The secret of the 7th row - visually explained
In 1995 I published an article in the Mathematical Intelligencer. This article was about giving the ultimate visual explanations for a number of stunning circle stacking phenomena. In today's video I've animated some of these explanations.
Here is a copy of a preprint of the Intelligencer article:
www.qedcat.com/misc/stacks.pdf
And here are links to a few beautiful interactive animations of the circle stacking marvels that I talk about in this video (on the Cut-the-knot site):
www.cut-the-knot.org/Curricul...
www.cut-the-knot.org/Curricul...
(check for more links to related animations at the bottom of these pages)
As usual, many thanks go to Marty for all his help in getting this presentation just right and Danil for his Russian subtitles. Also, thank you very much dad for your help with building the stacking machine that features at the end of this video.
Enjoy :)
13:28 "Are you proud of me?" I am ! That is amazing !
Yes, I am also proud.
I like how the word 'Geometry' on your shirt is written so that it is symmetrical.
suddenly i can read it XD
14:07 “I just spent 5 weeks in Germany...” Wow, you picked up an accent pretty quickly!
Hmm, I think it's more likely that I picked up the accent while growing up in Germany :)
Funny XD
funnies thing I've read all month
There is no German accent. As a German I can assure you, he speaks perfect English. ;)
@@Sgt__Hawk yes, perfect english, but it's not accent-free. those things do not contradict.
"Do you like this proof?" Yes I do, Mathologer
:)
Practical Application: How to secure a stack of round pipe on a flat-bed trailer so that it is stable over the road, using lengths of boards (standard 2x4 dimensional lumber) and tie-down straps. Determine how tall the stack will be.
Though stacking like this does not extend to 3D with spheres, it can be useful when leveling things via successfully rigid rods, as they will have a cross section that will always remain level. Though I think this is close to just being a scissor lift then.
14 feet or less due to state laws lol.
How big is the diameter of the pipe? How big is the truck bed? That needs to be known, yes?
@@easyjdier As long as all the pipes are the same diameter, the outer pipes are tangent to the walls of the truck bed, and the pipes are sufficiently close together so as not to allow higher layers to slip through, then it will work. Now the specific number of pipes/height will change depending on the width/diameters, of course.
This might be the first Mathologer video I've watched that made 100% sense to me. Not a criticism of Mathologer, his videos are excellent and I'm an idiot. But this was beautiful and so intuitive even a moron can follow. That was rad. Great video.
That's great :)
Hey, Folks! - Don't sell yourself short.
beautiful AND practical.
@@geezermann7865 I appreciate it but I don't think of it as a negative. Being an idiot means I've got lots of places to go. Lots of things to learn. Just means sometimes I slip into something like Mathologer and 3Blue1Brown and think "I know English but I can't comprehend these sentences." Still have a great time and I do love Mathematics. So it ends up being a win for me regardless.
Like my grandfather always used to say, the more you know...the more you know you don’t know!
One application could be the chemistry of crystals. You always try to find for example unit cells and symmetries to determine which packing fits and/or which crystal system the crystal structure at hand belongs to.
Yeah, this could potentially explain how crystals form; building up uneven layers until a flat layer is formed and then regular crystalline structure taking over from there.
My mind immediately went to solid state physics - in fact that's what I initially thought the videos might have been about.
Yes it might explain in part the parallel sides of large crystals. Also might explain strong planes and weak shareable planes in crystals. Also electrical conductivity in metals might flow preferentially along these parallel planes.
Also also, materials that undergo phase changes under pressure - the force effectively makes the "spheres" of the atoms clip through each other and shift to a new stable arrangement.
I agree that practical applications could be found for surface chemistry in the context of stacking atoms or molecules on a surface and could make use of this neat property. One problem that I can see is that there is a bit of a limitation as to how big can the surface be (i.e., width of the jar) relative to the particles (if I got this right, it would affect what layer will be the horizontal one, in our case study, a bottom row of 4 elements meant 7th row was "magic"). Another idea this gave me was to use the trick instead in the modelling of crystal structures where computation effort can be prohibitive depending on model assumptions. What is neat there is that, despite we have non-ideal stacking, we can still derive some symmetry property and help reduce the cost of computation. I like how this video shows how balls would adjust to different non-ideal configurations as the walls are moved while retaining that C2 symmetry element. Pretty neat. Maybe there is something to do about it to study particle interactions and lattice vibrations. I wonder how this would look in 3D ...
Let the magic number be M, and number of balls in the bottom row be n. M = 2n - 1
your answer although says what the number in question is, still lacks explaining why it is what it is. this can be easily understood by looking at the following picture: oooo ooo oo o oo ooo oooo what we seek is the number of rows 'M' in this structure and what we start with is the number of balls 'n' in its bottom row. this structure is build of two pyramids such as this: o oo ooo oooo except one of them lacks the top row. and since the number of rows in a pyramid is equal to the number of balls in its bottom row we have: M = n + (n-1) = 2n -1, as you claimed.
@@michalbotor thank you for that explanation 👍🏼
My intuition (M=5) was right in case of 3 balls. Couldn't find the formula though... Thanks for sharing it!!
I saw 4 and 7, went "2n-1?", saw 6 and 11, went "Looks like. So probably a symmetry around the center line" This video was a weirdly affirming experience 😅
If you look at the picture with the gray squares (e.g. at 8:12), there have to be n-1 horizontal rhombuses to hold the n circles. And then there have to be an equal number of vertical rhombuses since they are made by rearranging the horizontal ones. Each rhombus has the vertical vertices offset by two rows (with the horizontal vertices being in the row between) so this gets us 2(n-1), which is off by one, which we add in by noting that the bottom-most vertical rhombus has its bottom vertex coincide with the first row's horizontal vertex, so it's 1 + 2(n-1) which is 2n -1 by some algebra. Alternatively, we could just say that it's the 2(n-1)th row and consider the initial row the zeroth row which IMO showcases the truth better.
Despite being a Physics graduate, I had never seen this. And the very first time you built up to the 7th tow, at the start of the video, I actually let out a little involuntary gasp of delight. I LOVE it when this happens. Thank you, Mathologer!
This is the only channel that can get me to watch math at 7 am, I wish I had a teacher like him in highschool/college tbh
That would be quite waste of life for the teacher, if it was government school...
kzhead.info/sun/e6mfeKqrnXmOnas/bejne.html
Georg Wilde I grew up in Everett so most of the teachers weren't really trying because 90% of my school was wannabe gangsters and the other 10% was weebs doing jitzus in the corner lmao
HYEOL I was expecting that link to go to the rickroll or big chungus sicko mode but I was pleasantly surprised, thankyou sir! (Not that I don't enjoy my fair share of spicy memes)
we need 24/7 schools and elimination of the idea of "earning a living"; that's the dream
"Here is the secret of the 7th row!" "Figure out the math in the comments" Fine. Keep your secrets.
N x 2 - 1 ( where N is Number of circles, multiply by TWO, then subtract ONE.) EX: 4 x 2 = 8 - 1 = 7 EX2: 6 x 2 = 12 -1 = 11.
Line 1 + line 2? That’s why 6 balls was 11 lines?
@@eduardocorrochio208 Same number of balls touching all walls. Four on the bottom, four on the sides, four on top, so 4 'main' rows of 4 balls. Then 'filler' rows between them, n-1 filler balls per row in n-1 rows. So n+n-1 rows total. With 6 balls filler rows are 5 balls, 5+6=11.
@@user-sb3wh3dd4v lol, I am a year behind, you beat me to it. I was going to show this prediction.
g it’s really (n-1)*2, since it repeats every (n-1)*2 rows. The need to add/subtract one (depending on fencepost counting) is arbitrary.
14:30 ...that photo of Mr Mathologer Senior building such a tidy device answered so many questions. Please tell me that you made a video while he was building it... and upload it!, we want to meet your dad!
He's pretty photophobic :)
@@Mathologer well, please just tell him that it's a gorgeous model he built. (Or however you wish to translate that into German. ;) ... sehr schön.)
This was so beautiful. Thank you for taking the time to generate these gorgeous animations, graphics, and the physical model. Your content is the quantum gemerald!!
The device made out of wood and metal looks amazing! You should have joined the top with a long wooden bar to show that it's always a straight line.
That is a Do Nothing machine that is actually entertaining.
@@arikwolf3777 Then it does do something. It entertains you.
Nicey
There should have been a spirit level on the top row!
Mirko Junge level, or parallel to bottom row? Not remembering if horizontal means same as level.
Does it work with spheres? (outside of 1-wide depth)
Sadly no :)
Maybe it does if they're only stacked ontop of each other, with only 1 sphere of depth
@@dashua1735 But isn't that analogous to having 2-D circles?
It is, that's why I think it might be the only way for that to work
Only if it’s one row of spheres in a box as wide as one ball.
Everyone else: watches video and enjoys Me: confused by his T-shirt the whole time
I love how your father has the same hairstyle as you :D
class Father { public readonly HairStyle hair = null; } class Son : Father { }
@@nitsanbh it wouldn’t be read only because you can get a haircut, or dyed, or a wig. It’s definitely not immutable.
the proof is presumably useful in stacking problems such as atoms in a crystalline structure.
Yes ! Specially for study of anomalies caused by foreign atoms in the structure, ex. semiconductors in electronic devices (transistors)
Unfortunately, sphere stacking is very different than circle stacking, however on one dimension sheets of atoms such as Graphene, it might still have some use, maybe? I still can't really think of any :(
I actually thought of something, but it is really obscure and not significantly useful. It could be used to simplify simplex noise mapped to a cubic grid. Which could simplify the storage size and generation costs on the CPU, but that is a very niche use case.
Atom stacking is not so similar to sphere stacking as there are interactions (forces) between them. You can't just pull them apart arbitrarily to juggle the whole crystal structure as you can do with the device we saw in the video.
Wouldn't that (with in tolerances allowed by atomic forces) mean crystals have room for elastic deformation?
This reminds me of the packing of sand grains in soil mechanics. Unfortunately, we don't tend to deal with perfectly round particles (sand grains just don't look that way), but sometimes a very useful approximation can still be found by making that assumption. Also, we never deal with systems where all particles have the same size (again, sand grains). I'm interested in your bit at the end where you showed two different sized circles. I'm looking forward to playing with this some more. I don't see any really direct application right now, but sometimes that's where the coolest applications come from!
Exactly :)
Engineer: "I have a problem to solve, are there any solutions?" Mathematician: "I have a beautiful solution but... are there any practical applications?" never change...
The only application that springs to mind is packing pens or other cylindrical objects in odd sized boxes, but then there's no guarentee they will be level at the top of the box if it has a random height. If you could design a box specifically then you would just make it the closest packed solution, and each row would be level anyway.
Or the fair ground / fete game of estimating the number of sweets in a jar... I never won it so I guess I’m still bitter
Magic number for n circles in the bottom is (2n-1). Practical application : efficient tuna can stacking
Did you say “wobbly wine-sight”? I love that phrase, I will definitely use it!
3blue1brown recommended this video!!!
:)
Well - he usually does since both channels appreciate math presented in a cool way.
I have a small winery and one of our biggest problems is stacking the bottles without falling. So yes, I found your video quite enlightening and will put it to practical use immediately, it's also very interesting to see that I can use a pyramid of wine bottles to find the centre of a wall. With Covid-19 and the hotels closed, all the wineries here had a huge drop in sales, which means we have most of last year's stock plus the new wine we made this year that has to be squeezed into our tiny spaces. We are already at full capacity and try to find new ways to stack bottles. So thanks for the video, it was exactly what I needed!
I can see practical applications: transport of a heavy object that has to remain level. The balls could act like bearing. I have a question: How does a force exerted from the top actually transfer through the whole thing?
You could break it down by considering individual ball-ball and ball-wall interactions: start from the top, see how (*Edit: a ball) can get the support it needs from the bottom while balancing left&right forces, then slowly work you way down. When static, a system like that won't have any friction, so it should be relatively simple to calculate just by hand (but somewhat tedious)
How does a force exerted from the top actually transfer through the whole thing? nice point. Lets say you were transporting something. With another something on the top layer. And since it will always be a flat surface. You would be taking advantage of that. How much force could you exert on the edge, before the surface all flips up. Or is the force at the edge the same as the force at the middle. And how much of that answer depends on the friction between the balls? Like, could your bearing balls be a couple of shovels of field rocks? Or would they need to be silicon balls?
But the object will remain level only till the ground ( on which u r rolling the object ) stays level. If the base layer is not horizontal, it won't work. Plus side walls have to be perfectly vertical as well. . But lets say, if we are able to achieve this. My question would be what downward force difference would it make while comparing it with when the object is placed on the base horizontal layer. . That means if we are able to transfer a considerable amount of force on the side walls, this idea might work. . I need to think more 🤔
Ever seen a scissor lift?
The magic number (m) for any number of circles (n) at the bottom seems to be (2n - 1) but it also seems that if you don't confine yourself to having the second row being (n - 1) then you just get the sum of the bottom two rows
the second tow or any other row like that, not being n-1 would be unstable
@@RaymondJerome alas, so it seems... that's why I left it as an endpoint to my comment and not my main answer though. But, in some strange alternative of the mechanics involved, maybe the fact that it's the sum of the bottom two rows could be a more useful answer?
oh wait, the second row could be n, but that would then make the bottom row change its configuration.
The trick is to see that the even numbered rows don't really exist. Or that the columns also have them.
Wow, another mind-blowing video from Mathologer I can share with people to share why math is wondrous. Your animations really helped as this proof would be so different in text form. The best part is the physical box you built with your father to demonstrate this effect in real time. Mesmerizing! Thank you.
great video! I had a thought that one practical use for it would be in designing seating layouts for concert halls and stuff.
14:06 “I just spent five weeks in juh-” [Loud noises in the other room drown out the rest of the word] My brain autocorrect: “JAIL? What did you do?!”
Christmas :)
Mathologer actually spent five weeks in jail -- for having divided by zero.
Amazing! Thank you Burkhard! -> What happens when the bottom of the glass/jar is uneaven? (got to try it out...)
If you are interested in more details, here is my writeup from a couple of years ago . www.qedcat.com/misc/stacks.pdf
In the desert, there are a number of "glass houses" made out of old bottles laid on their sides (effectively a circle with mortar holding them in place). Using the principles outlined in the video, build in rectangles in order to keep the windows and doorway rectangular, even if the bottles within the rectangle are "random".
This is by far my favorite video you have created. I learned something. Thank you!
I'm not yet finished but I'm charmed by this video. I think this could be one of your best videos ever! Thanks!
I'm thinking that this could be used as an attempt to tackle the problem of finding order within seemingly chaotic particle motion (with regard to temperature, fluids, etc.). Obviously, would tackle just a portion of those cases, but I think it maybe could be applied.
Magnificent video. As an engineer the model is appealing on a level that it can be used to teach others who are more tactile. I especially like how you included your father in the video. I'd like to see more people include either their father or mother in their videos. It shows a certain level of respect and humanity that seems to be slipping from our culture today.
Excellent - this is definitely a candidate for your best video yet.
Does this translate to 3 dimensional spheres as well? And how does that effect the proof? I would imagine that it should be pretty much the same since you mentioned that different sized circles keep its symmetry, and if you take a 2 dimensional cross section of any point of a box filled with spheres you should (I think) end up with 2 sized spheres as the plane cuts through spheres at different points not through the center.
I don't think an irregular arrangement would lead to all the "smaller" (actually just shifted in the 3rd dimension) spheres being the _same_ "smaller diameter", and he didn't say this would also work "smaller spheres" with any random diameter...
I was just about to ask this
@@AttilaAsztalos what I was thinking was sense they would shift along the y and z axis the regular sphere would look irregular if the plane cuts through the center of some of the rows. So it would get the whole sphere of one (or more) does and an off center of others (and possibly all others?). I'm inclined (no proof) to think this should be true of 3 dimensional spheres too.
@@AttilaAsztalos read that wrong, let me respond properly. Ur rightn he diesnt say tgat, but he does say "different sized circles for alternating rows" but doesn't, so my question should maybe have been a different phrasing. I meant more that if a box is packed and it follows that rule, and assembles into some amount of regularity, there would be those alternating smaller circles in the 2 dimensional plane. And then my question ends up being how does that (and 3 dimensions in general) effect the proof
No it doesn’t, unless the spheres are packed into a box 1 sphere thick.
9:54 -one circled moved! (left top yellow/brownish circle)- Edit: actually, few circles moved!
Applications: coil winding for maximum fiber density, understanding free space and sizing the bobbin. How bullets stack in a magazine and will or not feed properly. Foundations of metrology and finding center. You have a beautiful mind. Things like this seem to not have application but they do. And it’s why everyone should think like this and explore a topic that tickles the brain. I loved your video, thank you.
Great job explaining this amazing mathematical nugget! Also a very clever proof.
Holy shit this is amazing!
What arrangement/spacing of circles at the bottom row gives the max/min height of the top row?
I have the same question
My best guess is either bottom row X 2 - 1 or the sum of rows 1 and 2. EG 4 X 2 = 8-1 = 7 (6X2=12 -1 =11). Or 4+3=7 (6+5 = 11).
That physical model you made is wonderful! I want one! I would imagine there should be many practical applications, but I have not thought of any yet.
Incredible! We should have known about this 7 years ago. There is an application: melting ice blocks in a piston! As any physicists, we said that the blocks are discs. But we missed the essential! Thank you to your video!
Great point!
@@Mathologer More recently (not the very same geometry): journals.aps.org/prresearch/pdf/10.1103/PhysRevResearch.2.023070
Looks like you're developing a tetris variant.
Ok let me make this straight ! what you're saying is that the arrangement of some tangible coins in a tangible wood frame that I'm touching in this very physical plane obeying some kind of ethereal geometric pattern that flip and flop only in our imagination, totally seceding from our corporeal perception and it spans to infinity ? Oh crap my brain is burning !
Yes. Welcome to maths. (Yes it is very possible to get an education and never once experience mathematics.)
oily foods help lube the gears
This is so fascinating and amazing! I've never heard of this principal before
I hadn't considered the mathematical exploration of this - but I have noticed this in practice when packing steel tubes into poorly fitted crates. Very happy to see a proper explanation.
*Actually i see it can be applied in different engineering tasks:* *1) For storages of circular objects (wine storage etc) on shelves.* *2) For superheavy transport vehicles (moving rockets to launch site etc)*
I'm actually working on a project what problem statement is somewhat as follow: "Optimize the loading of cylindrical (pipes) shaped object in a TL2." If people could show me some papers about this, I would ne very happy!
R=U/I
@Abdelaziz Zariouh What’s a TL2? Edit: forgot question mark
Well, they don't really stack those things on top of each other. There is usually something in between to keep them from hitting each other during transport
@@toshiro0o If those somethings are equally sized, e.g. same sized wedges, the distances between centers remain equal. Everything works fine
Magic number 7
Awesome video, that contraption you made is amazing!
For some reason, this is reminding me of a polyrythem in music where 2 rythems of equally spaced beats but different time signatures are played against one another. At a perodic point the two rythems converge so that the downbeat of each rythem is played at the same time. A 3:4 polyrythem for example, where 3 a beats per measure component rythem is played against a 4 beat per measure component rythem. The convergent beat happens every 6 poly-beats, or at beat 1 and 7, etc. Without doing any of the math and making a guess, it looks like a polyrythem eqivelent to the lower beat value rythem of the polyrythem (3 in my example above) being represented by the number of circles and the higher beat value rythem (4 in the above example above) being represented by the walls. Normally, when you play a polyrythem, you play an equeally spaced beat in each of its component rythems. (1 2 3 vs 1 2 3 4) Without actually tapping it out, this is suggesting the possibility of constructing a polyrythem where the lower value rythem is not equal spaced and is not repeating within its cycle but is symecrical around the same point that the equal spaced rythem, the unequal lower beat value rythem would still converge with the equal spaced higher beat value rythem at the same point it would if the lower beat value was equal spaced. Taking this a step further, you could extend this example to 3 dimensions and this would be equivelent two mapping a pattern of 3 or even 4 rythems playing against each other, resulting in a very powerful convergence beat. And to be completely random, extending this to general waveforms and construction, you'd have to be careful not to construct a building or a bridge without frequencies in the design which apear to be random, with the goal to iliminate any wave combination effects (like in the Tacoma Narrows bridge) but actually set up a multi-component polyrythem with a long convergent point that could converge and snap with no aparent warning and seem to be a random occurance but actually mathematically predictabe given that all the relevant equally spaced and non equally spaced component rythems were identified.
I think I have an application; in metals the atoms are stacked this way! You should talk to a quantum physicist specialised in metals. :)
Atoms in metals don't usually have rigid perfectly straight walls constraining their movements.
@@ergohack yet quantum physicists use square wells with infinite thick walls to understand quantum behaviour -> en.wikipedia.org/wiki/Particle_in_a_box
thats a box of potential energy, there aren't actually walls..
Atoms are spherical. This doesn’t work in higher dimensions.
Willem van de Beek that’s a toy model we use. Infinite square wells don’t really exist.
Let n be the number of circles in the first row The pyramid in the first step will have n layers The final construction has 2 times that size -1 as it's the pyramid on top of itself with the tips overlapping Thus the 2*n-1 th row would be level again
Your real life model brought back memories. My old Mecano set. Mecano was 50s and 60s toy stuff for very young inspiring engeneers and mechanical technicians. It consisted mostly of strips and plates of metal, dyed green mostly, with holes on regular intervals that made it possible to make joints with screws and bolts. It was way cool for a kid in those days to have mecano set. It had the allure of wealth and intelligence in one. If you had it, your classmates would envy you! 😁
I thought I was too old for it.. but MIND BLOWN!!!
2x-1=y X=amount of circles on the bottom Y=the magic row that is level
7th!
Very funny :)
THIS IS SO SATISFYING! I had a feeling that I knew what was going to happen after he said "Let's move the bottom circles" and I was so happy when it happened
This is fabulous, the behavior looks very mysterious and has an easy proof. This is the sort of thing that everyone should learn in middle school to show them what math is all about.
Also it seems like there are a lot of other ways to approach proving this, which is cool. If we view this as a system of levers and joints, we can talk about the number of degrees of freedom in the system and get the result that way. What was initially surprising to me was, this is a seemingly continuous system behaving like a discrete system. IE, if we were stacking pixelated objects, we would have a guarantee than eventually the stacking pattern would repeat due to the finite set of possible arrangements. I wonder if there's a way of viewing this as a discrete system and proving it that way.
4:17 Mathologer: "Let's begin with a visual proof from the book" Me: But is the proof small enough to fit inside Fermat's margin though?
Evariste Galois BOI
It was probably Erdös's book which has extra pages at the end.
2:33 n circles, 2*n-1 rows
thank you very much!! it shows that though it still dont have one material applications, the math patterns behind commons geometrics objects are one great source of inspiration for those know how to contemplate.
Love your work and that is a nice model that you created. I imagine what it would be like when we try to get these circles in a circular shaped box. Nice explanation on the limit to what extent the circle from above can drop to.
Because every square is a rumbus You gave out the answer 2 minutes earlier if anyone actually payed atention
Notification squad!
Sorry. The entire squad was killed over the weekend. You'll have to play somewhere else I guess. I heard Facebook was looking for stupid little bastards to holler "notification squad" every time a new cat video shows up.
So great to see you and your Dad applying your minds to an intellectual challenge. You are both lucky to have each other
love your videos. always so inspired, thanks for keeping me busy at home!
n*2-1?
Yup, since a circle-pyramid with a base of size n also has n rows (row with n circles, n-1 circles, ..., 1 circle) we can rotate it 180 degrees and get 2n rows except for the center circle which overlaps with its own rotation, giving 2n-1.
It does not work for his last example.
Like for more mathologer videos 👍
Well, I've got a lot of nice ones planned for 2019. Cannot wait to work on them :)
The height of the pyramid is N where N is the number of balls in the bottom. If we invert the pyramid it's height is also N, so if we have them share the top circle, their total combined height is N+(N-1). So for any N circle width at the bottom, the layer that is N+(N-1) above will be level. Going further, if you continue stacking circles, it will become crooked again until you reach N+(N-1)-1, since the whole system will share the same base. This can be simplified to say that for any given level base with N circles, you are guaranteed to have another level base 2*(N-1) levels after the base. (Not including the current base) This will repeat forever.
Yep, that's fine. Another nice way of looking at all this is to ponder this kzhead.info/sun/ltGdf9GHepilbGg/bejne.html and to realise that the stack is really a large "square" consisting of n long rows and, nested inside a smaller square consisting of n-1 short rows :)
Fantastic video! Thanks you so much. It's great when someone like me who loves math but doesn't have all that much education in it can completely understand some phenomenon like this.
You remind me of my calculus teacher at BU....I learned so much from him. He explained math in a way that was really easy to comprehend.....thanks
I don't really know why, but this is blowing my mind right now. Fascinating
That's what I thought the first time I stumbled across this. Maybe also check out some of the interactive animations that I link to in the description :)
I remember long long time ago when I was in primary school, they still thought geometry as part of math. There was one particular problem, similar to what you've jsut talked about, to solve and prove - how to arrange balls inside box to fit maksimum number of them inside. There are 'practical' applications used already - in supermarkets they arrange layers of bootles on d-pallets in that particular way that we were to find at achool
This video explains mathematically the solution of Mr. Puzzle's challenge of fitting some coins inside a rectangular frame. Amazing!
An application could be earthquake prevention platforms under buildings. Although I think it would need spheres, it could use cylinders that are the length of the building, because the whole function of this is to maintain level bases while things are shifting. It would be interesting to see how a building with this kind of base upholds under a shake test.
Thank you for this really interesting topic. Very well explained
I had always wondered about- rogue waves & the seventh wave - while walking on the seaside. This vid goes a long way to explaining - quasi crystal lattice interactions; as well. 3d - 7nth wave action - must also include the sub surface topograghy chaos, & current direction energy..
Makes me feel really engaged. Good Food for the mind. With many thanks for your personal effort.
I really like maths concepts but studying them is usually kinda boring, you make it really easy to follow and go about it in such a creative way! Great video
Great, just great. I can't wait for a follow-up.
:)
Hello I was recently in the JMSS talk and was the person who took a photo with you at the end. Thank you so much for coming and I hope that I will see you again!
Nice video, as usual. Great visuals too, I see you put a lot of work in it :-)!
My first thought was to simulate this on a computer, but now I think it would be more interesting to simulate the mechanism shown at the end! What a dad, no wonder you are like you are!
So much symmetry from something that appears so assymetric!
Cool stuff! Watching that grid move was so satisfying :D
Yes, that metal grid apparatus really worked out incredibly well :)
The math was awesome, the description is great. But the physical model built with your father is simply magical!
I've felt like mirroring/copying/rotating the pyramid structure three times to form a rectangle makes me understand what is going on much better. This leverages the fact that a rhombus has equal opposite angles.
Nice video! The stacked circles made me think of the positions of atoms in crystals. A possible application of the theorem might be related to creating well-defined small structures that have nice straight edges even in the presence of crystallographic defects... Anyway, cool theorem and beautiful proof!
Enjoyed my first Mathologer video. After five weeks in Germany you've picked up quite an accent! 😁
I love your explanations, thank you.
when you said suggestions. I do not know how this applies in SHIPPING. They had t oFIGURE OUT how best to transport spheres, balls a MAXIMUM number to transport in a LIMITED volume in a ship. All of your videos are EXCELLENT. thank you
If you are a non-math individual like me, I still encourage watching these wonderful videos. I get lost on the details but find the presentation fascinating... and who knows-maybe I'm training my brain to do a bit more!
Beautifully done
When I get so frustrated I could scream, I play one of your proof. Thk goodness my brains are not big enough to focus on my frustration & your most excellent proof. Thks
Although every Mathologer video is outstanding, I really enjoyed this one extra. Thank you for showing the physical model you built with your father as well.
You explain things really well.
Beautifully demonstrated, thanks.
Best math channel