The Insane World of Polygon Packings
Рет қаралды 52,205
Күн бұрынLet me show you some surprising things about how squares and other polygons fit into different spaces!
0:00 - Homemade Demonstrations of the Square Packing Problem
7:06 - Digital Images of The Craziest Square Packings
14:24 - Packing Other Types of Polygons
Here are the websites that I used images from:
erich-friedman.github.io/pack...
kingbird.myphotos.cc/packing/...
kingbird.myphotos.cc/packing/...
In the images I included, there are some notes attached (from the original websites). To clarify some terms: they use "s" to describe how many times longer the big square's side-length is compared to the small squares (or equilateral triangles), and use "r" to describe the radius of the larger circle in terms of smaller shapes. Most of the images also clarify whether they have been proven, or if they've just been found (which means they are the current best-known configuration but another weird arrangement could possibly beat them later).
Special thanks to my current supporters on Patreon!
Max, George Carozzi, Chandler Smith, Eric Brodeur, Henry Spencer, Jon Mandarin, Mitch Harding, Tybie Fitzhugh, Joshua S, Julius 420, Peter Offutt, Quinn Moyer, Beugul, Stan Seibert, Dave Brondsema, Florian, Jan Bosenberg, Mathias Ermatinger, terry, William Hawkes, Christopher Masto, Claudio Fanelli, Craig Butz, Harry Cruse, Joost Boesburg, Kali OmegaRogue, Michael Friemann, The Green Way, Mack, cat, Stephen Davies, and Mackenzie Newman!
If you want to get your name on that list and help support this channel (which helps me keep making videos without any brand advertisements) and get some bonus content, check out the Patreon here: / comboclass
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If you want to try to help with Combo Class in some way, or collaborate in some form, reach out at combouniversity(at)gmail(dot)com
In case anybody searches any of these terms to learn about them, this episode is about some crazy geometry that some may consider cursed (and/or beautiful), related to the most efficient ways to pack certain polygons into shapes, such as square packings, triangle packings, circle packings, cube packings, combinations of those, and more!
This episode was directed/edited/soundtracked by me (Domotro) and was filmed by Carlo Trappenberg.
Disclaimer: Do NOT copy any dangerous-seeming actions you may see in this video, such as any actions related to fire.
@ComboClass 15 күн бұрын
This episode is about some surprising ways that polygons fit together. I think it's a very fun and interesting underrated topic, so I hope you enjoy! See below for timestamps of the video’s three parts, and see the video description for more info. 0:00 - Homemade Demonstrations of the Square Packing Problem 7:06 - Digital Images of The Craziest Square Packings 14:24 - Packing Other Types of Polygons
@Bibibosh 15 күн бұрын
who is walter trump?
@adamredwine774 15 күн бұрын
Maybe underrated by the general public but it’s super important in the study of crystals and physical chemistry.
@davidbrinnen 14 күн бұрын
After years of UV mapping, I have had to learn to intuit the best selection of polygons and connected polygons to pack into the 1x1 UV space, with the aid of packing software. My natural tendency for the packing to be geometrically "tidy" is often offended by some of the more space efficient solutions the software offers after iterative packing.
@dyllanusher1379 12 күн бұрын
I am hoping that in ur next video you can repeat "combo class" 17 times at once so that I can remember your God damn channel name because this is gold and ive tried recommending it to friends several times and have forgotten the name.
@craiggersify 15 күн бұрын
Building a time machine to torture Pythagoras with pictures of cursed minimal square packings
@lukahutinski9075 15 күн бұрын
worse than beans!
@Rando2101 15 күн бұрын
pictures of irrational lengths of lines
@williamcompitello2302 14 күн бұрын
"That doesn't count!"
@element1192 14 күн бұрын
All you'd need is the theory of transcendental numbers to drive him crazy lol
@palmberry5576 14 күн бұрын
@@Rando2101lmao, my first though when I saw op’s comment was, “huh, idk how py guy would feel about irrational numbers”
@Ganerrr 14 күн бұрын
non-ideal packing of 17 squares: cuddling someone ideal packing of 17 squares: wearing someone's skin
@solveforx314 15 күн бұрын
optimal packing of 17 squares my beloved
@fractai. 15 күн бұрын
This channel gives me that funny learning feeling whenever I watch a video and i don't know what it is. Definitely one of the most unique places on KZfaq.
@JBiton 14 күн бұрын
The funny feeling of watching an educational video from 10 years ago
@joshuasims5421 15 күн бұрын
This topic perfectly captures the chaotic energy of this channel. Also, I nominate 10 & 27 triangles in a triangle as the best looking of all of these. Almost makes up for the horror of 50 squares in a square.
@NickiRusin 15 күн бұрын
when Jerma peeps the horror that's what it looks like
@eyesicecold 15 күн бұрын
I haven’t watched to that part yet, but I think 29, 17 and 11 are worse Edit: *oh my god!* What’s wrong with 50? Did you mean 51 or 55?
@joshuasims5421 15 күн бұрын
51 and 55 are messier objectively, but 50 feels like a betrayal of some sort. It’s double one square and half of another, and an important number in our base 10 world. It just feels like 50 should have had a strange but elegant packing, like 5 does. (That’s what I meant, anyhow.)
@eyesicecold 15 күн бұрын
@@joshuasims5421 ok
@CFGalt 6 күн бұрын
Timestamp in case anyone’s curious -> 13:35
@lucyinchat 15 күн бұрын
Every time I watch these videos I can’t help but think this is the chaotic energy that math just naturally has and ComboClass is just capturing it.
@elitettelbach4247 15 күн бұрын
I'd recently seen the optimal way to fit 17 squares inside a larger square and was like "huh, I'd never really thought about it before but that looks strange and interesting." Perfect timing for this video! I love your delivery of everything and the practical examples at the beginning were very helpful.
@TorgieMadison 15 күн бұрын
5:05 The value pi is for circle-y things, and root-2 is the value for square-y things. That's such a great way to put it!
@bensmith3890 13 күн бұрын
Pi is also a clue in the formulas for things. If you see pi, you know that whatever you're doing must be related to circles somehow.
@RepChris 12 күн бұрын
@@bensmith3890 The only question is how far hidden the circle is. Sometimes it is hidden within hidden³
@versacebroccoli7238 15 күн бұрын
I love this channel. The absolute chaos of your intros is fire.
@chiaracoetzee 15 күн бұрын
I mean, literally.
@jkid1134 12 күн бұрын
The absolute chaos of fire is your intro, really.
@LiamLimeLarm 15 күн бұрын
ive never seen this math channel before and opening it to see your backyard was on fire was certaintly something i wasnt expecting
@ConManAU 15 күн бұрын
Stick around for long enough and you’ll get so used to it you’ll barely notice.
@rooster5572 15 күн бұрын
So beautiful, as an engineer in the industrial field I can say this subjet definitely has real world applications and I've worked it in very non mathematical ways unfortunately
@geekjokes8458 15 күн бұрын
"get in!!! get IIIIIINNN!! jusT FUCKING GET IN! FUCKING SQUARE SON A FUUUUUCK!"
@estherstreet4582 15 күн бұрын
Some parts of maths are inherently beautiful and elegant. Polygon packings shows that maths can also be the opposite.
@tonydai782 13 күн бұрын
Math need not fit the very human perception of beauty. We find patterns beautiful because evolutionarily, pattern recognition helped us survive. Polygon packing just happens to fall outside the patterns that we are able to comprehend, and thus it appears “cursed”.
@ExzaktVid 15 күн бұрын
Erich Friedman practically made this video on his own.
@rujon288 15 күн бұрын
My goat 🐐
@aknopf8173 15 күн бұрын
He was only beaten by Mr. Trivial. He always found the neatest packings!
@ComboClass 13 күн бұрын
Erich Friedman is awesome and compiled the original images, but he didn't discover them originally, and he also didn't create the higher-def images of the squares (all those different types of people are credited in this video). And compiling good data is a different thing than making an explanatory video about that data.
@ExzaktVid 13 күн бұрын
@@ComboClass so did he ‘invent’ the packings or just compile them into a list? And I wasn’t saying that finding packing is like creating a video, I was just making a joke because so many of the ones mentioned here have him credited for it.
@smoceany9478 15 күн бұрын
ive actually recently gotten interested in the maths of optimal shape packing, glad you made this
@smoceany9478 15 күн бұрын
packomania has circles in squares to 1000
@jamesyoungquist6923 15 күн бұрын
Why did you actually recently get interested in shape packing?
@smoceany9478 15 күн бұрын
@@jamesyoungquist6923 cause its a really interesting subset of math that we know shockingly little about
@bestintheband5114 15 күн бұрын
Babe wake up new combo class video dropped!!!
@bestintheband5114 15 күн бұрын
Babe wake up again he loved my comment!!!
@chiaracoetzee 15 күн бұрын
I think one way I try to get insight into why these packings are so weird is to imagine that you put a bunch of dice in a square dish and just shook it until they all landed flat in the same layer. Most of the time you would not get a regular arrangement of any sort, but rather they would get stuck in sort of random places. If the dish is large enough, you could move them around and rearrange them into a more regular arrangement, but if it's small, they're just going to be stuck like that. I realize I am talking about a local minimum here rather than a global minimum, but it gives some idea at least.
@Ashebrethafe 15 күн бұрын
That suggests another question: How large does the dish need to be to guarantee that, after shaking it, you can rearrange the dice to make room for one more? I also noticed that the packings shown for 5, 11, and 40 squares were labeled as “rigid”, meaning that they have no wiggle room at all; and I’m guessing that the lock emoji indicates packings that have been proven to be optimal - I saw that other packings said “proved” instead of “found”, but maybe the ones with the lock were found by one person and proved by another.
@masapallo 14 күн бұрын
@@Ashebrethafe The lock emoji was always inside the side length: S = lock = approximation. I read it as the exact size being unknown (since in the neater solutions there'd be a nice little expression with some square root in it at the same place).
@bensmith3890 13 күн бұрын
@@AshebrethafeIntuitively, I'll say S+10% for "sorting room" since that's what disk defragmentation programs usually needed.
@adamredwine774 15 күн бұрын
For anyone interested in playing with this idea for kids, a good way is to use four of those big fat zip ties to make an adjustable square and then pack die or square legos or something in them.
@RAINBOWEXPLOSIO 15 күн бұрын
These videos are the best, I'm commenting for the algorithm.
@jamesyoungquist6923 15 күн бұрын
Comments are the carrots rewarding creative content. Keep 'em coming
@DurianFruit 14 күн бұрын
good idea. also gonna comment for the algorithm
@SongSeeker7 15 күн бұрын
This gives me some ideas about writing a computer program to shake a number a smaller squares inside a bigger square as it tries to shrink the container.
@satibel 15 күн бұрын
the issue is local minimas, you can't be sure it's the smallest one.
@xusv-hi4kl 14 күн бұрын
@@satibelI imagine you could just run the program for many iterations until you settle upon a very small container size. Of course, you wouldn’t be sure that said size is the smallest, but it may still tell you something.
@justsomeguy5628 15 күн бұрын
I have been waiting so long for someone to make a video about this!
@Bingcenzo 15 күн бұрын
4:41 - 5:18 continuous footage, steady hands. Bravo.
@ProSocialEntertainment 15 күн бұрын
Finally, a real fucking uv packing tutorial.
@miroslavzderic3192 14 күн бұрын
lol this is how Minecraft textures are packed
@davidbrinnen 14 күн бұрын
Aye, the hours and hours I've spent with UV layout packing rotating, repacking - going back to slice up some quads to see if it can get up over 80% space utilisation. I'll settle for 70% if it seems like it is taking too long to solve, but real happiness is 96% plus pack.
@publiconions6313 15 күн бұрын
Excellent video sir!! Its always a treat when one of these pops up. Thank you Domotro and Carlo!
@iamdigory 15 күн бұрын
Oh I got an idea, we could try to fit fractional squares inside a bigger square, it could be interesting to watch it go from one solution to another as the size of one square grows
@crypt8919 14 күн бұрын
Domotro's videos are the most fun videos out there: nature, numbers, animals, fruits, and fires. They are consistently great. I share them with my tutees.🍏🐦🔥
@RollcageTV 15 күн бұрын
This video oddly reminds me of the "moving sofa problem". I think that might make a good video topic. It doesn't look like it's been covered on this channel. What do you think?
@dominichill8037 14 күн бұрын
This is the most enthusiastic explanation of geometry I’ve ever had.
@brandontrevor2306 15 күн бұрын
interesting topics and i love the informality of it. good channel
@AntimatterBeam8954 6 күн бұрын
I have such a habit of trying to see patterns in everything, this is definitely something to explore
@Kwauhn. 15 күн бұрын
I ❤ Domotro. One of my favorite creators, and probably my top math youtuber right now. You always go above and beyond what other math communicators do. You're in the same league as 3B1B IMO.
@Manabender 14 күн бұрын
4:41 I was surprised to see you didn't derive this value (the 5-square packing that fits in 2+root2/2). I paused and derived it myself just before this. Logic followed: The diagonal of the big square includes, colinearly and one immediately after another, in this order: A diagonal of a small square, a line parallel to the side of a small square, and a diagonal of another small square. The diagonals are root2. The parallel is 1. Thus, the large square's diagonal is 2*root2+1. The large square's side is therefore (2*root2+1)/(root2). This simplifies to 2+root2/2
@huleboermannhule44 15 күн бұрын
I recently came over this topic and it was nic to see a video about it. I must however say that the presentation style is very different from other maths channels, and you burning things in your backyard was not expected.
@AlexBaklanov 15 күн бұрын
I imagine how we can fit infinite amount of squares into a cube, right? So my question is. Can we also pack an infinite amount of cubes into a hypercube?
@AlphaFX-kv4ud 15 күн бұрын
Yes
@ND62511 15 күн бұрын
I find it so interesting that a lot of the non-square number packings have a lot of numbers with very similar patterns. Makes me wonder what the process used to generate them are
@fire17102 14 күн бұрын
Would love it if you tackle 3d packing next ❤ this class is 🔥
@aze4308 15 күн бұрын
The Preposterous Planet of Perfect Polygonal Packings
@tierfreund780 15 күн бұрын
as a 3d artist these are the questions that keep me up at night
@vilvd3934 15 күн бұрын
Reminds me of uv unwraping of 3d surfaces. This video is so lit man
@TheStormingmonkey 15 күн бұрын
this is the most bonkers channel i've ever seen. I love it
@Rockyzach88 15 күн бұрын
For some reason I always forget the name of your youtube channel when I want to share one of your videos.
@samueldeandrade8535 15 күн бұрын
Crazy Chaos, Combo Class, Coming Up, Open your mind. Hahahahahaha.
@pennywerner9192 15 күн бұрын
You can always search Domotro. There’s only one.
@stickmandaninacan 15 күн бұрын
I'd be really interested to learn the process of how people come up with these, and what mathematical methods you can use to try and maximize the best result possible
@ExzaktVid 15 күн бұрын
Ive noticed that there are a lot of patterns in the sqaure packings, slightly over a square number has one square in each opposite corner, a diagonal string of squares between that, then just staicases in the other two corners.
@personperson278 15 күн бұрын
An interesting note on this topic - we know the general optimal packing for spheres only in dimensions 1, 2, 3, 4, 8, and 24.
@dinofx35 13 күн бұрын
Spouse: What did you do at work today? You: I discovered a new way to pack 272 squares into a larger square
@Very.Crazy.Math.Pistols 10 күн бұрын
Good evening, very interesting ! Now I wouldn't be surprised if upon entering a mathematician's house I found his living room tiled in one of those ways 😅.
@NickiRusin 15 күн бұрын
Triangle in circle good, square in square bad, got it
@fettisaurier8030 15 күн бұрын
username checks out
@fettisaurier8030 15 күн бұрын
picture*
@X3MgamePlays 14 күн бұрын
With higher dimensional fitting. Things get really weird with a certain ammount of dimensions for spheres and boxes. This is an awesome video. I am missing the pentagons though. Wasn't something weirder going on with them? Or was it another figure?
@geekjokes8458 15 күн бұрын
im surprised numberphile doesn't have a video on this yet
@soninhodev7851 15 күн бұрын
Awesome video like always, i have never considered this problem, its fascinating! i always liked the problem about how to tile a plane with more than one shape, like with octagons and squares. if memory serves, you already made a video about a similar topic, didnt you?
@ComboClass 15 күн бұрын
I made a video on this channel before about a new discovery in tessellating irregular shapes (the "aperiodic monotile") which is different but related :)
@josephbrisendine2422 15 күн бұрын
Love it, have a free engagement boost!
@TiagoTiagoT 14 күн бұрын
For any squared multiples of already found ideal packings of squares, would tiling the smaller patterns be guaranteed to produce the ideal packing for those numbers? Or the extra amount sometimes/always adds enough freedom that a tighter packing can be found? And similarly, can infinite ideal packings be generated from already found ideal packings by fractally replicating the ideal packings inside each square piece, or does it have that issue of additional freedom from greater number of squares? edit: Ok, at least the fractal idea seems busted, at least as a general rule; tried 25, and the holes allowed for a tighter packing than just doing the fractal replacement; and while I haven't ran the numbers, eyeballing it, it looks like it doesn't fit tighter than just plain 5x5 stacking even with the squeeze.
@mmfpv4411 15 күн бұрын
This is a great video. You find great topics to discuss. Found myself curious to hear more about how mathematicians come up with these patterns. Like practically speaking, what are the mathematical techniques used for optimizing packing configurations?
@henrysaid9470 15 күн бұрын
I saw a video about this before but I dont remember the KZfaqr but this went more into depth Edit: nvm it is square packing by Andy Math
@trentcard 14 күн бұрын
YES YES YES YES YES IVE BEEN WAITING FOR SOMEONE TO MAKE A VIDEO ON THIS TOPIC
@jacobbaer785 13 күн бұрын
Because i want to, the square pattern in notation: For any number of squares "n" where s² > n > s²-s, the optimal packing square will have side length equal to s. Only numbers that fall outside these bounds can be optimized.
@GhostZeroGZ 15 күн бұрын
This is amazing
@thethug2169 14 күн бұрын
perfect chaotic topic for a chaotic channel
@iispacedustii 14 күн бұрын
yooo this is fire! literally!!!
@Untoldanimations 14 күн бұрын
im publishing my first paper and it's about soft sphere packing :) i attribute my interest to results like these ones
@manuelsuarez7521 15 күн бұрын
amazing!
@element1192 14 күн бұрын
"sacred geometry" fans when Fritz Göbel and Erich Friedman show up with the cursed geometry
@Iiochilios1756 14 күн бұрын
Hey, did you ever applied to r/tree admin position?
@RafaelAcurcio 15 күн бұрын
What a happy accident was finding this KZfaq channel!!
@axelkeithgranath 14 күн бұрын
9:00 Ptsd-ed into old days trying to find optimal ways to fit squared in circle and vice versa! Me dead need sleep now. thx
@gamingwithspeedy858 15 күн бұрын
Very cool
@XDarkxSteel 15 күн бұрын
Been having dreams about that one everyone's been talking about lately after getting a package handler job lol
@ShadowKestrel 15 күн бұрын
now i want to use my computer to find optimal regular pentagon packings
@eliasross4576 14 күн бұрын
I was looking up ways to pack cylinders inside a larger container. There’s a lot of web sites for calculating this. Circle packing it is called. There’s circle packing into squares as well as squares into a circle.
@maynardtrendle820 15 күн бұрын
I think that the word 'ridonculous' might be the appropriate mathematical term here. Also...'schlopp-tastic'.🗿
@hughwilliamson2190 13 күн бұрын
The 272 square example is interesting. Seems to be the first time that n(n-1) squares require a side-length less than ns.
@kurtu5 13 күн бұрын
I am reminded of Eric Weinstein's recent steel manning of Terrence Howard's attempts to tesselate R3, and that there are gaps in the tilings that need a 'Pythagorean comma' in it. I always knew of imperfect tessellations, and only recently learned of this comma notion that arose in music.
@dinnertonightdinner7923 15 күн бұрын
very cursed, but that's what we're all here for
@owensthethird 13 күн бұрын
I'd be interested to know the "energy" required to transition from one solution to another, if you introduce some thermal jiggling
@DigitalJedi 15 күн бұрын
I would love to see the cube packing be expanded to tetrahedrons as well, being the 3D version of the triangle and all.
@SeanCMonahan 12 күн бұрын
Do the optimal packings always have a square tucked perfectly in each corner?
@moth.monster 15 күн бұрын
I designed a 3D printed puzzle version of the 17-square packing. It is deeply unsatisfying to solve, and I love it for that.
@ronaldc8634 15 күн бұрын
Thank you
@lumipakkanen3510 15 күн бұрын
You may not like it, but this is what peak packing looks like.
@gideonroberts885 12 күн бұрын
The fire made me chortle
@MandrakeFernflower 12 күн бұрын
This is explosionsandfire's DMT addicted brother
@king_james_official 2 күн бұрын
lmao
@VocalMabiMaple 15 күн бұрын
I love 39 squares minimally packed. Something about it just calls out to me At 13:20. It is just so randomly thrown together that it ends up whimsically efficient.
@samueldeandrade8535 15 күн бұрын
Domotro is carrying Math KZfaq on his shoulders! Hahahahaha. Great video, Prince of Chaos. Great video.
@MattHudsonAtx 15 күн бұрын
babe, wake up, combo class just dropped a new video
@MattHudsonAtx 15 күн бұрын
the intro is fire
@markm1514 15 күн бұрын
Coming to an Amazon warehouse near you...
@memzdotexe4127 15 күн бұрын
insane video
@KingJAB_ 12 күн бұрын
I think this is the first time I’ve seen a computer visualization on combo class
@videoDemon 15 күн бұрын
_how did the square become a circle?_ _( … _*_it was caught cutting corners_*_ )_
@sajrra 15 күн бұрын
Could you find out what is the least amount of snooker balls (33/16 inches or ~5.24cm diameter) would fill the D-Shape on a Snooker table (Semicircle of 2 feet or 60.96cm) So how many balls does it need to make it impossible to fit another ball into the shape without moving a ball? A ball counts as inside or the D if the center is inside. Not the entire ball needs to be inside the shape. I was wondering this for soo long, but I don't know how to takle this question. It is like answering the opposite question, what is the biggest space a quantity of objects can obstruct. 1 Square can obstuct a shape of 2.999... its size if you place it into the very middle with parallel orientation there it not enough space to place another square into it.
@qbasic16 15 күн бұрын
Mind = packed 🔥
@marasmusine 15 күн бұрын
Regarding hypersphere packing, am I recalling correctly that you get this weird situation where the total volume of all the small spheres exceeds the volume of the bounding shape?
@VidNudistKid 15 күн бұрын
Certainly not. But I think in really high dimensions the volume of the small spheres as a fraction of the volume of the container gets ridiculously small. Kinda the opposite of what you said, but just as weird.
@pepebriguglio6125 14 күн бұрын
Between 2² and 3², only 5 squares fit inside a square less than 3²: 9-4=5. So n²-a works for n=3 and a=4. So my question is, is a always 4 for all values of n? Or can a be 3? Or does it sometimes need to be 5 or 6 for the squares to fit inside a square smaller than the next square number?
@LakeGameCreepr 15 күн бұрын
optimal packing of 9 squares inside of a square
@tamnker8465 13 күн бұрын
I wonder if you could outsource this to the public by making it a game and offering rewards to anyone who finds better packings. It wouldn’t be likely to work, but with enough trials you could maybe get some valuable data.
@mezzanoon 11 күн бұрын
What about packing circles in hexagons? Having a terrible time finding any packing figures of it
@caicai-kd8ch 14 күн бұрын
Good fucking video man
@TymexComputing 15 күн бұрын
Hi there - recently i am observing so many bigger problems being brought to tesselation theory - couldnt this video be made in two parts?
@ComboClass 15 күн бұрын
This episode was mostly about a specific polygon-packing problem, but the concept (as well as tesselations) will return in later episodes.
@TheRealFOSFOR 15 күн бұрын
Improving the audio on these videos would probably make them even better.
@tamnker8465 13 күн бұрын
16:27 Crystal structures!
@nickwilliams2745 13 күн бұрын
I’d be really interested in seeing how closely the densest packing of spheres (and spheres of multiple set sizes) in cubes connected to crystalline lattice cells On the surface it seems like they’re trying to accomplish the same thing
ALEX LAB
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