The Best Way to Pack Spheres - Numberphile
The Best Way to Pack Spheres Numberphile

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Featuring James Grime... Check out Brilliant (and get 20% off their premium service): (sponsor) More links & stuff in full description below ↓↓↓ Sphere trilogy: Strange Spheres in Higher Dimensions: Earthquakes and Spheres: More James Grime on Numberphile: James Grime website (you can book him for talks): Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science. And support from Math For America - NUMBERPHILE Website: Numberphile on Facebook: Numberphile tweets: Subscribe: Videos by Brady Haran Editing and animation in this video by Pete McPartlan Patreon: Numberphile T-Shirts: Brady's videos subreddit: Brady's latest videos across all channels: Sign up for (occasional) emails:


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

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.

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

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

Tanav2903 : 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?

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"

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.

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.

Science with Katie : Handy information for jugglers. 😉

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

John Galois : "...looked for the best way to pack his CANNIBALS." Oh, i think i misheared that part.

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

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)

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

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 ;)

Andriy Makukha : Will you mention Viazovska's recent result in 8- and 24-dimensional space?

Aditya : 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?!

jérôme Muffat-Méridol : Had real trouble following this video as I couldn't help worrying about this unsolved cube... Leaving unsolved cubes lying around is a sin :)

MrVipitis : I remember when you pack this in 4D or even like 10D you can pack a larger sphere inside a sphere.

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

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

Anna Walker : 2 James video in a row *I HAPPY*

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

PowerMad : pack 'em deep is the best way

Freedom : 3D spheres in 4D is different.

Ethan Letzer : I love the way he says problem, he sounds French but just on that one word lol

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

Martim Cunha Rocha : Honestly I already knew this from inorganic chemistry classes and I hate it so much!!

IllidanS4 : A hint on combinatorics, Hamming distance, and error detection and correction?

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

tim ng : ask the material scientists or crystallographers, they can pack spheres in 27 ways

bartman999 : James STILL hasn't hung up his pictures!

Galakyllz : The animations are perfect - they clarify what's being said so well. Great video.

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

jj zun : What about packing n-spheres in the n-th dimension? Recently read that there is a hypothesis, that this kind of packing is not the best in higher dimensions...

Bassem B. : Oh that "to be continued"! Cheeky!

DoctorMaxMoebius : Lover your work. Bucky Fuller was a “closest-packed sphere” expert. Surprised you didn’t mention his work. Also, Penrose was big into tiling space, so not being an expert but a dilettante in all their work, I would’ve thought they might have addressed some of these ideas. You should do some videos on their work.

Kartik Sharma : Please solve that rubix cube lying over there for centuries ..

Attila Morvai : I love how easy you explain everything..always learning something new! Thank you!

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

TehJumpingJawa : Packing..... In an infinite space. Surely a vital consideration when considering the efficiency of the packing solution, is the shape of the bounding volume?

J LR : Sphere packing in n dimensions is my favorite problem in mathematics. Thanks for covering this!

Sgt Sphynx : I see that hexagonal packing formation and I see an FCC iron crystalline structure

men bls : Hahaaha yes, of course! (nervous American laughter) That gosh darned Sir Arthur Rowl--I mean Raleigh! Who else would it have been?

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

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.

Matts Utube : The first of the trilogy in the SPCU (Sphere Packing Cinematic Universe)