New sphere-packing record stems from an unexpected source
quantamagazine.org
quantamagazine.org
There are three choices, really:
You can give a quick explanation in terms they understand, which makes your job sound easy and makes them wonder how anybody gets paid to do it.
You can explain what you do and why it's important in terms they understand, but it'll take so long they'll get bored and wish they hadn't asked.
Or you can give a quick explanation using jargon that they don't understand, which will leave them bored but impressed, which is the best of the bad options.
I always opt for excruciating detail because it's what I enjoy the most.
Sounds like none of the people you answered, in excruciating detail, cared to warn other people about what would happen if they asked you.
You can give someone a simple explanation of quantum chromodynamics and have them walk away feeling like they learned something, but only by glossing over or misrepresenting critical details. You’d basically just be lying to them.
Only in the end to reveal the belt is truely conceptualized and does not formally exist. The belt is an accurate visual representation and teaching tool, but the actual mechanics emerge from data latches and the timing of releasing the data, etc.
I thought it was helpful.
Is this an asynchronous architecture CPU?
The belt moves once per cycle, if that wasn't clear? He says the word "cycle" (and measures latency in cycles) a lot.
https://www.nature.com/immersive/d41586-025-00269-y/index.ht...
Also, it’s hilarious to get comments like this voted down by non-experts who assume this must be an outsider’s uninformed point of view.
I have a physics degree and I studied the origins and history of quantum mechanics. Its “founding fathers” all admitted that it’s a bunch of guesswork and that the models we have are arbitrary and lack something essential needed for proper understanding.
The math that describes it is known precisely. Specific implications of this are known. There's no information transfer, there's no time delay, etc.
And yet lay people keep incorrectly thinking it can be used for communication. Because lay-audience descriptions by experts keep using words that imply causality and information transfer.
This is not a failure of the experts to understand what's going on. It's a failure to translate that understanding to ordinary language. Because ordinary language is not suited for it.
> Its “founding fathers” all admitted that it’s a bunch of guesswork and that the models we have are arbitrary and lack something essential needed for proper understanding.
We don't have a model of why it works / if there's a more comprehensible layer of reality below it. But it's characterized well enough that we can make practical useful things with it.
> We don't have a model of why it works / if there's a more comprehensible layer of reality below it.
Counterpoint:
You’ve just admitted they don’t understand what’s going on — they merely have descriptive statistics. No different than a DNN that spits out incomprehensible but accurate answers.
So this is an example affirming that QM isn’t understood.
But it sounds like your objection is that reality isn't allowed to be described by something as weird as complex values that you multiply to get probabilities, so there necessarily must be another layer down that would be more amenable to lay descriptions?
My point is that their models are fitted tensors/probability distributions, often retuned to fit new data (eg, the epicyclic nature of collider correction terms) — the same as fitting a DNN would be.
Their inability to describe what is happening is precisely the same as in the DNN case.
As you increase the detail of a description, it reaches a point where nothing is missing.
In the same way that a high number of epicycles was evidence our theory of geocentrism was wrong — even though adding epicycles did compute increasingly accurate results.
This is rather a problem of the standard model. Physicists will immediately admit that something is missing there, and they are incredibly eager to find a better model. But basically every good attempt that they could come up with (e.g. supersymmetric extensions of the standard model; but I'm not a physicist) has by now (at least modtly) been falsified by accelerator experiments.
Entanglement is just a statistical effect in our measurements — we can’t say what is happening or why that occurs. We can calculate that effect because we’ve fitted models, but that’s it.
Similarly, to predict proton collisions, you need to add a bunch of corrective epicycles (“virtual quarks”) to get what we measure out of the basic theory. But adding such corrections is just curve fitting via adding terms in a basis to match measurement. Again, we can’t say what is happening or why that occurs.
We have great approximators that produce accurate and precise results — but we don’t have a model of what and why, hence we don’t understand QM.
Bell's theorem was a prediction from math before people found ways to measure and confirm it. A model based on fitting to observations would have happened in the other order.
We’d already had models which said that certain quantities were conserved in a system — and entanglement says that is true of certain systems with multiple particles.
To repeat myself:
> Entanglement is just a statistical effect in our measurements — we can’t say what is happening or why that occurs.
Bell’s inequality is just a way to measure that correlation, ie, statistical effect — and I think it’s supporting my point the way to measure entanglement is via statistical effect.
ER=EPR is an example of a model that tries to explain what and why of entanglement.
But it is impractical to use in most situations so major simplifications are required.
The correction factors that you mention are the result of undoing some of those simplifications, sometimes by including more of the basic theory and sometimes by saying something like "we know that we ignored something important here and it has to have this shape but we can only kinda sort measure how big it might be because it's too hard to actually calculate".
So, what's a horse? Well, you look at it: it’s this big animal, standing on four legs, with muscles rippling under its skin, breathing steam into the cold air. And already — that’s amazing. Because somehow, inside that animal, grass gets turned into motion. Just grass! It eats plants, and then it runs like the wind.
Now, let’s dig deeper. You see those legs? Bones and tendons and muscles working like pulleys and levers — a beautiful system of mechanical engineering, except it evolved all by itself, over millions of years. The hoof? That’s a toe — it’s walking on its fingernail, basically — modified for speed and power.
And what about the brain? That horse is aware. It makes decisions. It gets scared, or curious. It remembers. It can learn. Inside that head is a network of neurons, just like yours, firing electricity and sending chemical messages. But it doesn’t talk. So we don’t know exactly what it thinks — but we know it does think, in its own horselike way.
The skin and hair? Cells growing in patterns, each one following instructions written in a long molecule called DNA. And where’d that come from? From the horse’s parents — and theirs, all the way back to a small, many-toed creature millions of years ago.
So the horse — it’s not just a horse. It’s a machine, a chemical plant, a thinking animal, a product of evolution, and a living example of how life organizes matter into something astonishing. And what’s really amazing is, we’re just scratching the surface. There’s still so much we don’t know. And that is the fun of it!
The quip you're referring to was meant to be inspirational. It doesn't pass even the slightest logical scrutiny when taken at its literal meaning. Please. (Apologies if this was just a reference without any further rhetorical intent though.)
It's like claiming that hashes are unique fingerprints. No, they aren't, they mathematically cannot be. Or like claiming how movie or video game trailers should be "perfectly representative" - once again, by definition, they cannot be. It's trivial to see this.
It's easy to give directions to somewhere near where you currently are -- "Just head down the road, it's the second left, then 3 doors down".
When giving directions to a far-away place you either have to get less accurate "it's on the other side of the world", or they get really, really long. Unless of course they already know the layout of the land -- "You already know Amy's house, over in Algebra Land? Oh, then it's just down the road, fourth left, six doors down".
People often seem cleverer because they know the layout of some really obscure land, but often it's just because people have never been anywhere near it. I have a joke about my research where I say, "A full explanation isn't that hard to explain, it's just long. About 4 hours probably. Are you interested?" So far, I've had 3 people take me up on that, and they all seemed to have an understanding once I'd finished (or, they really really wanted to escape).
"I teach computers what sounds different aminals make."
What is the problem with this?
Most jobs, when simplified, sound like "anybody can do it". I think it's generally understood among adults who have been in the workforce that, no, in fact anybody cannot do it.
A somewhat ungenerous characterization of the attitude may be something like the Rocket Scientist vs Brain Surgeon sketch - https://www.youtube.com/watch?v=THNPmhBl-8I
But we should also acknowledge that there's an entire culture built around valuing people and their time relative to one's perception of their "importance", that this culture can influence one's earning potential and acquisition of material possessions, and that many people do care about things like "seeming important" or moving upwards in this hierarchy as a result.
The first option (explaining things simply) might make your job sound easy to a very small minority of extremely uneducated, under-stimulated people, who also have unaddressed insecurities around their own intelligence. But that’s not most humans.
Moderately-to-very intelligent people appreciate how difficult (and useful) it is to explain complex things simply. Hell, most “dumb” people understand, recognize, and appreciate this ability. Honestly, I think not appreciating simple explanations indicates both low mathematical/logical and social/emotional intelligence. Which makes explaining things simply a useful filter for, well… people that I wouldn’t get along with anyway.
With all that said, I prefer to first explain my job in an “explain like I’m 5” style and, if the other party indicates interest, add detail and jargon, taking into account related concepts that may already be familiar to them. If you take them into account, they won’t get bored when you go into detail.
This isn't going to be most humans you encounter if you're in the HN demographic, but that's a bubble. It does describe most people in the world.
For me, it's quite the opposite: such a choice demonstrates that they their prior is that I'm sufficiently smart and knowledgable to be likely able to understand this explanation - which I rather consider to be a praise. :-)
Sometimes, explaining your issue to a random person leads you to a solution. They don't even need to have any experience with the same or a similar issue; indeed, sometimes it's better if they don't! Often turns out you don't need someone who can respond, so explaining to an object like a rubber duck will do.
Intelligence, in the traditional sense, also involves understanding when to give up. Part of "emotional intelligence" is judging whether the other party actually cares about what you're about to say.
It is just your choice. I'd prefer a short answer full of jargon. It gives people the opportunity to clarify what they want to ask. Do they really want to know details? Or they want a rough idea of an answer? Or they just filling silence with small talk?
Though other times, when I really want to talk about it, I'd go with some ELI5 explanation, while watching people, are they interested or not?
> Honestly, I think not appreciating simple explanations indicates both low mathematical/logical and social/emotional intelligence.
It can be. But mostly it is not. People are sending signals by choosing one form of the answer or another, you just need to decode their signals. And it will be better, if you don't jump to conclusions about their persistent psychological traits, based on the first impression.
What's wrong with asking their level of experience with the topic?
Sure, with parents you know the level. I'm talking "other strangers" you meet outside of a context where some familiarity would be expected (like at a conference one might assume at least some form of knowledge and ability to just have the other person ask about specific jargon they don't know).
But at the parents dinner party, that other guy may or may not be in your line of work. Just ask them.
Nothing. That's precisely the point. Giving a wall of jargon, isn't asking if someone is familiar.
meta: I hate how hn culture lately makes people assume that what was obviously a statement in support of a premise was somehow an argument. I get it, around here it is rational enough. Still weird.
https://web.archive.org/web/20230228154639im_/https://assets...
1) Cats do not really think that much about us at all, except for thoughts like - "oh no! it's about to attack! wait no, it's fine, relax..." or "will it feed me if I shout at it?" or "it's sitting down, perhaps I feel like sitting on it"
2) Their thoughts about us revolve around our weird lack of fur, the strange way we never clean ourselves by licking, and how bad we are at catching small animals to eat.
Or some of us with our tail on backwards.
I wonder if it’s anything like what I feel when I watch them sniff the same bush for a seemingly endless amount of time like it’s the most interesting thing in the world.
That is 90% of the professors I asked questions to. If they go full jargon and don't want to explain any of it, they don't want you near them ( or they want you to improve before even having a conversation ).
The counter-camp is "What would Landau and Lifshitz do?" :-)
---
For those who are out of the loop:
> https://en.wikipedia.org/wiki/Course_of_Theoretical_Physics
"The presentation of material is advanced and typically considered suitable for graduate-level study."
That remark reminds me of all the praise heaped by commenters onto videos that explain complex topics glibly. Like "I've been struggling to understand this for 20 years, until this video", etc.
In other words, ICP were not so far off base when they asked “magnets, how do they work?”
To me, it rather tells: "I consider you to be likely to be sufficiently smart and knowledgable to understand this topic if you put in some effort: do you want to learn some cool stuff which otherwise would demand a lot of literature research to learn? And since I already hinted that I consider you to be smart and knowledgable: would you like to teach me some cool, complicated stuff, too?"
But it is by far more common that it means "piss off you insect".
This is always the right answer. It is the only answer that respects the listener and contains a seed to further conversation.
"I'm a mathematician, I study how shapes fit together, which surprisingly, is being used for new methods of secure communication by so and so university, but I just love the math"
https://duckduckgo.com/?q=dance+your+phd&ia=videos&iax=video...
There is another:
Give away as little information as you can about it.
Don’t say or agree that it’s secret or that you can’t talk about it- just be tight-lipped, and don’t divulge.
If you do it right, you will seem mysterious.
If you do it wrong, they probably won’t talk to you much again.
Win-win.
— What’s your security clearance?
Eg you can focus on what you actually do, or you can focus on the benefits you bring to other people.
What's wrong with this? Making it look easy is why you get paid for it.
- Some people will not care / be dismissive of what I have to say. I probably don't want to talk to these people much.
- Some people will be interested! I probably will like these people.
If I use technical jargon, I am optimizing to impress people I don't really care about impressing - and I will be pushing away the people that I would actually be interested to spend time with.
If I speak respectfully, i.e. the simple explanation, it will resonate more with the people I like. I will push away the people who don't care, but I didn't really want to talk with them anyways.
Yes, don't fall into this trap. The other two options are still better. Everyone says, "no, no, I really want to know" and then tunes out two minutes later; then four minutes later they start doing the George Carlin lean: "Surgery! I am having my ears sewn shut!".
Blowing smoke around simple things to gatekeep them is not impressive and not cute.
I personally say I work on Bluetooth support for Google Home assistant devices. "It's like Alexa, but Google.
Even if you work on some absurdly down stack thing, this seems to work. You work on making sure the internet is as fast as possible, or files are stored in the cloud properly, or the graphics on your computer are displayed correctly.
For reasons that I care not to ask people get seriously annoyed by that.
People that are interested can ask either to give more details on what I have explained, or what about the rest. If they are not interested, they say something and I usually ask what about them, no hard feelings.
It works smoothly for me.
(not sure about convex shapes)
Convex shapes, well, annoyingly it's too broad. It has way more applications than sphere packings but it's hard to pick a good example. It's like trying to explain you design screwdrivers to someone who doesn't know what a screw is.
You ask me what it is I do. Well, actually, you know,
I'm partly a liaison man, and partly P.R.O.
Essentially, I integrate the current export drive.
And basically I'm viable from ten o'clock till five.
I have yet to figure out a way to tell people what my business is in a way that is even slightly accessible. Everything about it is so esoteric and multiple steps removed from regular life. It's not necessarily complex, it just contains a ton of details that the average person has no familiar contact with, and don't really have everyday analogues.
> I have yet to figure out a way to tell people what my business is in a way that is even slightly accessible.
You ... just did? In a remarkable short, concise, and very accessible way. I can ask as many follow up questions as I want and we might even have an engaging conversation. Sounds interesting!
I also obfuscated it a bit by giving the most general name just for privacy reasons since not many people do it. But rest assured it is a "Retro Encabulator" type machine, and as you add details it just becomes more and more alien.
This is not at all what I do, but its similar esoteric-ness to "I make differential gear sets for calibrating ion trap interferometry systems". A collection of words where every one of them the average person struggles to place.
Really if we're at a party that's more than enough unless I want to ask you more and you want to talk more about it. If you were a lawyer I'd probably ask what area of law that I probably stop and talk about something else. So I agree with others that you said was a very good distillation of what you do to the level that most people probably care about
How do I concisely describe the long chain between a fundamental research and something tangible?
This is a suitable description of possibly 70 % of all jobs.
convex hulls your car
Explain it from the perspective of, "well, in order to get XYZ done, we are frustrated by it being hard, so we make an easy guess .. we try thinking about the problem in this crude way way because that's easy to think about, and then we make ABC because we know about ABC's ... and we are excited when using it gets us closer to working than anything else we've tried before".
Emotion-laden explanations are a viable way to explain to non-techs. They may be more comfortable thinking emotionally, whereas we are steeped in the logic and sometimes mathematics of our practice. So we must reintroduce emotion into the explanations.
It worked for me, explaining to my family, they followed on and actually understood.
Nice example of how weird large-dimensional space is. Apparently, when smart minds were asked to put as many 100-dimensional oranges in a 100-dimensional crate as they could, so far, the best they managed to do was fill less than 1% of its space with oranges, and decades of searching couldn’t find a spot to put another one.
For this aside I crave a citation.
When n=1 the sphere fit is 100% as both simplex and sphere are congruent in that dimension. And dismissing n=0 as degenerate (fit is undefined there I suppose: dividing by zero measure and all that) that (first) dimension should be maximal with a steady decline thereafter thus also monotonic.
This relationship doesn't happen to the hypercube btw. Really, it is about the definition of each object. The volume of the hypercube just continues to grow. So of course the ratio is going to explode...
As an extra fun tidbit, I'll add that when we work with statistics some extra wildness appears. For example, there is a huge difference between the geometry of the uniform distribution and the gaussian (normal) distribution, both of which can be thought of as spheres. Take any two points in each distribution and draw a line connecting them and interpolate along that line. For the unit distribution, everything will work as expected. But for the gaussian distribution you'll find that your interpolated points are not representative of the distribution! That's because the normal distribution is "hollow". In math speak, we say "the density lies along the shell." Instead, you have to interpolate along the geodesic. Which is a fancy word to mean the definition of a line but aware of the geometry (i.e. you're traveling on the surface). Easiest way to visualize this is thinking about interpolating between two cities on Earth. If you draw a straight line you're gonna get a lot of dirt. Instead, if you interpolate along the surface you're going to get much better results, even if that includes ocean, barren land, and... some cities and towns and other things. That's a lot more representative than what's underground.
Citation(s):
> how weird large-dimensional space is
I really like the animation in this Math Overflow post[0], because it has a lot of hidden complexity that most people don't think about. The animation is actually an illusion, and you are "hallucinating". That top image projecting a cube down onto a plane? Well... that isn't a cube. We've already projected the cube into 2D! Technically this is 3D. But the 3rd dimension isn't a spacial dimension, it is a time dimension. Which itself is a helpful lesson in learning about the abstraction of dimensions! So we hallucinate a cube, rotating, and then see the projected image on a plane, which we hallucinate as a square that isn't skewed but instead has depth. This is all rather wild in of itself.
The truth is that we struggle to imagine 2D! And most people will claim to be able to visualize 2D and the claim will go uncontested.
If you haven't read Flatland[1], I'd encourage everyone to do so. A lot of people get it wrong. They read it as an analogy 1 dimension down. Where we 3 dimensional creatures are analogous to the 2D creatures and a 4D creature would be as baffling as a 3D creature is to the Flatlander. While that is true, there is a trick being played on you. You think understanding 2D is really easy. But I guarantee you what you're visualizing is inaccurate. Frankly, the book isn't perfectly accurate either.
But really put yourself in the Flatlander's shoes. In a real Flatlander's shoes, not the ones of the book. Be the Square Flatlander and imagine yourself looking at a Triangle. What do you see? I'm betting it is a line? But this is incorrect. You've given it thickness, you've given it a third dimension. Try this again and again, adding more depth and challenging yourself to imagine a real Flatland. You'll find you can't.
Instead, we can visualize and reason about a 2D space embedded within 3D. You might say I'm being nitpicky here, but if I weren't then it would be perfectly fine to say that this[2,3] is a 4-dimensional hypercube instead of a representation of a 4D hypercube.
I actually think understanding this goes a long way to help understanding very high dimensions. If you are forced to face the great difficulty of accurately visualizing one more or one fewer dimension, you are less likely to fool yourself when trying to reason about much higher dimensions.
And as Feynman once said:
The first principle is that you must not fool yourself and you are the easiest person to fool.
[1] http://www.geom.uiuc.edu/~banchoff/Flatland/
[2] https://en.wikipedia.org/wiki/Tesseract#/media/File:8-cell-s...
[3] Good video of Carl Sagan where he holds a 3D projection of the hypercube. The shadow. But anything I show you has to be embedded in 2D... He picks it up at 6:20 https://www.youtube.com/watch?v=UnURElCzGc0
EDIT: groped -> grouped
Here's some work on low-latency neural compression that you might find interesting: https://arxiv.org/abs/2107.03312
Another thing that I'm sure you explored, and I'd love to hear how it went, would be to rearrange the elements in the vectors such that perhaps the denser parts could be more contiguous, and the sparser parts could be more contiguous, on average. That sounds like something that would be easier to compress. Were the distributions such that a rearrangement like this might have been possible? Or were they very evenly distributed?
I.e. could you have rearranged a Gaussian-like distribution into a Poisson-like distribution?
I also remember trying to fit a distribution so that I can generate synthetic data (not for a lack of data, but more for understanding the problem space better). The synthetic data quantized pretty differently - my guess is that it's because of random areas of density and sparsity.
I'm not quite following your exact rearranging idea though. Not sure if the above answers the question.
E.g.: Suppose the data has high-order structure but is locally uniform (very common, comes about because of noise-inducing processes). Compute and store centroids. Those are more uniform than your underlying data, and since you don't have many it doesn't really matter anyway. Each vector is stored as a centroid index and a vector offset (SoA, not AoS). The indices are compressible with your favorite entropic integer scheme (if you don't need to preserve order you can do better), and the offsets are now approximately uniform by assumption, so you can use your favorite sphere strategy from the literature.
But I can imagine that drawing connections between different branches of maths would be especially powerful, yes
Try doing that in the modern academic environment tho..
Those numbers sound wild. For various comms systems does this mean several orders of magnitude bandwidth improvement or power reduction?
So it's only helpful for naturally high dimensional objects. Digital objects do not have a natural dimension (byte length), so you can choose a small dimension.
Due to the added variable aka axis, you have increased the size of complexity.
2d shapes packed in a 2d boundary vs 3d objects in 3d space. The difference is fitting quarters on a paper vs marbles in a perfect cube.
Now imagine having to find the most optimal method of packing for objects of Nth degree in Nth constraint. For example packing object that have 256 dimensional or variables into a constraint that also is 256 of complexity.
I feel as your dimension aka variable increases....the amount of information to compute grows quite quickly.
We can rule out some things as non optimal or non perfect, but we can also get close to perfection via trial and error. I see this as an example of the traveling salesman issue.
Do you stick with a randomly selected answer, do you go with the current most optimal solution, or do you invest time and effort into finding a new solution but you risk finding a worse solution. At the end of the day is the packing efficient gains worth the computational complexity of N dimension of N constraint given it will take an unknown amount of time to find a more efficient packing solution, and the new solution could be anywhere from 0.1% to 80% more efficient.
Also, the toolkit one uses in convex geometry, especially some of the harmonic analysis tools are quite handy in the study of sphere packing.
So "unexpected"? Not quite.
I agree with this and I'm not even a mathematician, I've seen convex hull algorithms pop up in unexpected places to solve problems I would never have thought of using convex hull algorithms for, like a paper on automatic palette decomposition of images.
https://www.rose-hulman.edu/class/cs/csse451/Papers/DILvGRB....
It is conjectured that in higher dimensions, the densest packing is always non-lattice. The rationale being that there is just not enough symmetry in such spaces.
See also https://www.ams.org/journals/notices/201702/rnoti-p102.pdf
For other dimensions, this is an open question; it seems unlikely to be true in general. For some dimensions the densest known irregular packing is denser than the densest known regular packing.
I thought that was one of the important results from the paper, the most efficient packing for all dimensions is symmetrical again and this increase was significant enough it seems unlikely that existing non-symmetrical methods will be able to beat it.
> His result has also revived a debate in the field about the nature of the optimal packing in arbitrarily high dimensions. For a while, mathematicians considered highly symmetric, lattice-based packings to be the best way to arrange spheres as densely as possible. But in 2023, a team found a packing that didn’t rely neatly on a repeating lattice; before Klartag’s result, it was the record to beat. Some mathematicians saw it as evidence that more disorder was needed in the search for an optimal sphere packing.
Clearly, this improvement doesn't apply to the few dimensions for which we already proven optimal packing, but the proof was general.
This is different from asking about the best-known specific packing in n dimensions when n = 7, 10, 19, or whatever.
In a few of the specific low dimensional cases, my impression is that the best currently known packing is not regular, and I don't see any evidence that this has changed. Perhaps this new method will give people a way to beat those records, or perhaps not.
The comments pointed out that anthropologist did not know that boiling was possible before the invention of pottery. Another comment pointed out that science teachers knew that it was possible because that was something they would do in class.
Final comment was about how people ReDiscover things in different fields - - like the trapezoidal rule for integration being discovered by someone studying glucose.
This is just yet another example of how bringing expertise from a different area can help.
1. The 2015 paper doesn’t indicate that it was unknown that you can boil water without pottery. It is meant to alert people to an already known fact. Maybe silly, but I do think there’s a distinction here — it’s not that the pinnacle of human knowledge was missing this, it’s that common sense was missing it. I’d compare this to the article “things SDEs assume about names”. Edit: then again, playing devils advocate against myself, the first paper does say “…sub-boiling temperatures than _thus far acknowledged_” (emphasis mine), which would seem to indicate that this was more than just a common misunderstanding.
2. The 2015 paper is not about boiling water in general, it’s about boiling it in a specific way. (Whereas the initial discussion was about boiling water by any method)
Someone can take this challenge to provide a more secure and reliable communication systems hopefully with more energy efficiency, very much an exciting research direction.
[1] https://en.wikipedia.org/wiki/Post-quantum_cryptography
[2] https://en.wikipedia.org/wiki/Random_close_pack#For_spheres
But he did it in a better way, randomly expanding ellipses. And when you're dealing with hundreds of dimensions, it's incredibly easy to be montecarlo-ing in the wrong directions...
The article does mention that phenomenon, deceptive simplicity.
In context of packing problem, it's a bit meta to me...
An LLM contains a k-dimensional packing of known knowledge. This packing is highly inefficient because it has holes and unbridged dimensions. By injecting random seed (prompts) into the LLM probability space, it gets perturbed. Sometimes this perturbation fills a hole in the packing and/or connects two adjacent units in way nobody thought of before because it wasn't fashionable any more, or wasn't top of mind. Thus new knowledge is created within the same k-dimensional box through a novel joining-of of existing know-how.
From the article:
> Klartag had broken open a central problem in the world of lattices and sphere packing after just a few months of study and a few weeks of proof writing. “It feels almost unfair,” he said. But that’s often how mathematics works: Sometimes all a sticky problem needs is a few fresh ideas, and venturing outside one’s immediate field can be rewarding. Klartag’s familiarity with convex geometry, usually a separate area of study, turned out to be just what the problem required. “This idea was at the top of my mind because of my work,” he said. “It was obvious to me that this was something I could try.”
Sounds like the technique is for high-dimensional ellipsoids. It relies on putting them on a grid, shrinking, then expanding according to some rules. Evidently this can produce efficient packing arrangements.
I don't think there's any shocking result ("record") for literal sphere packing. I actually encountered this in research when dynamically constructing a codebook for an error-correcting code. The problem reduces to sphere packing in N-dim space. With less efficient, naive approaches, I was able to get results that were good enough and it didn't seem to matter for what I was doing. But it's cool that someone is working on it.
A better title would have been something like: "Shrink-and-grow technique for efficiently packing n-dimensional spheres"
I think something like "Hypertopological Constriction-Expansion Dynamics in Quasistatic R^n-Ball Conglomeration" would be even more apt.