Calculating the optimal sphere packing density: with oranges

[@daseinbot]

Steve posted a video "with Matt Parker" and Matt posted "with oranges"

[@SamuelBoshier]

+

[@SteelSkin667]

Mouldy oranges

[@jamesbeanmachine857]

Steve and Matt play around with lattices Matt and oranges play around with Steve

[@SteelSkin667]

+JamesBeanMachine Oh my

[@apothecaryjohn]

Matt is a narcissist

[@Jahu-qs2us]

1:25 Normal person: "because your hands are too small" Mathematician: "because your hands are finite"

[@fruitshuit]

I started watching this video, but the intro made it sound like it was a follow-up to the video on Steve's channel. So I paused and went and started watching the video on Steve's channel and he made it sound like it was a follow-up to this video. The infinite loop made my brain crash, thanks.

[@b1odome]

Which came first? The chicken or the egg?

[@Pyronaut_]

I was fine with that because it seemed to me that Steve's video came first, but then my brain crashed when I reached the end of Matt's video and found myself back at the beginning of Steve's.

[@PlonkapplePrequel]

fruitshuit The egg because the first chicken has to have been born somehow.

[@raymondlee1024]

b1odome the egg because the chicken's ancestors were borne of eggs

[@jonathanfowler2932]

Same! Steve's is the first, though.

[@Ameto]

When life gives you oranges, make spherical lattices with them

[@tylercrowley2559]

Next life quote

[@youreviltwin]

Life is like a box of oranges. You spend all day trying to figure out the optimal method for packaging them.

[@tylercrowley2559]

KZbin comment sections actually inspire me so much

[@fanrco766]

when life gives you oranges, spend months attempting to prove 11 dimensional oranges efficiently pack in not lattice structures

[@Rayblx]

no, don't (hmm, i guess heads are spherical enough that they stack in a lattice...?)

[@jamiesmith8220]

very simple and elegant work. "Let me just rotate by tau/2 radians", I could feel the frustration in your voice Matt

[@CrushOfSiel]

"If feel like I've been accepted by your culture." "You haven't." LOL that was the best nonchalant burn ever...

[@SchutzmarkeGMBH]

"Okay, so this is a bit awkward, but we're gonna try something even more awkward now" Story of my life.

[@jamalhalili2173]

Simon T haha true

[@hebl47]

I love how us normal people go: "Oh, ok - so this is the best way to pack spheres (in our physical world)." And then end it there, but a mathematician goes: "Ah! But what about n-th dimension? Let's see how they stack in 23 dimensions."

[@Speed001]

Yeah, but I'm sure it has some application in engineering or science with lots of variables.

[@aryaaswale7316]

@@Speed001 nope. we live in three dimension ya know

[@massette43]

@@aryaaswale7316 they're only fourth dimensional but quaternions are a thing. living in three dimensions doesn't stop us from conceptualizing things in higher dimensions (even when it really should)

[@estebanmarco8755]

@@aryaaswale7316 n dimensionnal spaces are quite common everywhere in engineering, or everywhere else. In the physical world, the dimensions for our space may be how much do you like or dislike some things (for example a list of activities), and now suddenly you have a space with hundreds of dimensions and are trying to see whether people are close to each other and how to separate them.

[@AndrewKay]

The orange companion cube will never threaten to stab you, and in fact, cannot speak.

[@mal2ksc]

[♥] [♥] [♥] [♥] [♥]

[@1224chrisng]

+Scott Sakurai your comment have only 5 Hearts if it has 6 Faces (obviously its a Cube (of Companionship) itd have 6 Faces) then itd have 6 Hearts

[@1224chrisng]

When life gives you Oranges, don't make Orangade. Make life take the Oranges back! Get mad! I don't want your damn Oranges, what the hell am I supposed to do with these? Demand to see life's manager! Make life rue the day it thought it could give Patt Marker Oranges! Do you know who I am? I'm the man who's gonna burn your house down! With the Oranges! I'm gonna get my engineers to invent a combustible Oranges that burns your house down!

[@buttonasas]

@@1224chrisng You assumed a companion cube has 6 faces with hearts. But have you ever rotated the cube to look underneath it? What if that face is actually missing? What if the cube then really threatens to stab you?

[@doublespoonco]

@@sus-kupp yes

[@samsupke1142]

i finally found the guy from my math problems.

[@MerthanE]

XD

[@daemonCaptrix]

I died!

[@CraftQueenJr]

The “A man has forty oranges and has to pack them in a crate as efficiently as possible, how does he do it?” guy?

[@inigo8740]

"When life gives you oranges, make a tetrahedron." -Matt Parker, Things To Make And Do In The Fourth Dimension

[@nisargbhavsar25]

It was basically a FCC lattice

[@NaNAmbient]

The chemistry between these guys always makes me just smile as wide as I can without thinking about it :D

[@jimmysol]

I was sitting in my Asphalt Pavement DOT class this morning where we are learning about packing theory for aggregation and all I could think about was this video from 5 years ago. Thanks Matt and Steve for helping me pass my certification.

[@jwhite973]

Matt likes pi so much he's shaved it into the top of his head 1:11

[@mcinacio8323]

I was gonna like your coment but then I realised how unbalaced the number of likes wold be, that was a close one

[@LeBartoshe]

Oh, man... That was brutal xD

[@panda4247]

I thought it's the logo of the channel Real Engineering

[@OwlRTA]

that was so brutal that he shaved his head

[@michaelcolbourn6719]

20 people need to unlike this so it's at 314 likes

[@whitherwhence]

Looked it up. Christingle is a thing. The orange represents the world, the candle represents Jesus, the red ribbon represents Jesus's blood, and candy represents the fruit of the Earth. People do weird things.

[@Dragon-9000]

I used to do them

[@timothybexon6171]

I never understood them. I'm a Christian, but the church I went to never did them. But the church near my school did. So we did them on school trips. They make no sense.

[@Jimi4256]

@@timothybexon6171 we did it in my junior school. We had a bunch of ceremonies at the local church but christingle was the best one, cos you got to eat an orange and 4 sweets afterwards... I was laughing so hard at the jokes they made about it XD

[@woutervanr]

@@timothybexon6171 "they make no sense" so pretty on brand for religion then :p

[@timothybexon6171]

@@woutervanr True.

[@Gorgoj]

loved that they fumbled with the balls in their hands while having a perfect example in the box.

[@veggiet2009]

"My studio, my circle constant"

[@amyshaw893]

veggiet2009 oh my god it's you again. you, with the profile picture that looks like nerdcubed's eye

[@simor879]

Like this guy who tried to define Pi by law...

[@amyshaw893]

?

[@mackrac]

"Let me just rotate through half tau radians"

[@alandouglas2789]

veggiet2009 James

[@joea8426]

Not only do I love this video because of how interesting and informative it is, but also because of how entertaining it is. Here I am coming back to it 4 years later for the n-th time for the relentless sarcasm and great chemistry. We love you guys!

[@Chris_Cross]

I know Matt's thing is maths, and Steve's thing is physics, but I absolutely love the chemistry between them. It's hilarious to watch them rip back and forth. And I'm really starting to wonder if that Methodist orange stick lolly candle thing is actually real or not...

[@EmberLeo]

Wikipedia seems to think they are: en.wikipedia.org/wiki/Christingle

[@frognik79]

I love how you both cut each other off at the end of your videos.

[@SlyMaelstrom]

They did it because they're trying to create an illusion of a loop in the videos. You might notice that the beginning of Steve's video is the same as the end of Matt's video and vise versa where the script suggests that they just did the other person's video. So which video did they do first, really? Hint: One has an obvious insert before going back to the two shot.

[@Dankey_King]

That cutaway at the end xD

[@jamesbeanmachine857]

Matt's revenge against Steve for being rude and using the "wrong" circle constant

[@savage1267]

JamesBeanMachine They pick up in Steve's video.

[@TheTombot]

That cutaway is the transition to Steve's video. Then Steve's cutaway is a transition to this video ;)

[@AnonymousFreakYT]

I LOLed.

[@VeryLostFisherman]

Oooh clever transition.i thought he was being savage 😂😂

[@fakjbf3129]

FYI the Christingle is a British thing, not a Methodist thing. It's used by many Christian denominations but very few people outside the UK do it.

[@RedWurm]

Yep, I attended a couple in a C of E church when I was little. I still remember toasting grapes over the candle.

[@austinfernando8406]

I was brought up catholic and i'm pretty sure we did it at school, so it's british not just methodist

[@maghouinbeg5011]

The first Christingle was in the Moravian Church in Marienborn (Germany). It's history can be found at: www.moravian.org.uk/index.php/the-moravian-church/moravian-christingle

[@aspden8809]

Oh wow the memories... I forgot the Christingle even existed until he mentioned the sweets on sticks and then it all came back to me. Nostalgia hit hard there (for my childhood, not for the religion).

[@B3Band]

I like how Steve's video says "with Matt Parker," while Matt's video title says "with oranges." Says it all, doesn't it?

[@WONMARK]

And points at Steve in the thumbnail

[@timw1971]

Love the way you guys made this video as if it were after the one on Steve's channel, then made the one on Steve's channel look as if it was after this one.

[@ntsure2436]

Really enjoyed watching this video, guys. Thanks for the collaboration. If more people knew how much math could make you laugh, there might be more engineers in the world. :)

[@WarMage]

I simply love the editing that allowed you to post the two videos in such a way that they cause an infinite loop...

[@trucid2]

"Edible spheres" That's what I'll ask for next time I'm in the grocery store.

[@orangemage9522]

Genuine hilarity when you cut him off in the middle of plugging his own video. And in the video you filmed for your channel in his studio no less. Brilliant!

[@earthbjornnahkaimurrao9542]

MFW I am stuck in a loop trying to figure out which video to watch first because both videos reference the other at the start making me think I should watch the other video first but then the other video references this video at the start making me think I should watch this one first!

[@Moley1Moleo]

The first scene of one is the final scene of the other. This is true for both videos. !!!

[@savage1267]

Moleo And it is wonderful!

[@alecwhatshisname5170]

Earthbjorn Nahkaimurrao möbius videos

[@cipollinodan]

"Your hands are finite." may be my new favorite sentence.

[@CR0SBO]

Nicely done with having each video follow the narrative of the other! I never thought of using oranges for this! Diagrams provide fewer sticky fingers (not guaranteed).

[@BloCKBu5teR]

I don't care how geeky this sounds, but i genuinely share their enthusiasm.

[@neshploda17]

Now we know how Matt will calculate pi next year. I image some sort of juicer will be involved.

[@jonz2055]

Turns out the packing of spheres is the basis of an entire engineering branch: materials engineering! The understanding of how (at the atomic level) the packing of spheres/atoms and interactions thereof govern how different materials work! Most commonly studying metals, ceramics, and polymers as materials disciplines. This exact problem is the majority of a intro materials class, crystal structures and lattices! As a materials engineer these videos were lovely to expose this to more people! Well done.

[@Archiekunst]

I laughed so hard when Matt abruptly cut off Steve's segway into his channel. Love these chaps' banter.

[@adnamamedia]

It's incredible how fast your channel has grown. I have loved your channel since the first day I found it a while ago.

[@prothstein]

you got to love the cut off at the end. almost an FU to physics vs applied mathematics

[@Thea1d2r3i4a5n6]

I know little to no math and I'm not that bright BUT you guys are entertaining to watch and the math, references and such go way over my head but you do so nonchalantly and speak of it easy that it doesn't make me feel dumb or hurts my brain, thanks you and have a great day

[@Soliloquy084]

I remember working this out in a first-year chemistry lab. Was a good time.

[@Bobbyo2014]

Love that you guys made the videos match up no matter the order you watch the two videos.

[@Noremaad]

"Do you have an eighth?" is a real Parker Square of a drug deal. Everyone knows you don't buy partial pills, Matt. That's how you get cheated.

[@earfolds]

Plus, wouldn't you need the whole needle, not just an eighth of one, to inject a marijuana?

[@tdawson198]

I assume if 7 eights are equally taken off of each separate part of the needle, one might inject an eighth of a marijuana

[@fanrco766]

Not to be the killer of the joke, but an eighth is usually used to describe 3.5 grams of any illicit substance (3.5g is an eighth of an ounce)

[@titanmcrolland6877]

fanrco "Any illicit substance" No one has ever came up to me asking for an "eighth of coke". Usually only marijuana that deals in ounces. Harder drugs are done by grams if purchasing locally and kilos if your importing. I'm European though so I can't speak for imperial measurements. Though I can't recall at any point in my life people asking for an eighth of any substance other than marijuana.

[@fanrco766]

Noremaad You're totally right, I've never heard it used in any other scenario other than marijuana (mostly because I try to avoid anything outside that scope) so I just assumed the phrase spanned to everything else

[@savage1267]

Nice job, guys, making the videos link up in a cycle. I'm impressed.

[@AnonymousFreakYT]

"I feel like I've accepted by your culture." No hesitation at all "You haven't."

[@JayJay64100]

That "the outro is the intro of the other video" thing in both vids is great. Really great.

[@ElaaxV]

22:20 DENIED

[@calebdoner]

That spontaneous conversation about the Christingle was hilarious!

[@olivialambert4124]

This sums up why I ended up in Physics rather than maths too. I always preferred the Mathematical aspect of Physics, but because A) my organisation is an utter mess and B) I prefer the answer to the beauty I went towards Physics. Here we have Steve Mould with a "messy" sheet of working out on all directions (look OCD to mine) and Steve wanting to finish at the result rather than the beautiful simplified result. Its something I've noticed throughout my career, Mathematicians want utter perfection and Euler's Formula like beauty, physicists would rather take a few assumptions like sin x = x, and any simplification is to ease remembering. And that is also why Mathematical Analysis never agreed with me, a branch of Mathematics so anal as to demand a proof for "1+1=2" with a Physicist's constant reasoning "because it is" for any semi-obvious mathematical assumption. There's your answer, nobody cares why, lets get to the fun stuff. On a side note, I've always tended towards watching maths videos. Not sure if its because I simply prefer listening to Maths, if its because it gets difficult to find a video with enough Physics left to learn, or if its because the physics I do have left gets a little too in depth for easy watching. Either way, I could watch you two collab forever. Physicist and a mathematician, but you certainly didn't lack any chemistry. Yeah I know, women aren't funny.

[@trulyUnAssuming]

As someone who would rather just do the beautiful maths, what you said is pretty accurate 😂. Although it differs quite a bit by field. Applied mathematics like numerics and statistics can be complete abominations, while purer counterparts like analysis and stochastics are generally very pretty. And if everyone knows that everyone knows how you would prove something, you can leave it out as trivial. You just don't allow early semesters to do that. 😉 I guess most mathematicians end up doing the pretty stuff for fun and the applied counterpart for fun-ding

[@oliviabruner5824]

Very much enjoyed the "working it out" intermission in the video, and rather proud of myself for getting the right answer (even if it is just simple calculation) on my own. Thanks for the great video.

[@YarianZy]

Nice video

[@monchytales6857]

two dudes hanging out doing math and physics for fun this is friend goals

[@freshrockpapa-e7799]

14:28 "that is the least cool use of the phrase have you got an eighth" Damn, wasn't expecting that kind of joke from parker at all lol

[@digestivedunker2044]

I don't normally do the working out parts of these videos to be honest, but this time I thought I'd give it a go. Turns out I got the right answer. I know it isn't exactly like I've discovered e=mc², but it's always good to know that you have worked something out correctly.

[@stocktonjoans]

*Waits for Steve to try and pour Matt out of a beaker*

[@Gillysaurxx]

These guys are answering questions literally no one has asked before

[@GlizzyTrefoil]

So, can we call this 6:12 a Parker-Packing?

[@Moley1Moleo]

I like the illusion that these videos looping with each other. For both videos, the final scene is the same as the first scene of the other video.

[@grzegorzcichosz8240]

*When you should revise for your physics test that's tommorow but Steve Mould and Matt Parker have both published new videos just 2 minutes after each other*

[@automatedminer7158]

Close enough

[@RyanDB]

At least one of my exams is on crystal lattices, but I get you.

[@cosmicjenny4508]

+Grzegorz Cichosz I also have a Physics test tomorrow. Are you sure we're not all from the same school? Lol

[@laionneves3477]

Funny story, my tomorrow's exam includes crystal lattices so this is actually a good review for me

[@joe9832]

My Physics exam was last week, thank God it was easier than half of the years worth of material we had to learn. Having said that, don't be like me, if it says "End of exam" at the bottom of a page, check the back anyway... I lost 10 marks on it...

[@yuvalne]

I thought this was a follow-up video to Steve's, so I watched his first. I couldn't figure out which video his one was following up, so I continued, and then went back to yours. Well played you guys, well played.

[@derekantrican]

"Two guys get together and play with balls"

[@Vulcapyro]

They try really hard to get their balls to touch.

[@mrphlip]

Mathematicians tried for centuries to kiss as many balls as possible.

[@bwayagnesarchives]

This shouldve been the title 🤣😂

[@c4oufi]

These two videos recall all the memories on the metallurgy class back on college.

[@nikolausengh6630]

0:21 I thought the video was stuck for a moment there XD

[@aaronallblacks]

10:03 "A mathematician buys 12 oranges. He gives 3 of them to his friend Steve. How many does he have left?" These problems are real

[@zionj104]

9.2 thousandth view! YYYYYAAAAAAAAAAAAAAAAAAAAYYYYYYYYYYY!!!!!! I'm surprised this never happened: Matt: There it is! Pi over root 18! Steve: Tau over root 72. Matt: Jerk.

[@mihast3273]

There is something so cosy about watching two men building a spherical lattice with oranges and toothpicks

[@DrSnap23]

You love to struggle arranging stuff in a square on this channel, don't you ? =D

[@zatarraagain7496]

The fact that I studied that all you have showed here has an application in chemistry and crystallography makes it even more fascinating

[@nickchampion8392]

ALL HAIL THE GLORIOUS PARKER SQUARE

[@RomanQrr]

...get out... there is nothing parker about that square!

[@alexkatz9047]

It was created by Matt Parker. So it's Parker's square

[@derekantrican]

Is it now a "Parker Cube"?

[@imusthegreat]

It's about 74% of a full cube, so I guess it IS a Parker cube ...

[@SlyMaelstrom]

Parker lattices have an amazing 93% packing density... except some of the component spheres are much smaller and others are made of a very malleable clay. Matt still thinks it's pretty great, though.

[@oniondesu9633]

This is the most oddly romantic lattice stacking video I've ever seen.

[@Kostchei]

21:28 Says "do it properly", but cuts the video anyway xD

[@stevemcintosh9381]

It kind of both amazes me that this video leads strait into the next one which leads strait into this one, perfect loop.

[@Porkey_Minch]

Which came first? The "Calculating the optimal sphere packing density: with oranges" or the "3000 ball bearings show crystal defects with Matt Parker"?

[@mitchbottema1317]

You can demonstrate that so much easier with the use of round magnets Preferably round colored magnets to make it even more noticeable.

[@kleko]

brb popping down to the shop for some oranges

[@davidonfim2381]

lol I just watched Steve's video, and I love how both of you cut off the other person at the end before they could talk about their channel.

[@invyspirit]

You guys should make an infinite number more collab videos together!

[@jonathanfowler2932]

6:50 The spheres mark the vertices of a cuboctahedron, if anyone was wondering.

[@PlasmaHH]

I would really love to hear about the properties of the "jumbled packagings" in higher dimensions, it is rather uinintuitive to imagine there are some that are not regular, so maybe the (prime?) properties of the actual dimension play a role in here?

[@heaslyben]

Love it! Thanks for the fun crossover!

[@silpheedTandy]

god i want to ship them so hard. they're both gorgeous, personable, funny, and smart, and they both totally make it a fun way to pass the time thinking about mathematics. i love it when they show up in videos together! that chemistry and banter that they have, so delightful and dreamy. *faints*

[@disnecessaurorex4908]

i know the comment is more than a year old but omg i thought the same and saw no else talking about it e_e they are cute together

[@Auchioane]

Really great videos. Nice editing, love how you can watch either video first and it works :)

[@CarnelianUK]

Sweets on a Christingle? Bah! In my day we had to make do with cloves!

[@fasfan]

Brilliant how you managed to somehow make two separate videos that blend seemlessly with no actual beginning or ending. #Mindblown

[@lyradawn4176]

this is the most adorable math vid ive ever seen. boys playin with balls and fumblin around. yall r cute

[@drandrewsanchez]

I'm sorry but I don't think I followed the 'tetrahedral Christmas gift' explanation at 2:30

[@magicicle]

Steve x Matt I ship these 2

[@cee_yarr]

Serene Grace oh gosh the shippers have arrived

[@silpheedTandy]

lol it's fun when i write a comment, then scroll down and see that others feel the very way i do :D

[@insertcreativenamehere492]

This is the second infinite video loop I've watched today. I just finished the vsauce/minutephysics one and now im here.

[@MrCyanGaming]

The amount of spheres that can touch a sphere in any dimension is: n^2 + n 0: 0^2 + 0 = 0 1: 1^2 + 1 = 2 2: 2^2 + 2 = 6 3: 3^2 + 3 = 12 4: 4^2 + 4 = 20 ...

[@Kalumbatsch]

No. For n=4, it's 24, and in higher dimensions your terms are lower than the known lower bounds. Look up "kissing number".

[@Chezboi30003]

21:45 you should do another collab with the Hydraulic Press Channel about packing of spheres in 3D.

[@charlien.5841]

The square base is not the same as the tetrahedron on its side (though the spacing is the same) as a tetrahedron is platonic and so is the same when placed on any face. There are no Square faces, and the square base has one, so the two shapes aren't congruent.

[@stevethecatcouch6532]

True, a square based pyramid is not congruent with a tetrahedron, but how is that relevant to the video?

[@aoifebakunin1966]

The tetrahedral arrangement has 6+3+1 or 10 spheres, the square one has 9+4+1 or 14 spheres. Take four spheres out of the square arrangement without moving any of the others and you get the tetrahedral arrangement.

[@LokNWykLeer]

5:07 I've been trying for YEARS to figure out the arangement of planes in a multiverse for a fictional world I'm working on, so this helped me actually. Thanks!

[@toucaninterieur8011]

15:45 NOICE

[@tcf70tyrannosapiensbonsai]

6:50 The regular pattern, Steve expected, is an icosahedron. If Matt took three red balls in the left hand with the green one on top, as he did, and the three other hands of the two Smarties would carry three red balls each, they could arrange the triplets as a tetrahedron. Et voilà, an ikosahedron with twelve corners. From that perspective it's hardly imaginable, that anyone considered 13 balls could ever touch the center ball. I really like watching the two of you educating us math and physics with such joy!

[@Smitology]

Well yeah depending on whether you consider each sphere a face or a vertex it's a dodecahedron like he said or an icosahedron you you say

[@alan2here]

hypertetrahedral = simplexal?

[@twichnitr0]

Christingles. Man did not expect to be reminded of that nugget of child hood in a maths video.

[@alvatopia]

Please watch this video on 0.5 speed and try not to laugh.

[@dramawind]

This works for every video actually. You can turn anyone stoned as fuck.

[@bellemyers8776]

jesus they sound drunk and stoned hahaha

[@griffinbeaumont7049]

two guys high as a kite working out how best to stack oranges LMAO

[@alvatopia]

Yes it does but if you take a look at the content you get not only two stoned guys but two stoned guys playing with balls and stacking Oranges.

[@Godram]

i love that they both cut each other off at the end.

[@seanm7445]

Tetrahedral numbers are cool, but they’re not quite as cool as 90,525,801,730 -packing!

[@ffggddss]

At the working-out pause: Let each sphere have unit diameter (easier than using unit radius). Then each sphere has V = ⅙π. The 'cube of sliced oranges' contains 8 ⅛'s and 6 ½'s of spheres, which is 1 + 3 = 4 spheres, whose volume is 4V = ⅔π. The cube's face diagonal has length ½ + 1 + ½ = 2, so the edge = √2, and the cube's volume is 2√2. So the filling fraction is f = ⅔π / (2√2) = π/(3√2) which is also π/√18, as Matt likes to put it. Incidentally, it's interesting to note that since these spheres have unit diameter, they will, when cubically packed, have a filling fraction of V = ⅙π ≈ 0.52360 while the filling fraction we just found for the optimal case is π/(3√2) = ⅙π√2 ≈ 0.74048 or exactly √2 times a cubic-grid packing.

[@TS-yb2lo]

pi > tau

[@The0Skeleton123]

pi > 2pi? There is something off with your logic :P

[@Doobs110]

therefore, 1>2

[@foundleroy2052]

tau > pi ;)

[@jamesbeanmachine857]

how is 1/2 turn greater than 1 turn?

[@raptokvortex]

I agree with the principle, not the mathematics.

[@jmo235]

Props on the never ending loop between videos