, like a long stick, you can get a much higher kissing number, in any dimension.

[@alexpotts6520]

1) Basically, the more dimensions you extend to, the harder the equations get to solve. The higher the dimensions get, the more ways you have of shuffling your kissing spheres around to try and fit an extra one in (both because there are more spheres to shuffle around, and because there are more dimensions in which you can move them). That means it gets harder and harder to prove that a given kissing number is indeed the maximum 2) He actually said "irregular pattern". That means that the pattern of spheres doesn't have nice symmetries. For example, in the packing of 3D spheres, with a kissing number of 12, you can carry on the pattern infinitely, so every sphere in your lattice can be kissed by 12 others. (In other words, every sphere in this pattern is equivalent to every other one.) For the 9D example they mentioned, you could fit 306 spheres around the central sphere, but then you couldn't carry on that pattern for every other sphere - only the original central sphere could have 306 neighbours, the other spheres would be forced to have fewer. (That is, the spheres are not symmetrically interchangeable.)

[@Zarunias]

1) the formula only gives you a maximum number for the kissing number. For 3 dimension we know that it can not be bigger than 12, and luckily we found an arrangement that gives us this number, so we know it. Similar we know that in 4 dimensions the kissing number can not be bigger than 24 and we also know an arrangement for this. If you extend the formula to 5 dimensions it will give you a maximum number. But we don't know if there is actually an arrangement for this number or if in fact the kissing number is smaller.

[@BubblegumTrollKing]

Pause the video at 8:35

[@nekogod]

Are they always multiples of 6?

[@stillagamer3603]

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

[@peglor]

It's a point contact, which has zero area in the maths world, in the real world it'll depend on the roundness of the material and the atoms it's made from.

[@alan2here]

2, 6, 12, 24, ?, ?, ?, 240 It's successive highly composite numbers? 🤔 🙂 2, 6, 12, 24, 36, 48, 60, 240 Except 4 is skipped, sorry 4. And then 196,500 completely breaks the sequence. Following the sequence there should be more.

[@dylanslingsby7643]

with such a limited number of kissing numbers its easy to see them in all kinds of places I started seeing them in number plates (only joking)

[@joryjones6808]

13 is the loneliest number.

[@Terrik240]

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

[@thomasborgsmidt9801]

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.

[@jeffreyalvarina8709]

remember me when you get famous 😀

[@wanderingrandomer]

Trying to visualise a 4th dimensional sphere gave me an anxiety attack

[@LieseFury]

Why would you use x1, x2, x3 when you could just use x, y, z?

[@dlevi67]

Because he's calling the point "X"... and it's a lot handier to use numbers to indicate the dimension/axis once you start talking of arbitrary dimension numbers.

[@AlexandrBorschchev]

I love this guy

[@stephenbeck7222]

For dimension 5, I can prove that the kissing number is between 6 and Graham's number, but that's as close as I can get.

[@dlevi67]

Go on, then.

[@stuberosum1]

When I saw the title 'kissing numbers I thought of the number pair 252 & 260

[@drewchristensen2432]

Whats your prof pic? I could swear I've seen that before.

[@stuberosum1]

jebidiah kerman

[@kingxdedede7327]

May he live on in all of us.

[@felixroux]

you can have infinite. you never said the spheres had to be the same size

[@rounaksinha5309]

what is its application

[@jeffreygalle5960]

Better mention my man Kenneth Rosen