I think something worth mentioning is that a lot of celestial bodies are really just collections of many smaller objects. For example, galaxies are collections of millions of star systems. Each star probably has a few planets. I wonder if how we classify what an “object” is affects the trend, or even causes it
I'm glad you mentioned that! I actually cut a section where I talked about that. The problem is especially hard when we talk about nebulas (is a dust particle one "thing"?), but the researchers ignored that. They may have double or triple counted many objects.
This is why philosophy is important, in particular the discipline of asking what is the nature of a thing and getting good definitions from that questioning. Also epistemology to define how we know a thing. Physics is meant to be paired with metaphysics.
I don't think this is by any design at LEGO, it would also be very hard to make sets with a "forced" distribution. I think it's a naturally arising property from the way things build up. I mean, you can only have a few big pieces (base blocks, big custom monoblocks, etc), while you need a ton of tiny things to fill the gaps, connect blocks, add decoration, and so on. This is why it's so amazing, I think this arises organically.
Prime numbers play a role too I believe. The composite-ability of the world. There’s even been a couple videos recently on tiles and the history of these geometries in sacred religious temples. Iirc isn’t this curve similar to Pareto distribution?
I think it also just has to do with the nature of the toy. Too many tiny pieces would be difficult to build. Too many big pieces would be boring. It's not necessarily something the designers would think about, but it's something somebody would think about before retail
This is kind of reminiscent of Zipf's Law. Something to keep in mind is that a relatively small variation from -2, to, say -2.5, is a bit bigger than we might think as it is on a log scale.
It also reminded me of a study of the distribution of scene lengths in movies, or maybe shot lengths, to be more precise. Lots of little shots, like glancing over to a character's face to see their reaction, quite a few shots longer than that, and some long shots, maybe a slow pan over beautiful scenery. No, I'm afraid I don't remember the exponent; I think it wasn't highlighted in the article so much as the exponential decay shape.
I'm a video game designer and when I make levels, adding clutter to the environments follows the same curve of object sizes for the map to feel "Right" It might be some sort of fractal pattern that we all have a deep intuitive understanding of, that's why the legos follow the same rule. Maybe even our bodies follow the same rule: bone sizes vs quantity; it could potentially apply to anything.
I don't think this curve shows anything about the nature of the universe but about how we see/label things. There are no hard rules in the universe about when something counts as a new separate thing, except for maybe the tiniest particles that make up the fabric of reality. At certain points our brain decides that a collection of smaller things or different characteristics is worth distinguishing as a new separate thing, giving it a name and so on. This experiment shows that there is an interesting curve in how we do it. It makes total sense that the designers of lego bricks would feel the need to create bricks at sizes where our brains like to distinguish things as separate or you would feel the need to place objects on the map like that. Pretty cool.
Humans have a preference for environments that are between 1.3-1.5 fractal dimensionality. So I think you may likely be right about that. Maybe Gaussian clutter?
i had something similar in mind. what if it's some sort of natural distribution that we're instinctually accustomed to and that often appears in nature?
Astrophysicist here, I'm getting my Ph.D. studying galaxy formation/evolution. I wanna start out and say, I really enjoyed the video! It's really interesting to think about how mass is distributed in the Universe and if the same laws of nature that apply to galaxy clusters apply to LEGOs :). First is in regards to the initial mass function (IMF, as we typically call the mass function for stars) you describe here. Indeed the Salpeter (1955) IMF is a power law with an index of -2.35. However, more recently increasingly complex IMFs are used to describe the distribution of stellar masses of stars (see, e.g., Kroupa 2001 or Chabrier 2003) which are more than just a power law. These distributions are more like log-normal distributions, they have a plateau and turn over at lower masses. Even further than "Is it a power law?" it's actually a super open question in Astrophysics right now whether the IMF has evolved throughout the lifetime of the Universe. Second (sort of related), I think your intuition about gravity and the r^-2 power is really awesome, but the dense, star-forming interstellar medium is a little more complex than that. You have to take into account turbulence (collisions within the gasses), stellar feedback (other high-mass stars exploding in the neighborhood), etc. So I am skeptical as to whether or not that's where the Salpeter power law index comes from. It's an interesting connection nonetheless! Finally, at 10:07, I take exception to the statement "...galaxies are formed by stars clustering together into bigger pieces." Galaxy formation is a direct effect of dark matter overdensities, which gravitationally attract baryonic (non-dark) matter to their center, not necessarily groups of stars in space that all coalesce. The stars have to form from the gas reservoir, which needs to be sufficiently dense. In the absence of large dark matter potential wells the diffuse intergalactic gas simply doesn't get dense enough (or if, by some wild chance it would, not enough of it would in order to form galaxies of the mass we see today!). Again, really enjoyed the video! I hope you find the comment helpful 😀 EDIT: formatting (thanks @Speed)
*Formatting Astrophysicist here, I'm getting my Ph.D. studying galaxy formation/evolution. I wanna start out and say, I really enjoyed the video! It's really interesting to think about how mass is distributed in the Universe and if the same laws of nature that apply to galaxy clusters apply to LEGOs :). First is in regards to the initial mass function (IMF, as we typically call the mass function for stars) you describe here. Indeed the Salpeter (1955) IMF is a power law with an index of -2.35. However, more recently increasingly complex IMFs are used to describe the distribution of stellar masses of stars (see, e.g., Kroupa 2001 or Chabrier 2003) which are more than just a power law. These distributions are more like log-normal distributions, they have a plateau and turn over at lower masses. Even further than "Is it a power law?" it's actually a super open question in Astrophysics right now whether the IMF has evolved throughout the lifetime of the Universe. Second (sort of related), I think your intuition about gravity and the r^-2 power is really awesome, but the dense, star-forming interstellar medium is a little more complex than that. You have to take into account turbulence (collisions within the gasses), stellar feedback (other high-mass stars exploding in the neighborhood), etc. So I am skeptical as to whether or not that's where the Salpeter power law index comes from. It's an interesting connection nonetheless! Finally, at 10:07, I take exception to the statement "...galaxies are formed by stars clustering together into bigger pieces." Galaxy formation is a direct effect of dark matter overdensities, which gravitationally attract baryonic (non-dark) matter to their center, not necessarily groups of stars in space that all coalesce. The stars have to form from the gas reservoir, which needs to be sufficiently dense. In the absence of large dark matter potential wells the diffuse intergalactic gas simply doesn't get dense enough (or if, by some wild chance it would, not enough of it would in order to form galaxies of the mass we see today!). Again, really enjoyed the video! I hope you find the comment helpful 😀
I knew someone had higher knowledge (no hate towards the bird, he’s great and being outside of physics n astrology they’re awesome to keep my interest peaked) but the second Mr bird asked for Lego workers I had a feeling if not them YOU’d show n ya did :) thanks for the insight but, and excuse my ignorance forgive my incorrectness, has dark matter been found? I mean i know have strong proof it’s there, with galaxies being heavier than expected n what not but we have yet to capture/replicate/find proof aside from implication of its existence? Again I’m just an onlooker, but dark matter really fascinates me as a concept and I’d like to know more about it if possible Really interesting to read your take as a soon-to-be Ph.D !
OOOO OOO ALSO again not too well versed in any of these fields a t a l l but wouldn’t we need to include dark matter in the IMF to make the calculations more accurate??
@@trashmanjacobs7180 Great questions! First, yes we have fairly direct evidence for dark matter (on astrophysical scales). A quick example of this is orbits within galaxies: the (visible) matter within galaxies orbits faster than can be held together by just the (visible) mass within it, therefore there must be some missing (invisible/dark) matter that we cannot see. Whether or not we just missed normal matter in observations, it is some exotic particle, or we need to re-write physics (see "Milgromian physics" if interested, there's a really great video by "PBS Spacetime" explaining it) is still a manner of debate. With that being said, I would say there is a majority who believe it to be exotic particles that don't interact with light. Second, by construction the IMF sets out to measure the distribution of masses of stars. So no we don't explicitly need to account for dark matter within it since stars are made of normal matter. Galaxy-wide scaling relations, however, can (and do) include dark matter.
The -2 coming from gravity makes good sense, but if a similar pattern exists in a wide range of different settings wherein things are built from combinations of other things (e.g. LEGO sets, IKEA furniture, laserjet printers, motor vehicles, etc.), that would suggest that it's a statistical property instead. I'd be interested in seeing a data where rather than "mass" being the x-axis, it's "mass of component divided by mass of the final product" wherein components of a wide range of different things are included, rather than just being a single category of objects, like LEGO. I think that distribution might provide further insight into this.
i think what’s more likely is that the -2 comes from the dimensionality of our universe. the -2 in the gravitational equation comes from the same place: the inverse square law. which means it might be worth exploring 2d or 4d video games
I love that this concept of smaller parts summing to equal bigger parts is kind of intuitive while being simultaneously mysterious. The universe is a strange and magical place.
Great video! I’m an acquaintance of the author of the paper. I have a school-aged son and a BrickLink store, so he sent me the paper to read, and then later sent me this video. You did a great job. He’s really impressed, and my son has become a big fan of yours (I already was). If you need any help with Lego, give me a hollar. Or if you’d like to get in touch with Stefan, I know he’d be happy to speak with you. Keep up the great work!
That makes me very happy to hear - both that the author likes it and that you and your son enjoy the videos! I thought this was a fun topic so I made it with all ages in mind.
I like your theory about why -2 for astronomical objects. Edit: Though actually, I think I just thought of a handwavy variation that doesn't depend on the gravitational constant. Let's say we start with a bunch of really small things (rocks, particles, etc.). Some might merge together into particles that are twice as big, and some might not. then, out of those, some might merge again to form particles of the next order of magnitude, etc. If, at any level, the probability of merging vs. not merging is about even (for any reason at all, which may be different for different types of "things"), then I think the exponent will be roughly around -2. For LEGO bricks, I have a different (half-baked) model that might be worth exploring. We can view a LEGO model as trying to approximate a specific 3D shape using a minimal(ish) number of bricks. With some additional constraints, e.g. there's a limit to how large (or at least how massive) the bricks get, and also limits on individual dimensions (e.g. normal bricks are only so thick). So, they fill in the rough shape with big bricks, then they start using progressively smaller bricks for the details, then even smaller for the tiny details, etc. At some point it bottoms out because they either give up on the level of fidelity, or make bricks that are the exact right shape they need (e.g. human head, flower, window, ...). I feel like these constraints will at the very least naturally produce a power distribution. It would also be interesting (and relevant) to find out whether brick volume is proportional to mass (or maybe larger bricks are less dense, for example)
I'm a Lego fan and custom design creator. Small pieces are used to create detail, as well as specific functionality. Since Lego designs typically reflect our world in some manner, it makes sense that lots of small pieces are required to match the fractal and chaotic nature of our world. An interesting research topic would be to do something similar to actual fractals. Or turbulent fluid flow.
I love your videos, the way you mix different subjects that at first might seem completely unrelated is fascinating to say the least Warmest regards and best of wishes🌹🌹🌹
I really want to point out that the arithmetic mean of all the twelve lego slopes you've show in the video is equal to (-2.13-1.95-2.16-1.98-2.04-2.05-1.40-1.76-2.091.98-2.39-2.2)/12 = -2.01083333... Pretty close, right? love the way you investigated the idea in the video, wish you all the best from Brazil 💚💛💙💛💚
the arithmetic mean of the slopes translates to a geometric mean of the original distributions, would we rather take an arithmetic mean of the distributions? then I don't know what the mean of the slopes should be (the critical problem is how we should think about the 'error')
I have two (very high-level and possibly very wrong)thoughts on this: 1. It could very likely be related to packing small and large pieces in bags(or even some sort of packing algorithm), as the design might be optimised for that. Or some obscure packing problem, for that matter. 2. Smaller pieces are spread out on the surface area of larger parts. So a graph like this is expected. Some more thought on the circley things might give more insights on the exact power of 2 point something. Random extensions to these: For 1. The density of smaller objects are larger because they have more bulky and dense edges compared to light flat surfaces. (I checked this for squares and circles) lego lengths have a higher variance wrt bag sizes, so it won't scale as much. Using this, I tried to see what would happen if we naively take n(legos) \propto vol(bag)/vol(lego), but was off by 0.5 or something. Without looking at the data as it's already 4 am lmao For 2. For VLSI, a design following a somewhat-similar principle, I'd say mass and surface area both scale according to the square of the side, as height does not vary too much. So it should not be very far from 1/mass. Legos often have tall pieces. idk what I'm talking about, though. Also, great video: Can't stop thinking about this :p
You want me to count every object in my house?! Uh, yeah, I'll get right on that. Respect to NSU to use a Ninjago set for the experiment. Because it shows they jumped up, kicked back, whipped around, and spun, and then they jumped back and did it again. If everyone did the Weekend Whip, the world would clearly be a better place. Nova Southeastern was originally a National Association of Intercollegiate Athletics (NAIA) institution back in the 1982-83 athletic season, which they would compete in their first conference affiliation home in the Florida Sun Conference from 1990 to 2002. The Sharks were originally called the Knights, which was from 1982 until 2004. In 2005, they unveiled the new Sharks logo and athletic mascot. The nickname was selected by the students.
3:04 need to say, this is actually called the initial mass function (IMF) . Slatpeter is just one of them, another really used one is Kroupa for example (also named after the scientist like the Saltpeter). Also for galaxies a nrw one tends to show up called the IGIMF which is integrated galactic initial mass function and comes from the behaviour of the initial mass functions of all the star clusters and stars inside the galaxy.
Playing leads to observation then leads to learning. I have never encountered boring triviality - only data my brain can't fathom. Thank you for your videos, it is so refreshing to have these higher level thoughts written so concisely! As Above So Below, from High to Low, Unified under the law that governs it all, whatever that may be.
This was a very interesting video! Also, as software engineer student, doing stuff like what you did in this video, just because you can, is what got me into this field.
This was such an enjoyably nerdy and poetic moment of science. You sound like such a fun friend to have. I hope you are able to make a living creating these videos.
I'm certain your channel will grow exponentially as you make these videos, they are very thorough and high level! Keep up the good work, I'll keep watching!
There is a likelihood that this has to do with a carefully selected balance of play-versatility for customers and manufacturing. In the 1990s Lego was going wild with lots of themed sets, but this resulted in manufacturing many kinds of larger, unique and niche-use pieces, like the rope bridge piece, boat hull elements, and the big ugly rock piece. Going into the 2000s, Lego was in financial trouble because these products didn't have the shelf life they wanted and they were expensive to manufacture. They switched gears and started making more sets that involved smaller more versatile pieces and more of them that were easier to manufacture, like cheese wedge slopes and common 1x2 plates. At the same time they started doing themes with other intellectual properties, like Star Wars, Harry Potter, and the NBA. These turned out to be the right decision. The major exception to this generalization is Bionicle.
My first guess for the distribution was a Zipf's Law curve, since we're talking about how common different groups of things are, but as you explained the actual answer, I realized that a Zipf curve doesn't actually give relationships in any ordered way, only between the 'most common' to the 'next most common' so it wouldn't really be applicable here. That said, a Zipf curve *is* pretty close to a power curve with a = -1, with the implication that that there would be half as many things which are twice as massive. If there are 4 times as many things that are half as massive, that means that the *total amount of mass* in each 'group' follows a power law of a = -1. There is twice much total mass in the universe made up of (~1kg) things as there total mass made up of (~2kg) things. I think part of this might be due to the way that big things are usually made of small things, but there are also small things that aren't also part of other big things, but that doesn't really have meaning when talking about discrete LEGO bricks, which are all separate, atomic objects for purposes of comparison - no LEGO brick is made up of smaller bricks (at least, not the way a galaxy is made up of stars). This raises further questions!
9:32 I think it's most of everything? For legos, there is most likely a balance between difficulty of producing a factory to make that piece of that size, and demand for that piece, and the raw plastic needed to produce X count of that piece, which leads to pieces of 1 gram being roughly 4x as common as pieces of 2 grams, and 1 gram little studs being 64x as common as 8 gram bricks, or 10,000x as common as 100g monster bricks (who wants a single piece that large??)
This gives me the same vibe as stuff like benford's law, it feels like there must be an underlying reason for this to happen even if its just the nature of things
My gut tells me you're correct that stellar mass following a square power is related to gravity, but there's no real reason to think the two distributions are related. Lego being hollow with close to uniform surface thickness could easily account for how reliably the mass distribution follows a square. I'd like to see what mass distribution holds for solid objects (like maybe rocks?). If it looks like a cube, then the question becomes: "What rules govern the size distributions of these objects?"
I instantly recognized the mascot at 4:43 because I literally live 5 minutes away from the NSU campus and have gone there dozens of times. that's amazing
I love these videos because they give me a mild existential crisis before safely bringing me back down to earth moments later. **Is everything just a power of 2?! Are we living in a simulation?!?! WHAT IS LEGO HIDING FROM US?!?!! WHAT DOES IT ALL MEAN?!?!!!!!!! I dunno, but this video was sponsored by Brilliant!**
You're videos are a rollercoaster. The initial title screams that you're on crack like Russian badger or Jeff. But then you're so incredibly calm as you casually explain everything on a level for a degree-less pleb like me can comprehend and understand. Bro you're awesome.
Hmm... In the context of mass distributions in the universe, you note counting planets vs stars vs clusters vs galaxies etc. So higher mass buckets include items from smaller buckets. However, with lego pieces this is not the case - why doesn't this cause a difference in the distributions?
my thought is universality. Parts that are smaller can be used in more situations - the giant ship haul would be very difficult to use outside of a boat context, whereas a 1x2 grate can be used in almost any context. This likely extends to the universe, where a star or a black hole can't really be used outside of that one role or purpose in the universe, whereas planets and asteroids have so much more diversity at their smaller sizes.
You have excellent timing! I was putting together a LEGO shopping list at work today. I work at a university, and I'm putting together a prototyping kit (trying to encourage the engineering undergrads to actually experiment with their designs, instead of mono-focusing on the first idea they have). I'm now thinking I need to order about 5x more of the fiddly little bits. 😅
I think it has to do with probability of interaction. If we interact objects 2 by 2, the probability of interaction is the newton binomial (n / 2) = n*(n-1)/2, i.e. "n choose 2". For n>>1, p ~= n²/2. If your system has a limited resource to share among all items, then if the number of items in a class (in this case size) goes up, the resources allocated by item has to go down (in this case mass). In the case of LEGO, I think they artificially choose each number to make it fun, since there's less and less use for bigger pieces; they end up dialing an optimum by experience. For gravity, the more objects you have, the more they will collide, and the bodies can 1) bounce, 2) break each other apart, or 3) lump together. The probability for each type of interaction depends on the size of the bodies: a big body is more likely clump smaller ones, small bodies are more likely to break each other apart. So, as bodies grow bigger, they will collide less and absorb smaller ones, reducing their number. Small bodies on the other hand tend to collide and crumble even more. So, why isn't there a void in the middle, for average sized bodies? Well, there kinda is... locally at least. Just look at the solar system: the sun is more massive than all planets combined. Around Saturn, the rings are pretty much powder orbiting a big mass.
Power law distributions are everywhere! Not every phenomena is power law distributed, but alot are. For instance, take the earthquakes: every day there are a ton of extremely small quakes, but very few are large enough to be noticeable (and the exponent is still around 2, which comes out everywhere you look with just slight variation). Not all events are power law distributed: when it rains, rain drop sizes are distributed according to (when the number of drops goes to infinity) a Gaussian. I think, but I can be wrong, that there's something to do with aggregation processes, when you create objects by combining others the exponent comes out. Maybe is due to the dimensionality of our world, as I think, but this is just speculation, that in a world with more than 3 dimensions, the exponent could be different. On the exponent itself, if I remember correctly, most sets have a slightly larger value than 2, like 2.3 or something (still its been a while since I've studied the subject in depth), if you want to know more I advise you to study power laws in the subject of complex physics
I play with game design as a hobby... one of the things I have learned is about Level of Detail (LoD). While LoD is used to performance... it kind of is with our brains as well. Say you are making a car out of legos. My first thought is about the blockiest shape of a car.. basically a box. Then start rounding off the edges and adding a of form. Basically each step, with finer and finer detail, cuts the size of the shapes we're working with in half. Same with LoD (though, that is a very simplified explanation of LoD)... so I think that's why it comes up in legos. Why it comes up in nature .. I can't even begin to hypothesis.
Ooh, this gives me ideas on things to count and see whether the exponent is the same. I feel like it has nothing to do with gravity, but rather just... how things are composed of other, smaller things.
Reminds me of Benford's Law, the underlying statistical fact that the Universe has a negative log distribution for all numbers beginning with the digit 1 (just over 30% of all numbers begin with 1) down to 9. Wiki has a good page on it.
"And the subject of their experiment is Lego Ninjago[...] I mean, what does that say about the state of scientific research?" Simple, it means we must be doing something right 😤
This is such a good channel. It is rare to see channels that not only explain complex phenomena but also expand upon it. Such a well-researched and laboriously crafted video. You just earned a subscriber
I think the best part about science is EVERYTHING is science if you look at it long enough. The reason science fairs and kids science experiments are so fun and interesting is realizing the degree to which science dictates and is dictated by everyday life, and things like this paper and your videos seem very silly but ITS THE SAME AS WHAT EVERY OTHER SCIENTIST DOES, but you aren't afraid to look at EVERYTHING, even if you think it would make you look silly to experiment or observe it
I think looking at megablox or other brands of definitely-not-LEGO would provide good evidence for if the -2 being intentional there. As for why it keeps popping up in the universe, It could come down to spacetime's shape being parabolic around particles, a.k.a. the force due to gravity and r^2. I think it is linked to entropy in some way too, or they are symptoms of the same mathematical fact of the universe
Pareto distribution also shows up in network theory (e.g. in distribution of node connectivity) and social science in general (distribution of wealth, settlement sizes, and more). There's definitely something about the way that things aggregate that leads to this behaviour. I feel like statistical mechanics will (one day?) have something to say about it.
this rule feels like something to due with fractal dimensions in 3d euclidean space. eg for an object to be considered an "object" it needs to be someone self-related [no treedogs here] and that generally includes a notion of "connectedness", and things are generally also made of several smaller things and that connectedness is related to how often the smaller things are by themselves or form a bigger thing [⅕ of the time it seems]
Predicting it before finishing the video, it's just the 80-20 graph where eighty percent of stuff is in the bottom 20% and vice versa. It shows up fricking EVERYWHERE.
Fun fact: the radius of moon impact craters, the chance of opening chess moves, and the rate at which we forget *all follow this exact same rule*. It's called Zipf's Law or the Pareto Principle. When using the power law formula, things in nature tend to have an exponent between 2 and 3. This is not a conscious decision made by someone, it's just the way natural data is distributed. Another way to think about this case with LEGO is using the 80/20 principle - 20% of the pieces contain 80% of the mass. Vsauce has a great video on this mind-blowing phenomenon.
Look into polymerization and how distributions of different lengths of mers form. It’s pretty much your rock explanation. A decade ago I wrote a program to find out the average lengths of polymer chains formed in a solution of mers for my materials science class. From what I remember, the rate of reaction between molecules of various chain lengths, like 1-mer and 1-mer, 2-mer and 1-mer, 3-mer and 2-mer and so forth were based on concentrations of each, and obviously the bigger a chain is, the rarer it is, and the less likely it is to react with other longer polymer chains. Anyway, there’s a lot of papers written about average length of polymer chains and mass distribution in polymer solutions - like, what range of lengths of polymer chains contains most of the mass of a cured polymer solution. Look into it, it’s interesting, and your video reminded me of it.
I personally think 1/m^2 is where -2 comes from. For those that don’t know, an exponent in the denominator is expressed as a negative exponent, so 1/m^2 is the same thing as m^-2. Considering that the process where this exponent came up is intrinsically important to explaining why there are so many small things, it makes sense that -2 would be from there.
Great video! I love the idea that random occurrences strictly follow some grand rule like this. Just another example of mathematics so shockingly representing reality.
This is probably one of the best video I have ever seen, I really appreciate that we start from a fact about universe, think of a way to experiment at our own scale, an draw conclusions from it. This is research done great. Also, it opens so much potential for experimentation. As you said, we should try this experiment with all kind of things, this is very exciting!
In a way, you could consider this phenomenon the universal distribution of particles through the form of charge force, magnetic force, and gravitational force. Perhaps because all matter consists of the same elements which are naturally occurring configurations of particles, it carries the same patterns throughout its spectrum of mass. Nonetheless, this finding is a testament to how fascinating and complex the universe truly is. Wonderful video :)
My first thought after never thinking about this problem until now is that this might be a trait of our human psychology not the universe. Our decisions of how we break things up and decide on categories could naturally fall into this pattern. That a descriptive venture, categorizing discrete objects, and a proscriptive venture, creating discrete objects, lends itself to the same answer leads me to think the measuring device is the culprit.
If I'm not mistaken I think organisms follow this similar distribution, like bacteria sized organisms to single celled organisms to small creatures and so on until the biggest animals.
Isn't a power law relation somewhat inevitable when trying to approximate a solid with a curved surface by using what's (mostly) a set rectangular cuboids? Approximate the center mass, then add more and more smaller cuboids to add detail. The exponent is measuring how "cube-like" the solid is: if it can be easily approximated with cubes then the exponent is very negative, if it cannot then the exponent is closer to 0. You can imagine a lego set where all the mass is in a single brick that's the largest in the set: it's the set you get when you buy a single brick. The other extreme is tons of tiny pieces and nothing else, which I suppose you'd get if you tried to approximate some purely thin-walled fractals with bricks. I am asserting without any testing, analysis, verification or really much thought that an exponent value of around -2 is what you get when approximating a sphere with progressively smaller bricks (or other spheres, or any shape?) when using most sane tactics to generate the subdivision. It is well known that all objects in physics are approximately spherical, so you get -2 everywhere.
i finish my undergrad degree this semester, and i probably have time to publish only one more paper. so far, i've been able to reference star wars and star trek in some of my papers, but today i learned that i need to somehow find a way to connect my research to lego before i graduate so that i can reference lego in my last paper
It's not just things subject to gravity. It also works for country sizes (area), town/city population sizes (population), number of organisms (body mass), you will get this bias: the curve of the measure will follow some power law with negative exponent. This happens whenever you chunk things, so this is a strong indication that it is the chunking that's the origin of this phenomenon. The following is a pretty intuitive way of making this make sense, if not a clinching proof. For each given amount of material, you can make a larger number of smaller things than larger things, so even if there is an equal distribution of total mass in each category, there will be a bias towards small things. For example, if we have a collection of Big Things and Small Things, where a Big Thing has twice the mass of a Small Thing, yet a given atom in this collection is equally likely to find itself in a Big Thing as it is to find itself in a Small Thing, then Small Things outnumber Big Things 2:1. For the even odds of each atom finding itself in a Big Thing and a Small Thing means that the amount of mass in all the Big Things equals the amount of mass in all the Small Things. But, of course, the amount of Big Things you can make with a given mass is half that of the amount of Small Things you can make with that same mass, so Small Things outnumber Big Things 2:1, and indeed, the distribution of the numbers of such objects goes as inverse of the mass.
This was an incredible video, thank you for making it. Your channel is quickly becoming one of my favorites. Just a point on what you said in the end, the models we have for galaxy formation actually are somewhat similar to star formation, in that they are both formed from clouds or roughly uniform clumps of matter that get too dense at one point and collapse into different sized pockets. For stars this happens in clouds of gas, but for galaxies it happens over the scale of the whole universe with dark matter halos. Halos look something like stars forming out of a nebula, and the regular matter collects in the center, forming galaxies and galaxy clusters.
I think for celestial bodies and objects it makes sense to think that the larger one is the more easily it is destroyed or converted into smaller objects. Just how when you bite your lip it swells, leading to it being more susceptible to biting again, which leads to a chain reaction, celestial bodies must act in the same way. The probability that they will be converted to smaller, less massive objects increases with increasing size.
At least three reasons for the LEGO: - Small pieces are more versatile than big pieces - Small pieces can combine to make the big pieces but not vice versa - There a lot more permutations of big pieces than small pieces and they need to make a separate mold for each piece
I wonder if the square-cube law somehow plays into how the lego sets' distribution occurs. The centre of the model is where basically none of the surface (surface area) of the piece is visible, so its only really being chosen for its size and ability to provide structure (and thus mass). But as you move out, the surface of each piece is more and more visible, and thus the detail of it is more important and its size and ability to provide structure less-so. I'm not really sure how the maths and algebra of the square-cube law could be manipulated to get the -2 exponent, but i feel like there might be something there
That's nuts. Even without weighing them I can tell the objects on my desk follow this rule. A small sampling: 1 desk 2 monitors a keyboard, a mouse, and 2 speakers 8 cups and containers two dozen pencils, pens, and markers a set of 42 screwdriver bits, two tubes of glue, 2 removable erasers for mechanical pencils, misc junk
I'm a physics student, so this law reminds me of one I'm very familiar with: Gauss's Law! Typically it's used to relate a conserved "quantity" within a boundary to a "throughput" passing through the boundary. Mathematically, this says: kQ = TA, where Q is the quantity within the boundary, T is the throughput, and A is the total area of the boundary. (k is a scaling constant) For "throughputs" that only depend on distance, the area can be taken to be the surface area of a sphere, proportional to r^2. This is precisely the inverse square law! Another form of this is the inverse cube law. Some examples of things that follow this pattern are fluid flow, light, the electric and magnetic fields, sound amplitude, heat flow, stresses in buildings, mass, and momentum. Conserved things *have to* follow the square cube law. This is because we live in a 3 dimensional space, so "areas" are the number of dimensions minus 1. To cancel out the flow spreading out over an area, its intensity must be proportional to 1/(d-1). We observe images of things as areas, and the universe has a fairly uniform density, so this result makes sense to me. By Gauss's law, I expect a classification of groups of things in a 2d universe to follow a 1/r distribution, and things in a 4d universe to follow 1/r^3. This is testable too, since we can directly measure 2d realms like the game Agario, or city plans and look to see if things fit that 1/r pattern.
Since there is a bunch of different designs for cutties, going from butter knives to swords, this could be interesting. Same with bowls in a restaurant. Or House size across a state/province.
I think the reason for the distribution is much as you said where we start with larger pieces and use smaller pieces to detail. You can picture most Lego construction like a fractal where the face of the shape has exponentially more pieces than the interior and you could theoretically scale these models infinitely and see the edge's detail grow infinitely.
What might be a similarity is the counting process. For stars, we are counting galaxies AND stars, which is essentially double-counting. While not true for legos, there is a somewhat similar process there, with small pieces often being parts of larger pieces (not exactly, but oh well dark matter is also a big mystery). Maybe that’s a similarity? Also: I wonder what the ikea catalogue looks like!
To me this sounds like a matter of chaos or entropy. With less chaos we see more organised structures (such as lego creations or solar systems), while more chaos will show less big structures and many more small things.
For the article from “The Pedagogical Representation of Mass Functions with LEGO and their Origin” I had the PI as a professor for undergrad and the Kyle was a friend of mine. We were both presenting at a conference at our school and he was telling me how many lego boxes he went through for his research.
Awesome video! Now I wanna weigh everything in my house and check the distribution lol. One thing that raises my interest is that many of the bigger objects at home for example are made out of smaller objects. What happens to the distributions if we only count the smallest separable pieces, what happens if we only count the final whole objects, and what happens when you count both, thus effectively counting small pieces as small objects and once as the bigger whole object
I'm pretty confident this has nothing to do with any design standard at lego. it's serendipity! I'd bet if you weighed everything you have in your house, or the sticks in your yard, or some other group of objects with sufficient variability in size, you would find similar results. just intuitively, small things are always more common.
love the video! I love fun little scientific explorations into things that I would never have encountered otherwise, especially in the format you present in!
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I think something worth mentioning is that a lot of celestial bodies are really just collections of many smaller objects. For example, galaxies are collections of millions of star systems. Each star probably has a few planets. I wonder if how we classify what an “object” is affects the trend, or even causes it
I'm glad you mentioned that! I actually cut a section where I talked about that. The problem is especially hard when we talk about nebulas (is a dust particle one "thing"?), but the researchers ignored that. They may have double or triple counted many objects.
All classification discussions will eventually lead you to a 45-minute VSauce video called What Is A Universe?
@@RaquelFoster "Do Chairs Exist?" is also relevant here
This is why philosophy is important, in particular the discipline of asking what is the nature of a thing and getting good definitions from that questioning. Also epistemology to define how we know a thing. Physics is meant to be paired with metaphysics.
I thought the same; was wondering how you could count Systems when you count objects alone
I don't think this is by any design at LEGO, it would also be very hard to make sets with a "forced" distribution. I think it's a naturally arising property from the way things build up. I mean, you can only have a few big pieces (base blocks, big custom monoblocks, etc), while you need a ton of tiny things to fill the gaps, connect blocks, add decoration, and so on. This is why it's so amazing, I think this arises organically.
Prime numbers play a role too I believe. The composite-ability of the world. There’s even been a couple videos recently on tiles and the history of these geometries in sacred religious temples. Iirc isn’t this curve similar to Pareto distribution?
I think it also just has to do with the nature of the toy.
Too many tiny pieces would be difficult to build. Too many big pieces would be boring.
It's not necessarily something the designers would think about, but it's something somebody would think about before retail
What he said
I wonder if it's like Benford's law in some way, too.
exactly. this feels incredibly intuitive that of course this would be a logarithmic distribution.
This is kind of reminiscent of Zipf's Law.
Something to keep in mind is that a relatively small variation from -2, to, say -2.5, is a bit bigger than we might think as it is on a log scale.
Not to even talk about the trend line error. In many cases it doesn't describe the data very well at all
It also reminded me of a study of the distribution of scene lengths in movies, or maybe shot lengths, to be more precise. Lots of little shots, like glancing over to a character's face to see their reaction, quite a few shots longer than that, and some long shots, maybe a slow pan over beautiful scenery.
No, I'm afraid I don't remember the exponent; I think it wasn't highlighted in the article so much as the exponential decay shape.
I came here to mention the Zipfyness, but I knew in my heart that someone already did 😄
Came here to say this
I'm a video game designer and when I make levels, adding clutter to the environments follows the same curve of object sizes for the map to feel "Right"
It might be some sort of fractal pattern that we all have a deep intuitive understanding of, that's why the legos follow the same rule. Maybe even our bodies follow the same rule: bone sizes vs quantity; it could potentially apply to anything.
Someone should count the objects in Katamari levels and see what the exponent is lol
I wonder if it has something to do with Weber-Fechner law - our senses are more logarithmic than linear.
I don't think this curve shows anything about the nature of the universe but about how we see/label things.
There are no hard rules in the universe about when something counts as a new separate thing, except for maybe the tiniest particles that make up the fabric of reality. At certain points our brain decides that a collection of smaller things or different characteristics is worth distinguishing as a new separate thing, giving it a name and so on. This experiment shows that there is an interesting curve in how we do it.
It makes total sense that the designers of lego bricks would feel the need to create bricks at sizes where our brains like to distinguish things as separate or you would feel the need to place objects on the map like that.
Pretty cool.
Humans have a preference for environments that are between 1.3-1.5 fractal dimensionality. So I think you may likely be right about that. Maybe Gaussian clutter?
i had something similar in mind. what if it's some sort of natural distribution that we're instinctually accustomed to and that often appears in nature?
Astrophysicist here, I'm getting my Ph.D. studying galaxy formation/evolution. I wanna start out and say, I really enjoyed the video! It's really interesting to think about how mass is distributed in the Universe and if the same laws of nature that apply to galaxy clusters apply to LEGOs :).
First is in regards to the initial mass function (IMF, as we typically call the mass function for stars) you describe here. Indeed the Salpeter (1955) IMF is a power law with an index of -2.35. However, more recently increasingly complex IMFs are used to describe the distribution of stellar masses of stars (see, e.g., Kroupa 2001 or Chabrier 2003) which are more than just a power law. These distributions are more like log-normal distributions, they have a plateau and turn over at lower masses. Even further than "Is it a power law?" it's actually a super open question in Astrophysics right now whether the IMF has evolved throughout the lifetime of the Universe.
Second (sort of related), I think your intuition about gravity and the r^-2 power is really awesome, but the dense, star-forming interstellar medium is a little more complex than that. You have to take into account turbulence (collisions within the gasses), stellar feedback (other high-mass stars exploding in the neighborhood), etc. So I am skeptical as to whether or not that's where the Salpeter power law index comes from. It's an interesting connection nonetheless!
Finally, at 10:07, I take exception to the statement "...galaxies are formed by stars clustering together into bigger pieces." Galaxy formation is a direct effect of dark matter overdensities, which gravitationally attract baryonic (non-dark) matter to their center, not necessarily groups of stars in space that all coalesce. The stars have to form from the gas reservoir, which needs to be sufficiently dense. In the absence of large dark matter potential wells the diffuse intergalactic gas simply doesn't get dense enough (or if, by some wild chance it would, not enough of it would in order to form galaxies of the mass we see today!).
Again, really enjoyed the video! I hope you find the comment helpful 😀
EDIT: formatting (thanks @Speed)
*Formatting
Astrophysicist here, I'm getting my Ph.D. studying galaxy formation/evolution. I wanna start out and say, I really enjoyed the video! It's really interesting to think about how mass is distributed in the Universe and if the same laws of nature that apply to galaxy clusters apply to LEGOs :).
First is in regards to the initial mass function (IMF, as we typically call the mass function for stars) you describe here. Indeed the Salpeter (1955) IMF is a power law with an index of -2.35. However, more recently increasingly complex IMFs are used to describe the distribution of stellar masses of stars (see, e.g., Kroupa 2001 or Chabrier 2003) which are more than just a power law.
These distributions are more like log-normal distributions, they have a plateau and turn over at lower masses. Even further than "Is it a power law?" it's actually a super open question in Astrophysics right now whether the IMF has evolved throughout the lifetime of the Universe.
Second (sort of related), I think your intuition about gravity and the r^-2 power is really awesome, but the dense, star-forming interstellar medium is a little more complex than that. You have to take into account turbulence (collisions within the gasses), stellar feedback (other high-mass stars exploding in the neighborhood), etc.
So I am skeptical as to whether or not that's where the Salpeter power law index comes from. It's an interesting connection nonetheless!
Finally, at 10:07, I take exception to the statement "...galaxies are formed by stars clustering together into bigger pieces." Galaxy formation is a direct effect of dark matter overdensities, which gravitationally attract baryonic (non-dark) matter to their center, not necessarily groups of stars in space that all coalesce.
The stars have to form from the gas reservoir, which needs to be sufficiently dense. In the absence of large dark matter potential wells the diffuse intergalactic gas simply doesn't get dense enough (or if, by some wild chance it would, not enough of it would in order to form galaxies of the mass we see today!).
Again, really enjoyed the video! I hope you find the comment helpful 😀
I knew someone had higher knowledge (no hate towards the bird, he’s great and being outside of physics n astrology they’re awesome to keep my interest peaked) but the second Mr bird asked for Lego workers I had a feeling if not them YOU’d show n ya did :) thanks for the insight but, and excuse my ignorance forgive my incorrectness, has dark matter been found? I mean i know have strong proof it’s there, with galaxies being heavier than expected n what not but we have yet to capture/replicate/find proof aside from implication of its existence? Again I’m just an onlooker, but dark matter really fascinates me as a concept and I’d like to know more about it if possible
Really interesting to read your take as a soon-to-be Ph.D !
OOOO OOO ALSO again not too well versed in any of these fields a t a l l but wouldn’t we need to include dark matter in the IMF to make the calculations more accurate??
@@trashmanjacobs7180 Great questions!
First, yes we have fairly direct evidence for dark matter (on astrophysical scales). A quick example of this is orbits within galaxies: the (visible) matter within galaxies orbits faster than can be held together by just the (visible) mass within it, therefore there must be some missing (invisible/dark) matter that we cannot see. Whether or not we just missed normal matter in observations, it is some exotic particle, or we need to re-write physics (see "Milgromian physics" if interested, there's a really great video by "PBS Spacetime" explaining it) is still a manner of debate. With that being said, I would say there is a majority who believe it to be exotic particles that don't interact with light.
Second, by construction the IMF sets out to measure the distribution of masses of stars. So no we don't explicitly need to account for dark matter within it since stars are made of normal matter. Galaxy-wide scaling relations, however, can (and do) include dark matter.
The -2 coming from gravity makes good sense, but if a similar pattern exists in a wide range of different settings wherein things are built from combinations of other things (e.g. LEGO sets, IKEA furniture, laserjet printers, motor vehicles, etc.), that would suggest that it's a statistical property instead. I'd be interested in seeing a data where rather than "mass" being the x-axis, it's "mass of component divided by mass of the final product" wherein components of a wide range of different things are included, rather than just being a single category of objects, like LEGO. I think that distribution might provide further insight into this.
It could definitely be that Lego seeks to mimic the constraints of gravity with its connections and structure
🙃🥸🤩🤓
i think what’s more likely is that the -2 comes from the dimensionality of our universe. the -2 in the gravitational equation comes from the same place: the inverse square law. which means it might be worth exploring 2d or 4d video games
Guess I'm a bird because the youtube algorithm decided this video was for me
same bro
Welcome to the flock
Birds of a feather.. 😅
Chirp chirp!!
Yea this cool
I love that this concept of smaller parts summing to equal bigger parts is kind of intuitive while being simultaneously mysterious. The universe is a strange and magical place.
Great video!
I’m an acquaintance of the author of the paper. I have a school-aged son and a BrickLink store, so he sent me the paper to read, and then later sent me this video. You did a great job. He’s really impressed, and my son has become a big fan of yours (I already was). If you need any help with Lego, give me a hollar. Or if you’d like to get in touch with Stefan, I know he’d be happy to speak with you.
Keep up the great work!
That makes me very happy to hear - both that the author likes it and that you and your son enjoy the videos! I thought this was a fun topic so I made it with all ages in mind.
I like your theory about why -2 for astronomical objects. Edit: Though actually, I think I just thought of a handwavy variation that doesn't depend on the gravitational constant. Let's say we start with a bunch of really small things (rocks, particles, etc.). Some might merge together into particles that are twice as big, and some might not. then, out of those, some might merge again to form particles of the next order of magnitude, etc. If, at any level, the probability of merging vs. not merging is about even (for any reason at all, which may be different for different types of "things"), then I think the exponent will be roughly around -2.
For LEGO bricks, I have a different (half-baked) model that might be worth exploring.
We can view a LEGO model as trying to approximate a specific 3D shape using a minimal(ish) number of bricks. With some additional constraints, e.g. there's a limit to how large (or at least how massive) the bricks get, and also limits on individual dimensions (e.g. normal bricks are only so thick). So, they fill in the rough shape with big bricks, then they start using progressively smaller bricks for the details, then even smaller for the tiny details, etc. At some point it bottoms out because they either give up on the level of fidelity, or make bricks that are the exact right shape they need (e.g. human head, flower, window, ...). I feel like these constraints will at the very least naturally produce a power distribution.
It would also be interesting (and relevant) to find out whether brick volume is proportional to mass (or maybe larger bricks are less dense, for example)
it is actually criminal with how small your channel is. Your content is fantastic and so well executed. I can't wait to watch you grow in the future.
I'm a Lego fan and custom design creator. Small pieces are used to create detail, as well as specific functionality. Since Lego designs typically reflect our world in some manner, it makes sense that lots of small pieces are required to match the fractal and chaotic nature of our world.
An interesting research topic would be to do something similar to actual fractals. Or turbulent fluid flow.
I love your videos, the way you mix different subjects that at first might seem completely unrelated is fascinating to say the least
Warmest regards and best of wishes🌹🌹🌹
Thanks! That means a lot
@@physicsforthebirds Plethora ❤
I really want to point out that the arithmetic mean of all the twelve lego slopes you've show in the video is equal to (-2.13-1.95-2.16-1.98-2.04-2.05-1.40-1.76-2.091.98-2.39-2.2)/12 = -2.01083333... Pretty close, right?
love the way you investigated the idea in the video, wish you all the best from Brazil 💚💛💙💛💚
Hey, it works out pretty well! I regret not calculating the error and doing this myself, so I'm glad people like you are doing it for me
@@physicsforthebirds it's a pleasure! I was too curious to hold myself back and don't go calculate
the arithmetic mean of the slopes translates to a geometric mean of the original distributions, would we rather take an arithmetic mean of the distributions? then I don't know what the mean of the slopes should be (the critical problem is how we should think about the 'error')
I have two (very high-level and possibly very wrong)thoughts on this:
1. It could very likely be related to packing small and large pieces in bags(or even some sort of packing algorithm), as the design might be optimised for that. Or some obscure packing problem, for that matter.
2. Smaller pieces are spread out on the surface area of larger parts. So a graph like this is expected. Some more thought on the circley things might give more insights on the exact power of 2 point something.
Random extensions to these:
For 1.
The density of smaller objects are larger because they have more bulky and dense edges compared to light flat surfaces. (I checked this for squares and circles)
lego lengths have a higher variance wrt bag sizes, so it won't scale as much.
Using this, I tried to see what would happen if we naively take n(legos) \propto vol(bag)/vol(lego), but was off by 0.5 or something. Without looking at the data as it's already 4 am lmao
For 2.
For VLSI, a design following a somewhat-similar principle, I'd say mass and surface area both scale according to the square of the side, as height does not vary too much. So it should not be very far from 1/mass. Legos often have tall pieces. idk what I'm talking about, though.
Also, great video: Can't stop thinking about this :p
#2 makes a lot of sense.
You want me to count every object in my house?! Uh, yeah, I'll get right on that. Respect to NSU to use a Ninjago set for the experiment. Because it shows they jumped up, kicked back, whipped around, and spun, and then they jumped back and did it again. If everyone did the Weekend Whip, the world would clearly be a better place. Nova Southeastern was originally a National Association of Intercollegiate Athletics (NAIA) institution back in the 1982-83 athletic season, which they would compete in their first conference affiliation home in the Florida Sun Conference from 1990 to 2002. The Sharks were originally called the Knights, which was from 1982 until 2004. In 2005, they unveiled the new Sharks logo and athletic mascot. The nickname was selected by the students.
3:04 need to say, this is actually called the initial mass function (IMF) . Slatpeter is just one of them, another really used one is Kroupa for example (also named after the scientist like the Saltpeter). Also for galaxies a nrw one tends to show up called the IGIMF which is integrated galactic initial mass function and comes from the behaviour of the initial mass functions of all the star clusters and stars inside the galaxy.
This is my new favourite Channel!
Playing leads to observation then leads to learning. I have never encountered boring triviality - only data my brain can't fathom. Thank you for your videos, it is so refreshing to have these higher level thoughts written so concisely! As Above So Below, from High to Low, Unified under the law that governs it all, whatever that may be.
This was a very interesting video! Also, as software engineer student, doing stuff like what you did in this video, just because you can, is what got me into this field.
This was such an enjoyably nerdy and poetic moment of science. You sound like such a fun friend to have. I hope you are able to make a living creating these videos.
Always a good day when one of your videos comes out!
I'm certain your channel will grow exponentially as you make these videos, they are very thorough and high level! Keep up the good work, I'll keep watching!
There is a likelihood that this has to do with a carefully selected balance of play-versatility for customers and manufacturing.
In the 1990s Lego was going wild with lots of themed sets, but this resulted in manufacturing many kinds of larger, unique and niche-use pieces, like the rope bridge piece, boat hull elements, and the big ugly rock piece. Going into the 2000s, Lego was in financial trouble because these products didn't have the shelf life they wanted and they were expensive to manufacture. They switched gears and started making more sets that involved smaller more versatile pieces and more of them that were easier to manufacture, like cheese wedge slopes and common 1x2 plates. At the same time they started doing themes with other intellectual properties, like Star Wars, Harry Potter, and the NBA.
These turned out to be the right decision. The major exception to this generalization is Bionicle.
My first guess for the distribution was a Zipf's Law curve, since we're talking about how common different groups of things are, but as you explained the actual answer, I realized that a Zipf curve doesn't actually give relationships in any ordered way, only between the 'most common' to the 'next most common' so it wouldn't really be applicable here. That said, a Zipf curve *is* pretty close to a power curve with a = -1, with the implication that that there would be half as many things which are twice as massive.
If there are 4 times as many things that are half as massive, that means that the *total amount of mass* in each 'group' follows a power law of a = -1. There is twice much total mass in the universe made up of (~1kg) things as there total mass made up of (~2kg) things. I think part of this might be due to the way that big things are usually made of small things, but there are also small things that aren't also part of other big things, but that doesn't really have meaning when talking about discrete LEGO bricks, which are all separate, atomic objects for purposes of comparison - no LEGO brick is made up of smaller bricks (at least, not the way a galaxy is made up of stars). This raises further questions!
9:32 I think it's most of everything? For legos, there is most likely a balance between difficulty of producing a factory to make that piece of that size, and demand for that piece, and the raw plastic needed to produce X count of that piece, which leads to pieces of 1 gram being roughly 4x as common as pieces of 2 grams, and 1 gram little studs being 64x as common as 8 gram bricks, or 10,000x as common as 100g monster bricks (who wants a single piece that large??)
This gives me the same vibe as stuff like benford's law, it feels like there must be an underlying reason for this to happen even if its just the nature of things
My gut tells me you're correct that stellar mass following a square power is related to gravity, but there's no real reason to think the two distributions are related.
Lego being hollow with close to uniform surface thickness could easily account for how reliably the mass distribution follows a square. I'd like to see what mass distribution holds for solid objects (like maybe rocks?). If it looks like a cube, then the question becomes: "What rules govern the size distributions of these objects?"
I instantly recognized the mascot at 4:43 because I literally live 5 minutes away from the NSU campus and have gone there dozens of times. that's amazing
Beautiful beautiful beautiful video. Can't believe no body would explain stellar mass distribution with legos. This what science is all about
I love these videos because they give me a mild existential crisis before safely bringing me back down to earth moments later.
**Is everything just a power of 2?! Are we living in a simulation?!?! WHAT IS LEGO HIDING FROM US?!?!! WHAT DOES IT ALL MEAN?!?!!!!!!! I dunno, but this video was sponsored by Brilliant!**
Once I finish high school watching these videos is how I'll "apply what I learned"
Love your slow calm relaxed speech
That's soooo cool -- plus a question I'd never stumbled upon before. Such a good video as always!
You're videos are a rollercoaster.
The initial title screams that you're on crack like Russian badger or Jeff.
But then you're so incredibly calm as you casually explain everything on a level for a degree-less pleb like me can comprehend and understand.
Bro you're awesome.
Hmm... In the context of mass distributions in the universe, you note counting planets vs stars vs clusters vs galaxies etc. So higher mass buckets include items from smaller buckets. However, with lego pieces this is not the case - why doesn't this cause a difference in the distributions?
These weird emergent properties that just seem to apply everywhere fascinate me. If you could do more on this, I would love it.
I have never played the game Katamari Damacy, nor thought about it for years, but for some reason this video reminded me of it.
my thought is universality. Parts that are smaller can be used in more situations - the giant ship haul would be very difficult to use outside of a boat context, whereas a 1x2 grate can be used in almost any context. This likely extends to the universe, where a star or a black hole can't really be used outside of that one role or purpose in the universe, whereas planets and asteroids have so much more diversity at their smaller sizes.
You have excellent timing! I was putting together a LEGO shopping list at work today. I work at a university, and I'm putting together a prototyping kit (trying to encourage the engineering undergrads to actually experiment with their designs, instead of mono-focusing on the first idea they have). I'm now thinking I need to order about 5x more of the fiddly little bits. 😅
Never thought I’d see the brick link api mentioned in a video about cosmic masses but here we are🙏🙏
2 minutes in - already subscribed and added the rest of your videos to my queue. this is great
I think it has to do with probability of interaction. If we interact objects 2 by 2, the probability of interaction is the newton binomial (n / 2) = n*(n-1)/2, i.e. "n choose 2". For n>>1, p ~= n²/2. If your system has a limited resource to share among all items, then if the number of items in a class (in this case size) goes up, the resources allocated by item has to go down (in this case mass).
In the case of LEGO, I think they artificially choose each number to make it fun, since there's less and less use for bigger pieces; they end up dialing an optimum by experience.
For gravity, the more objects you have, the more they will collide, and the bodies can 1) bounce, 2) break each other apart, or 3) lump together. The probability for each type of interaction depends on the size of the bodies: a big body is more likely clump smaller ones, small bodies are more likely to break each other apart. So, as bodies grow bigger, they will collide less and absorb smaller ones, reducing their number. Small bodies on the other hand tend to collide and crumble even more.
So, why isn't there a void in the middle, for average sized bodies? Well, there kinda is... locally at least. Just look at the solar system: the sun is more massive than all planets combined. Around Saturn, the rings are pretty much powder orbiting a big mass.
This video was published on my birthday! What a cool present
Power law distributions are everywhere! Not every phenomena is power law distributed, but alot are. For instance, take the earthquakes: every day there are a ton of extremely small quakes, but very few are large enough to be noticeable (and the exponent is still around 2, which comes out everywhere you look with just slight variation). Not all events are power law distributed: when it rains, rain drop sizes are distributed according to (when the number of drops goes to infinity) a Gaussian.
I think, but I can be wrong, that there's something to do with aggregation processes, when you create objects by combining others the exponent comes out. Maybe is due to the dimensionality of our world, as I think, but this is just speculation, that in a world with more than 3 dimensions, the exponent could be different.
On the exponent itself, if I remember correctly, most sets have a slightly larger value than 2, like 2.3 or something (still its been a while since I've studied the subject in depth), if you want to know more I advise you to study power laws in the subject of complex physics
This is beautiful, thank you for the video
This would be the science fair project of the decade
I play with game design as a hobby... one of the things I have learned is about Level of Detail (LoD). While LoD is used to performance... it kind of is with our brains as well. Say you are making a car out of legos. My first thought is about the blockiest shape of a car.. basically a box. Then start rounding off the edges and adding a of form. Basically each step, with finer and finer detail, cuts the size of the shapes we're working with in half. Same with LoD (though, that is a very simplified explanation of LoD)... so I think that's why it comes up in legos. Why it comes up in nature .. I can't even begin to hypothesis.
Good video, makes me think of Zipf's mystery, like what Vsauce mentioned a while back.
I learned about your channel from a talk from the professor who wrote this paper!
This is so cool!
Ooh, this gives me ideas on things to count and see whether the exponent is the same.
I feel like it has nothing to do with gravity, but rather just... how things are composed of other, smaller things.
Reminds me of Benford's Law, the underlying statistical fact that the Universe has a negative log distribution for all numbers beginning with the digit 1 (just over 30% of all numbers begin with 1) down to 9. Wiki has a good page on it.
"And the subject of their experiment is Lego Ninjago[...] I mean, what does that say about the state of scientific research?"
Simple, it means we must be doing something right 😤
This is such a good channel. It is rare to see channels that not only explain complex phenomena but also expand upon it. Such a well-researched and laboriously crafted video. You just earned a subscriber
I think the best part about science is EVERYTHING is science if you look at it long enough. The reason science fairs and kids science experiments are so fun and interesting is realizing the degree to which science dictates and is dictated by everyday life, and things like this paper and your videos seem very silly but ITS THE SAME AS WHAT EVERY OTHER SCIENTIST DOES, but you aren't afraid to look at EVERYTHING, even if you think it would make you look silly to experiment or observe it
I enjoyed a more speculative video for a change, good work!
This is a hecking brilliant video thank you
I think looking at megablox or other brands of definitely-not-LEGO would provide good evidence for if the -2 being intentional there. As for why it keeps popping up in the universe, It could come down to spacetime's shape being parabolic around particles, a.k.a. the force due to gravity and r^2. I think it is linked to entropy in some way too, or they are symptoms of the same mathematical fact of the universe
Pareto distribution also shows up in network theory (e.g. in distribution of node connectivity) and social science in general (distribution of wealth, settlement sizes, and more).
There's definitely something about the way that things aggregate that leads to this behaviour. I feel like statistical mechanics will (one day?) have something to say about it.
Your voice is bumpy but smooth, like a stone path. Soothing.
this rule feels like something to due with fractal dimensions in 3d euclidean space. eg for an object to be considered an "object" it needs to be someone self-related [no treedogs here] and that generally includes a notion of "connectedness", and things are generally also made of several smaller things and that connectedness is related to how often the smaller things are by themselves or form a bigger thing [⅕ of the time it seems]
Predicting it before finishing the video, it's just the 80-20 graph where eighty percent of stuff is in the bottom 20% and vice versa. It shows up fricking EVERYWHERE.
All your videos are so good, thank you very much ☺️
Man I'm a teacher and I'm flabbergasted about each one of your videos, I love them!!! So many creative ways of approaching classical topics
The Musik on this is 🔥
PLEASE upload your background music, it's so beautiful!!
Fun fact: the radius of moon impact craters, the chance of opening chess moves, and the rate at which we forget *all follow this exact same rule*.
It's called Zipf's Law or the Pareto Principle. When using the power law formula, things in nature tend to have an exponent between 2 and 3.
This is not a conscious decision made by someone, it's just the way natural data is distributed.
Another way to think about this case with LEGO is using the 80/20 principle - 20% of the pieces contain 80% of the mass.
Vsauce has a great video on this mind-blowing phenomenon.
Wow. I was doing bend tests with different lengths of flat bar, and none of the math I found online made sense, but my exponent was 2.37
Look into polymerization and how distributions of different lengths of mers form. It’s pretty much your rock explanation. A decade ago I wrote a program to find out the average lengths of polymer chains formed in a solution of mers for my materials science class. From what I remember, the rate of reaction between molecules of various chain lengths, like 1-mer and 1-mer, 2-mer and 1-mer, 3-mer and 2-mer and so forth were based on concentrations of each, and obviously the bigger a chain is, the rarer it is, and the less likely it is to react with other longer polymer chains. Anyway, there’s a lot of papers written about average length of polymer chains and mass distribution in polymer solutions - like, what range of lengths of polymer chains contains most of the mass of a cured polymer solution. Look into it, it’s interesting, and your video reminded me of it.
I personally think 1/m^2 is where -2 comes from. For those that don’t know, an exponent in the denominator is expressed as a negative exponent, so 1/m^2 is the same thing as m^-2. Considering that the process where this exponent came up is intrinsically important to explaining why there are so many small things, it makes sense that -2 would be from there.
Really cool paper and explanation
Great video! I love the idea that random occurrences strictly follow some grand rule like this. Just another example of mathematics so shockingly representing reality.
This is probably one of the best video I have ever seen, I really appreciate that we start from a fact about universe, think of a way to experiment at our own scale, an draw conclusions from it. This is research done great.
Also, it opens so much potential for experimentation. As you said, we should try this experiment with all kind of things, this is very exciting!
In a way, you could consider this phenomenon the universal distribution of particles through the form of charge force, magnetic force, and gravitational force. Perhaps because all matter consists of the same elements which are naturally occurring configurations of particles, it carries the same patterns throughout its spectrum of mass. Nonetheless, this finding is a testament to how fascinating and complex the universe truly is. Wonderful video :)
My first thought after never thinking about this problem until now is that this might be a trait of our human psychology not the universe. Our decisions of how we break things up and decide on categories could naturally fall into this pattern. That a descriptive venture, categorizing discrete objects, and a proscriptive venture, creating discrete objects, lends itself to the same answer leads me to think the measuring device is the culprit.
If I'm not mistaken I think organisms follow this similar distribution, like bacteria sized organisms to single celled organisms to small creatures and so on until the biggest animals.
Ziph! I just went on an old VSAUCE video binge ok? :)
What a great STUD-y!
Isn't a power law relation somewhat inevitable when trying to approximate a solid with a curved surface by using what's (mostly) a set rectangular cuboids? Approximate the center mass, then add more and more smaller cuboids to add detail. The exponent is measuring how "cube-like" the solid is: if it can be easily approximated with cubes then the exponent is very negative, if it cannot then the exponent is closer to 0.
You can imagine a lego set where all the mass is in a single brick that's the largest in the set: it's the set you get when you buy a single brick. The other extreme is tons of tiny pieces and nothing else, which I suppose you'd get if you tried to approximate some purely thin-walled fractals with bricks.
I am asserting without any testing, analysis, verification or really much thought that an exponent value of around -2 is what you get when approximating a sphere with progressively smaller bricks (or other spheres, or any shape?) when using most sane tactics to generate the subdivision. It is well known that all objects in physics are approximately spherical, so you get -2 everywhere.
i finish my undergrad degree this semester, and i probably have time to publish only one more paper. so far, i've been able to reference star wars and star trek in some of my papers, but today i learned that i need to somehow find a way to connect my research to lego before i graduate so that i can reference lego in my last paper
It's not just things subject to gravity. It also works for country sizes (area), town/city population sizes (population), number of organisms (body mass), you will get this bias: the curve of the measure will follow some power law with negative exponent. This happens whenever you chunk things, so this is a strong indication that it is the chunking that's the origin of this phenomenon.
The following is a pretty intuitive way of making this make sense, if not a clinching proof. For each given amount of material, you can make a larger number of smaller things than larger things, so even if there is an equal distribution of total mass in each category, there will be a bias towards small things. For example, if we have a collection of Big Things and Small Things, where a Big Thing has twice the mass of a Small Thing, yet a given atom in this collection is equally likely to find itself in a Big Thing as it is to find itself in a Small Thing, then Small Things outnumber Big Things 2:1. For the even odds of each atom finding itself in a Big Thing and a Small Thing means that the amount of mass in all the Big Things equals the amount of mass in all the Small Things. But, of course, the amount of Big Things you can make with a given mass is half that of the amount of Small Things you can make with that same mass, so Small Things outnumber Big Things 2:1, and indeed, the distribution of the numbers of such objects goes as inverse of the mass.
This was an incredible video, thank you for making it. Your channel is quickly becoming one of my favorites. Just a point on what you said in the end, the models we have for galaxy formation actually are somewhat similar to star formation, in that they are both formed from clouds or roughly uniform clumps of matter that get too dense at one point and collapse into different sized pockets. For stars this happens in clouds of gas, but for galaxies it happens over the scale of the whole universe with dark matter halos. Halos look something like stars forming out of a nebula, and the regular matter collects in the center, forming galaxies and galaxy clusters.
Oh wow I wonder if it’s the same for gundam kits too!! I’ll try and remember to grab my kitchen scale when I start a new set 👀
I think for celestial bodies and objects it makes sense to think that the larger one is the more easily it is destroyed or converted into smaller objects. Just how when you bite your lip it swells, leading to it being more susceptible to biting again, which leads to a chain reaction, celestial bodies must act in the same way. The probability that they will be converted to smaller, less massive objects increases with increasing size.
Economics student here. I tried it on different metrics like gdp, gdp per person, city size, gdp per city and it always turned out between 2 and 2.5
At least three reasons for the LEGO:
- Small pieces are more versatile than big pieces
- Small pieces can combine to make the big pieces but not vice versa
- There a lot more permutations of big pieces than small pieces and they need to make a separate mold for each piece
I wonder if the square-cube law somehow plays into how the lego sets' distribution occurs. The centre of the model is where basically none of the surface (surface area) of the piece is visible, so its only really being chosen for its size and ability to provide structure (and thus mass). But as you move out, the surface of each piece is more and more visible, and thus the detail of it is more important and its size and ability to provide structure less-so. I'm not really sure how the maths and algebra of the square-cube law could be manipulated to get the -2 exponent, but i feel like there might be something there
I've got no idea what dark magic you performed there with logarithms, but hey, cool video!
That's nuts. Even without weighing them I can tell the objects on my desk follow this rule. A small sampling:
1 desk
2 monitors
a keyboard, a mouse, and 2 speakers
8 cups and containers
two dozen pencils, pens, and markers
a set of 42 screwdriver bits, two tubes of glue, 2 removable erasers for mechanical pencils, misc junk
i was a subscriber from 300 subs! nice job growing your channel! :D
I'm a physics student, so this law reminds me of one I'm very familiar with: Gauss's Law! Typically it's used to relate a conserved "quantity" within a boundary to a "throughput" passing through the boundary. Mathematically, this says: kQ = TA, where Q is the quantity within the boundary, T is the throughput, and A is the total area of the boundary. (k is a scaling constant) For "throughputs" that only depend on distance, the area can be taken to be the surface area of a sphere, proportional to r^2. This is precisely the inverse square law! Another form of this is the inverse cube law.
Some examples of things that follow this pattern are fluid flow, light, the electric and magnetic fields, sound amplitude, heat flow, stresses in buildings, mass, and momentum. Conserved things *have to* follow the square cube law. This is because we live in a 3 dimensional space, so "areas" are the number of dimensions minus 1. To cancel out the flow spreading out over an area, its intensity must be proportional to 1/(d-1).
We observe images of things as areas, and the universe has a fairly uniform density, so this result makes sense to me. By Gauss's law, I expect a classification of groups of things in a 2d universe to follow a 1/r distribution, and things in a 4d universe to follow 1/r^3. This is testable too, since we can directly measure 2d realms like the game Agario, or city plans and look to see if things fit that 1/r pattern.
Since there is a bunch of different designs for cutties, going from butter knives to swords, this could be interesting. Same with bowls in a restaurant. Or House size across a state/province.
I think the reason for the distribution is much as you said where we start with larger pieces and use smaller pieces to detail. You can picture most Lego construction like a fractal where the face of the shape has exponentially more pieces than the interior and you could theoretically scale these models infinitely and see the edge's detail grow infinitely.
What might be a similarity is the counting process. For stars, we are counting galaxies AND stars, which is essentially double-counting. While not true for legos, there is a somewhat similar process there, with small pieces often being parts of larger pieces (not exactly, but oh well dark matter is also a big mystery). Maybe that’s a similarity? Also: I wonder what the ikea catalogue looks like!
To me this sounds like a matter of chaos or entropy. With less chaos we see more organised structures (such as lego creations or solar systems), while more chaos will show less big structures and many more small things.
For the article from “The Pedagogical Representation of Mass Functions with LEGO and their Origin” I had the PI as a professor for undergrad and the Kyle was a friend of mine. We were both presenting at a conference at our school and he was telling me how many lego boxes he went through for his research.
It would be awesome if this video reached any of them! Please let them know that there's a video almost entirely inspired by their paper!
@@physicsforthebirds I sent them a email and they finally got back to me the PI liked your video and wants to reach out to you!
Awesome video! Now I wanna weigh everything in my house and check the distribution lol. One thing that raises my interest is that many of the bigger objects at home for example are made out of smaller objects. What happens to the distributions if we only count the smallest separable pieces, what happens if we only count the final whole objects, and what happens when you count both, thus effectively counting small pieces as small objects and once as the bigger whole object
I would have to assume it's related to Benford's Law. Or at least, the data underlying Benford's Law has similar properties of distribution.
I'm pretty confident this has nothing to do with any design standard at lego.
it's serendipity! I'd bet if you weighed everything you have in your house, or the sticks in your yard, or some other group of objects with sufficient variability in size, you would find similar results. just intuitively, small things are always more common.
Fantastic video as usual!
love the video! I love fun little scientific explorations into things that I would never have encountered otherwise, especially in the format you present in!