The Best Way to Pack Spheres - Numberphile
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- Опубликовано: 23 сен 2018
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I once sat a physics exam that had a question that relied on this result, and it prefixed the claim with "as all physicists know, and many mathematicians believe, the most efficient way to pack spheres is..."
😂 Mathematicians are not satisfied by proof through examples
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.
But where would the world be if we didn't have the phrase "A parker square". Honestly i wish we had more video's of Matt and James working together, they have great chemistry on camera.
I feel I got cheated by the headline ' how to park spheres'. Thought it was actually a unknown way. But tell you what throw spheres into a box and the will arrange this way without your knowledge!!!!!
yes, we love ol' grimey.
But 'Doctor Grime' sounds more like some late-night TV window or drain cleaner.
A villain for a dish soap commercial on the telly.
"He invented the potato and other lies." True story.
Is the potato a lie? What happened to the Irish? Are we being fed lies? Is the government making us hallucinate? *puts on aluminium foil hat*
*tries to cook your head on a camp fire*
Now I’m confused, do people make lies about Walter’s inventions or did the he make the lies? Is the potato a lie? What about the cake?
are you being fed potatoes?
Cake is a lie.
Ah yes the grand mathematical properties of a ball pit
I don't think you'll be able to sink in a packed ball pit. I think you'll just belly flop on it and maybe bounce off.
Isn't the universe just a giant ball pit?
Bazinga!
Or literally anything spherical... yeah not important at all
If you pay extra you get 20 more minutes in the ballpit :p
I love how number 6 of Hilbert's problems just says "physics" 7:12
its meant to say can physics be axiomatized
Physics is problematic
One of them is buy milk
...and pay the bills! :D
Can you talk more about the formal mathematical language used for a computer to check a proof conclusively? A nice number/computerphile crossover
Agreed
This!
That would get me to visit Computerphile for sure.
This is some Hollywood level staff
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"
Farmers have been using this packing for as long as agriculture yields spherical edibles, but now it's actually proved in mathematics.
not just farmers anyone dealing with spheres
And here goes a Fields Medal!
I love how easy you explain everything..always learning something new!
Thank you!
I love Dr. Grime. Never fails to get me interested in the subject matter and he’s always a joy to watch.
The animations are perfect - they clarify what's being said so well. Great video.
The moment when numberphile comes in clutch for the chemistry test
Most annoying topic in inorganics
0:41 "slightly unsatisfying" - he must've just noticed the rubik's cube back there.
Honestly this video helped me understand my Gen Chem homework better than any TA ever could. And I could say the same about any video from this channel really. I love how they're filmed as if the instructor is talking directly to the viewer, it makes hard concepts really easy to understand
I love the animations in this video. They seem to be next level.
2 James video in a row
*I HAPPY*
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.
I'm so glad James is still around! I think he was in the very first Numberphile video.
Checking in to acknowledge how awesome James Grime is. Love his enthusiasm
“Everyone should have a mathematician...”
Oh, James!
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.
At least someone noticed
Thank you ^_^ I just did Crystallography in physics and was confused
Thank you for this, I was trying to find my course notes again. Isn’t there also the problem that this packing is optimal only if you consider over an infinite area? Like if you fix a 5x5 box there are better packings than this? Please correct me if I’m wrong
The video also claimed that the BCC structure had APF of 0.74, but it's actually around 0.68
NaCl actually is FCC because it's thought of as "Chloride in an FCC structure with each Chloride having a Sodium cation half a unit cell above it" or one of many other equivalent descriptions.
Surprisingly some of the most relevant content towards my major! I'm a freshman at university for materials science and engineering and this is exactly how we address close packed arrays of atoms in metal materials!
(Also for ceramic materials but those include different sizes of ions)
Very nice. I've learned some applications for this in a mineralogy class, but it's cool to see the mathematics behind it.
I watched this a couple times and only now am I beginning to understand.
I feel a lot denser than 74.05% so I guess Im not an assembly of spheres.
"...looked for the best way to pack his CANNIBALS." Oh, i think i misheared that part.
John Galois umm he said cannabis, dumbass.
I seriously love this kind of videos!
Nice to see another video with Ol' Grimey
Thank you so much, I learned so much from this clip!! How about sphere packing in higher dimensions? 4, 5? N?
An unsolved problem.
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.
Hence the formalisation.
I also assume the 5000 are subsets of the 100.
They probably showed that all possible local structures must contain one of the potential counterexamples
@@wierdalien1
A formalisation of this incomplete proof only serves to show that this incomplete proof is correct.
After all, no computer could go through all counter-examples in 15 years, if all counter-examples are not essentially the same as one of the given 5000.
@@jfb-
That would make sense, but then why is there no mention of that in the video?
This is necessary for the proof to work.
@@xCorvus7x because its a solid assumption that they had done that. You could also look up the paper
The "bit of Pythagoras":
Let's call the radius of a sphere 1. (Call it R if you are unhappy with this; it works out the same.)
First application of Pythagoras: the distance from the top center of the half-sphere to the corner of the box is 2 (it's two radii). The distance from the corner of the box to the midpoint of the edge is 1. So the distance from the edge midpoint to the top of the half-sphere is root 3.
Second application: The distance from the edge midpoint to the center of the bottom of the box is just 1. The previously result gave us the hypotenuse of this new triangle, so the height is root 2.
im so glad that james is back
please do a video on michael atiyah and the riemann hypothesis thing
Will you mention Viazovska's recent result in 8- and 24-dimensional space?
Wish you made this video a year ago. Would have helped me and friends with materials engineering since you’re describing different atom packing structures. Absolutely great video!!
Always happy to see James Grime :)
*I wonder why always, that rubiks cube always remain unsolved*
Why all the mathematics? Just look at my gut after I eat 12 bags of Maltesers.
you don't chew?
Maybe not eat the bags, but contents?
Take note of the 64% packing number that he provided without citing, unless the uniform diameter of the maltesers has changed it will not be as efficient. Please contact your local surgeon to schedule a repacking if your have OCD.
People eat those nasty things?
Oh no then you'd be only 74.05% full!
Sphere packing in n dimensions is my favorite problem in mathematics. Thanks for covering this!
I love your videos. Could you do one on how irrational sine values are found, both by ancients such as Ptolemy and Ulugh Beg, as well as by modern calculators using power series?
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 ;)
Metals are considered crystals? Whaaaaa
@@VictorTani Usually they are, yes!
@@Twewy13 oh my i learned at school that metals are just a uniform arrangement of atoms of any metal really. I dont know the correct names in english for the expressions my teachers used (im brazilian hue bolsonaro) but i remember something like "sea of electrons" when they would refer to the structure of metals.
@@VictorTani Ah, yes, metals do have the special property where electrons kind of flow freely in the entire material, kind of like a sea. But something being a crystal just means that the atoms are repeated in a pattern, like the tiles on a bathroom floor.
@@Twewy13 Oh ok thanks bro KNOWLEDGE IS NEVER ENOUGH
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?!
The proof is really that there are no counterexamples that are not equivalent to the ones tested. It's an indirect proof. First they proofed that these are the finite equivalent counterexamples and then tested them under the hypothesis that they are better.
@@weker01 hmm so if I understand you correctly they found that ALL counterexamples are equivalent to these 100 cases and then they manually checked these 100??
@@AdityaPrasad007 exactly!
Weker thanks for clearing that up. Much appreciated.
11:08
My parents when I try to explain how I was going to do my homework at 10:00 but can't because it's now 10:01
Can't wait for part 2 and 3!
Maryna Viazovska winning the 2022 Fields Medal brought me here.
Handy information for jugglers. 😉
I see you EVERYWHERE!! LOL
crusty jugglers...
The science of ball juggling and communicable diseases with Katie!? uh oh.
Thumbs up on the animations, really helped me understand what was going on.
heck yeah, can't wait for the next two :D
dump em in
let em roll
5:45 What is this "bit of Pythagoras"? - It's Monday morning and my brain hasn't woken up
It's the "height, h, of the cuboid is proportional to the diagonal [of the base] via the radius of the spheres" that I'm struggling to understand, how we get from this to the answer of sqrt2
I guess the key is that the top corners coincide with centres of areas not occupied by spheres. Each of these centres is located at an equal distance from the centres of all local spheres. We got such area centres in the top corners and in the middle of the diagonal of the base, therefore the height is one half of the diagonal.
Yeah I get that, I'm just unsure why the centre point of the base diagonal is the same as the height. I'll have to think about it some more. Thanks for answering though :)
Let the center of the top half-sphere be A. Drop a vertical line to the bottom of the box and call that point of intersection B. Take one of the bottom corners of the box and call that point C. Since angle ABC is a right angle, line AB is one edge of a right triangle formed by points A, B and C. If we know the lengths of the other two edges of triangle ABC, then we can calculate AB using the Pythagorean theorem. AC is two sphere radii, so its length is 2. But because of the symmetries of the packing, BC equals AB. So,
AC² = BC² + AB²
AC² = 2AB²
2² = 2AB²
2 = AB²
AB = √2
Thank you all. I read a comment from Bob Stein (before I saw the additional replies here) and all is now clear :)
The animation is on point in this video 💯🔥🔥
Blown my mind once again!!
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?
Yeah, that shoulda been ebayed!
10:00 What is the formal proof language they used? COQ or similar?
Stefan Reich succ 🅱️ig COQ 💯😂
ROFL... yeah, no, it's an actual software :)
Oh, that ending is pure gold, sooo sinematic.
James' smug grin at the end was amazing
I have a solution for problem #25.
Where can i collect my million dollar reward?
:D
7:10
Hilberts Problems =/= Millenium Problems
Just send an email to Mr. Hilbert. He'll be delighted and immediately wire you the prize in Reichsmark (especially if you mention that you are a Nigerian prince). Mostly for discovering 2 new problems he has, he wasn't even áware of having.
Did you have enough space on the margins to write it down?
Shove it all in until it works!
beautiful explenations !
great episode.
You are 🔥🔥🔥
Seeing you helping students in education has inspired me to come out of my comfort zone and start my own RUclips channel to help more students..
Love you sir
Maryam Study your channel is really helpful, ,, subscribed❤️
Well done
Thank you
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.
He didn't say it was table salt specifically, or that it was exactly 0.74 packing density. He just said that crystals _in general_ have a _general_ structure that is the same.
Chemist here. I took Crystallography, which was a grad-level Material Science(metallurgy) course, and from my experience, even material scientists have to tiptoe around the terminology of HCP (hexagonal close-packed,).I give him a bye.
I think he's not talking about bulk crystalline structure, but about the arrangement of subatomic particles in the nucleus, which does follow this packing scheme
Interesting, 67% sounds still quite close though
Nathan,
Then he would be extra wrong. Nucleons in the nucleus definitely do NOT pack like rigid spheres.
Wow this made me remember my course in material science back in the days. Some materials form other crystal lattices than others. The best ones, you guessed it, fill 74% of the space :)
Wow such a cliffhanger! Can’t wait for the rest of the saga!
Under 301 club
Limit: Well... 300
We have food and drinks, come on in and.. Party?
I remember when you pack this in 4D or even like 10D you can pack a larger sphere inside a sphere.
Love the Hakone puzzle box in the back, there.😉
When I saw that I would have to wait for a part 2 of this episode, I was sphurious.
11:08 That right there has meme potential.
How did they come up with the 5000 potential counter examples and how did they know that there weren't better alternatives out there?
permutations and combinations
There are only so many ways to pack spheres around each other.
There are only 5 platonic solids. Why aren't there any more? Surely we can use computers and stuff to find a bunch unknown to the Greeks, right? But there just aren't any more ways to form regular polyhedrons.
remember doing this at Uni, loved it
I learnt this in first year engineering physics. Something related to atomic packing, packing factor, simple cubic, fcc, bcc, etc etc etc.
12th chemistry
5h3r10ck h01m35 bro do you know why the height was root2? I can't figure it out...
Pick the unknown height connect the top of the height to the center of the upper square this upper line is half the diagonal so its radical 2,connect the center to the lower part of the height this connects the center of two spheres the center of the corner and yhe center of the half sphere this hypotenuse is 2,so in the end x^2 + 2 = 4,x = square root 2
At first, i heard "...packing cannibals."
Lol
If you simplify humans to spheres, it would be the same. However, then you're a physician and not a mathematician.
@@anlumo1 no, because the spheres would eat each other. 😎
After packing some cannabis it's perfectly understandable why.
I believe he was referring to the shrunken heads that cannibal tribes would make from whoever lost to them in battle.
the fact that he hanged the paper which graham wrote his number... amazing!
I love how James can tell a joke without breaking his explanation at all
You mean the best way to park squares, Parker squares that is
Yeesh this is forced. You don't need to shoehorn in the same tired joke on every single video when it isn't even tangentially related...
Haters gon' hate
Matthew Elder - I thought it was a *compact,* well *rounded* comment that *fit* in well here.
Matthew Zuelke It was a parker of a joke
@@pomtubes1205
Parker joke
Looks like a joke, but isn't
Make a video on the Riemann Hypothesis proof
They did, 4 years ago.
@Jeremy Shuler I don't think so. He is an eminent mathematician who came first in his class at Cambridge in maths. If he claims he's solved it I don't think he would have told the public unless he was sure it was right
sangon chokas He’s very old, so it might just be a desperate attempt to solidify his status as one of the greatest mathematicians ever.
That may be, but it still is interesting to see how he approaches the problem and what arguments he brings to the table.
Just learned about all this stuff in my materials science class!
The link between telecommunications and sphere packing could be either shannon s theorem of information or symbol mapping/modulation in digital telecommunications due to the nature of Gaussian noise.
Oi mate! 'Ave you got you'self a loicense fo them fancy maths bruv?
I suppose his doctorate counts 🤔 even after Unauthorized Information Online is banned, he should remain safe to make these videos
"24: Pick up dry-cleaning
25: Buy milk
26: Send invoice!"
Hahahaha
For those wondering, Hilbert only set 23 problems (only one of which is also among the seven Millennium Prize Problems, namely the Riemann Hypothesis). I don't think there's a monetary reward for the problems, especially since they're more of a guideline for where mathematical research should expand in the 1900s rather than concrete problems.
Grime's grin was great !
Yes! Sphere Trilogy!
Can someone please tell me how the height was root2 I can't seem to figure it out....
Make a vertical diagonal slice and see that the spheres are in contact
It's kinda hard to explain without a diagram, but it involves using Pythagoras (a²+b²=c²) twice because the object is 3D. I'm sure if you Google Pythagoras in 3-dimensions you'll find something that will guide you to solve this
@5:49 The distance between the centre of the red ball and the centre of the blue ball is 2 (you need to think about why this is true), and the distance from the centre of the red ball to one of the comer is root2.
SPOILER ALERT 5:38 The distance from the bottom corner (the center of the green ball) to the center of the top square (the center of the purple ball) is 2. The distance from the top corner to the center of the top square is root 2. By Pythagoras theorem the vertical side also has to be root 2.
Yes but without being given the answer for the height, u would need to prove to yourself that the distance u mention passes through a contact point. While correct, that's the more challenging part of the problem.
Pause the video and go to 0:00
lol
I noticed it too! He looks cute.
This answer always seemed intuitively correct to me.
You take a sphere, you bring a second sphere as close as possible (touching), then another as close as possible to both (triangle), then another (tetrahedron). That's the tightest small configuration you can make. Any additional spheres you add will at best locally replicate this tetrahedron, so the best you can pack them is in this repeated tetrahedron.
I'm stunned it took so long to prove and then an additional 15 years to _really_ be sure.
I remember learning all these kinds of packing and their relationship in high school chemistry.
First (sorry I had to)
but, did you?
Eloi Comabella No, but I got virtual likes that gave me a sense of pride and accomplishment from my firstness
enjoy!
That cliffhanger at the end!! OOOOOOOO
Pretty cool to see the signed Graham’s Number paper on the back wall :)
I read in some book a similar problem, where you pack spheres into boxes. Basically you wonder what is the smallest box that can contain n spheres.
For low number, square packing is best (i.e. 8 spheres are best fit by placing them in corners of a box), hexagonal eventually takes over.
But for something like 59 spheres, there is a proof that even better packing exists, some random-looking structure is supposed to be even more efficient, even though the proof does not create such packing, just says there must be one.
Best trilogy of all time:
-Halo-
-Star Wars-
-Hitchhiker's Guide to the Galaxy-
Dr. James Grime's Sphere Trilogy
Usually, when you combine factors into a number (for example: factors of 4: 41216), and divide it by the number, you get something based off of 1002, or 10002, etc
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.
i luv that 4D Sphere packing lark.
When I say the "To Be Continued" Roundabout just started playing in my head
(Which is kind of appropriate given how often mathematics pop up in JoJo)
"Can be applied to how we transmit messages on the internet today"
"I can't see any LINK there!"
so underappreciated
Now I have been playing with the problem in other contexts.
The problem of weather it is 12 or 13 that fits around a sphere is not trivial. Newton only counted 12 AROUND the central sphere; but if you count the sphere actually included (surrounded) in the ball it gives 13.
That is why Iron is the atom with the lowest binding energy in the nucleus of all!
Iron has the atomic number of 26 = 2 *13. The doubling of 13 is due to the fact, that the basic building block of atomic nuclei is not hydrogen, but actually helium with an atomic number of 2 and 2 neutrons, thus in essense an alpha particle. There is virtually no helium except helium 4.
Now helium is not 2 protons and 2 neutrons it is 12 quarks tied together by the strong nuclear force - which is not at all mystic - given a few basic assumptions. A regular dodecahedron has 12 faces. And could hypothetically contain another quark in the center, but as quarks don't exist in isolation and can only be torn out by pouring energy enough for a replacement (plus an anti-quark).
The iron nucleous is more lucky in so far as there is room for a helium nucleus in the center - hence 13 helium nuclei! 12 faces plus one in the center.
Now 13 times 4 is 52 and the - by far - the most common iron-isotope is 56 thus relegating 4 neutrons as hangers on - actually iron 54 (the second most common isotope) and the two neutrons i presume are as far away from each other as possible.
I could go into more detail, but won't as my work would be nicked and I would get no money.
Remember Newton? He got so pissed off by someone who cheated him for money that he got 26 forgerers executed under torture - not that he bothered seeing them squirm.
But then you would have to get a nuclear physicist in on the game as well, because it is quite illumination to the nature of the strong nuclear force.
I am not sure but I believe it wes Archimides who said,"if there is a best way it will be natures way" I note this in relation to the Bees" close packing of nectar within honeycombes.and also the answer to steiner"s problem,answered by plateau"s experiments with soap films