The Best Way to Pack Spheres - Numberphile

Share this video on

What's Hot

What's New

Top Grossing

Top of the Chart


Scanlaid : Can you talk more about the formal mathematical language used for a computer to check a proof conclusively? A nice number/computerphile crossover

Dom : "He invented the potato and other lies." True story.

teddy boragina : I love James Grime, he's one of my fav people in numberphile videos. You have to admit, though, that "Doctor Grime" would be an excellent name for a Captain Planet villain.

Caillouminati : Ah yes the grand mathematical properties of a ball pit

M. de k. : I saw someone do this with oranges once, I think he was also the inventor of the "parker square"

Science with Katie : Handy information for jugglers. 😉

Tanav Singh : 2 James grime video in a row It feels like heaven

Sebastian Elytron : Why all the mathematics? Just look at my gut after I eat 12 bags of Maltesers.

Paul Paulson : I have a solution for problem #25. Where can i collect my million dollar reward?

Richard Andersson : There are some inaccuracies in the video. The triangular pyramid is in fact exactly the same as the square pyramid, but they are not the same as the hexagonal one. The two types of packing have the same packing factor but a fundamentally different structure. Google FCC and HCP for more info. Also table salt, NaCl, is not fcc or hcp, it is a simple cubic lattice and is therefore not a perfectly packed.

Bo Do : Oi mate! 'Ave you got you'self a loicense fo them fancy maths bruv?

δτ : How does disproving a finite of number counter-examples count as a proof? They have to demonstrate first that any potential counter-example is essentially equal to one of the five thousand or one hundred. Have they? Edit: As has been pointed out in responds to this, it might be that Dr. Grime glossed over this point to keep the level of mathematics involved understandable to laypeople. I understand that and it is perfectly understandable and fine, I would have just preferred this detail at least to be mentioned in the video, considering that the reduction from infinitely many cases to finitely many is both a necessary condition and that it being possible is an interesting fact, if true. Instead not a single word is spent on wether these 5000 examples cover all cases. One sentence would have been enough, but this way there is something missing.

Matthew Zuelke : You mean the best way to park squares, Parker squares that is

Max Labb : As a material engineer I am a bit annoyed by the distinction between "aluminium and copper, or crystals like tablesalt". If aluminium or copper have a regular packing they ARE crystals ;)

Alex Johansson : At first, i heard "...packing cannibals."

cyancoyote : The computer screen displays a few lines from the first paragraph of the Wikipedia article "Sphere packing" in a hexadecimal representation. "In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing problems can be generalised to consider unequal spheres, n-dimensional Euclidean space (where the problem becomes circle packing in two dimensions, or hypersphere packing in higher dimensions) or to non-Euclidean spaces such as hyperbolic space. A typical sphere packing problem is to find an arrangement in which the spheres fill as large a proportion of the space as possible. The proportion of space filled by the spheres is calle"

BonZaiOne : Your statement that 74% is the value for structures like "copper and table salt" is pretty inaccurate; let me explain why. The max possible packing factor is 0.74 (or if you prefer 74% of the "structure" that contains the particles), this factor competes only to the monoatomic metallic elements like copper or aluminium (of course under some simplifying hypothesis). NaCl or "table salt" is a binary ionic salt formed by two different atomic species (sodium and chlorine) with opposite charges, so in order to stabilize the entire structure they will dispose themselves in different position leaving different voids. So in conclusion the packing factor isn't 0.74 , it's around 0.67.

Kumar Suyash Rituraj : please do a video on michael atiyah and the riemann hypothesis thing

Aditya Prasad : wait... you just make a finite list of possible counterexamples and just cause you could not find a better packing you conclude you found the best packing?!

raditz : Great talk, but IMHO you should have mentioned Laszlo Fejes Toth for "He also investigated the sphere packing problem. He was the first to show, in 1953, that proof of the Kepler conjecture can be reduced to a finite case analysis and, later, that the problem might be solved using a computer." (from wikipedia)

Jonathan Corwin : 5:45 What is this "bit of Pythagoras"? - It's Monday morning and my brain hasn't woken up

Funking Prink : How did they come up with the 5000 potential counter examples and how did they know that there weren't better alternatives out there?

Dalgaim : However, all this holds only if there are no boundaries. If there actually is a finite box we want to pack with spheres, it becomes much more complicated, especially if the dimensions of the box are not divisible by the size of the unit of the packing.

Calin Gligore : Make a video on the Riemann Hypothesis proof

Rahul sama : Can someone please tell me how the height was root2 I can't seem to figure it out....

Canai O Canal : Where was this 3 years ago when I had a math task to find out the best way to pack spheres.

Doug Rosengard : I’m curious about the framed paper on the wall in the background, was that a signed copy of the sheet used in a Graham’s number video?

Giantalfe : Shove it all in until it works!

Theo Godid : Might be a bit off Topic, wanted to write that under an older Video but then it wouldn't be seen by anyone. So i thought of an easier proof for infinite primes: if we assume that for every prime, we take a Portion of all numbers away (1/2)(1/3)(1/5)(1/7)(1/11)…..(1/n) and we assume we have a finite amount of primes this equation will be sth very Tiny, but above 0, and if we multiply that Tiny number with infinity, we get an infinite amount of numbers that are primes. Does that make sense?

MAD A NION : *I wonder why always, that rubiks cube always remain unsolved*

Stefan Reich : 10:00 What is the formal proof language they used? COQ or similar?

INTJ Mind : Where is he at? I love how simplistic it is. If that is his home this would be my dream. A brown simple home decorated minimally.

Igor Fedik : In this packing the centers of the spheres are close to the vertices and the center of regular icosahedron. But the radius of a circumscribed sphere is about 5% less that the edge length of a regular icosahedron. Therefore it is impossible to make a perfect 3D tetrahedral lattice. It may be a bit counterintuitive because it is possible to make a perfect infinite triangular lattice in 2D.

TheMakersRage : So Raleigh asked his friend to solve the problem and Raleigh gets the credit?

shomolya : Some of the seemingly most obvious things are the hardest to prove.

amira3333 : When is the video on the supposed RH proof coming?

alex paoki : Which number is largest? Tree(3) or G64! I have no idea😕😕

MrVipitis : Can't you proof this geometrically with a platonic solid? Insert the tetrahedron into a sphere... And you get 4 poles of equal distance, the maximum distance between each point. Those are basically the points where the packed spheres are touching. Now add a fifth pole(point on the surface) to the initial sphere and you are unable to get them all into equal distance because there is no platonic solid with five corners. If you go back up to six points you insert a octahedran and you are able to fit spheres of equal size on those poles again. And the pattern continues. Now check the other platonic shapes and it ends up as as a cube. Which is the best way to pack a rectangle itself. Or could you argument about the way circles pack in 2 dimensions?

L0j1k : "To be continued..." Me:

Iqbal Mala : Under 301 club Limit: Well... 300 We have food and drinks, come on in and.. Party?

jfitz0716 : What’s the best mathematical way to pack fudge?

cartoon molvies : Alguém quem BR?

Mikhail Kudinov : Make a video on congruent numbers plz!!!

5h3r10ck h01m35 : 12th chemistry

BluesyBor : So can we say that they have bruteforced the proof? ;)

Nuovoswiss : Table salt (NaCl) is not close-packed, it's simple cubic.

Quahntasy - Animating Universe : Two James grim video in a row. I am James grinning now. Sorry bad joke.

Commentah : Maybe you better use a tripod for your camera?

kustomweb : Octahedra and tetrahedra stacked together fill volumes in the most efficient manner. Read R. Buckminster Fuller Synergetics.

Denis Molla : Hahhahaha cliffhanger in a Pack Spheres video? Numberphile!