1/16/2025 at 11:34:57 AM
I'd really love to know what the mathematicians are actually doing when they work this stuff out? Is it all on computers now? Can they somehow visualize 24-dimensional-sphere-packings in their minds? Are they maybe rigorously checking results of a 'test function' that tells them they found a correct/optimal packing? I would love to know more about what the day-to-day work involved in this type of research actually would be!by danwills
1/16/2025 at 11:54:11 AM
> Is it all on computers now?Most modern math is certainly not "all on computers" and in general not even "mostly on computers". There are definitely proofs for things like testing large spaces exhaustively which are sped up by computers (see the https://en.wikipedia.org/wiki/Four_color_theorem) and definitely for things like visualization (probably one of the oldest uses of computers for math), but usually the real work goes into how math has always been done: identifying patterns and abusing symmetries.
For this one explicitly, if you read through the paper you'll find the statement that the main theorem presented here "does not depend on any computer calculations. However, we have made available files with explicit coordinates for our kissing configurations"
by terminalbraid
1/16/2025 at 9:31:37 PM
It really depends though. Even in something like knot theory, that one might consider to be a very "pure" area, there's still a lot of computation involved that can be automated by computers.by viccis
1/16/2025 at 1:24:09 PM
The kind of intuition you gain for higher dimension tends not to be visual. It is more that you learn a bunch of tools and these in turn build intuition. For example high dimensional spheres are "pointy" and most of their volume are near their surface. These ideas can be defined rigorously and are important and useful. For medium dimension there are usually specific facts that you exploit. In my own work stuff like "How often do you expect random walks to intersect" is very important (and dependent on dimension).by iNic
1/16/2025 at 2:07:24 PM
> For example high dimensional spheres are "pointy" and most of their volume are near their surfaceI had a visceral reaction to this. In what sense can a sphere be considered pointy? Almost by definition, it is the volume that minimizes surface area, in any number of dimensions.
I can see how in higher dimensions e.g. a hypersphere has much lower volume than a hypercube. But that's not because the hypersphere became pointy, it's because the corners of the hypercube are increasingly more voluminous relative to the volume of the hypersphere, right?
by david-gpu
1/16/2025 at 4:09:35 PM
There is a standard thought experiment where you start with a hypercube of side-length 2, centered at the origin. You then place a radius 1 sphere on each vertex of this hypercube. The question then becomes: what is the largest sphere you can place at the origin so that it is "contained" by the other spheres. As it turns out in like dimension 6 or so the radius of the center sphere exceeds 1. It will actually poke out arbitrarily far (while still being restricted by the corner spheres).by iNic
1/16/2025 at 8:34:27 PM
I hear this point parroted all of the time, but I think it is a misunderstanding and a poor visualization. Consider the same situation, but instead of focusing on the radius of the center sphere, focus on the distance between the spheres on the corners to the origin. For 1-dimension, these 'spheres' are unit intervals and so the distance is 1 (Central radius is 0). For 2-dimensions, these are circles at a distance of root(3) (Central radius is root(2)-1). 3-D: root(3) (Central radius is root(3)-1). Etc. So, it isn't the central circle getting more 'pointy' allowing the central radius to increase, but rather that the corner circles are getting further from the origin, allowing larger N-spheres (increasing proportional to the root of N). Thus, pointy is not the right way to conceptualize these spheres. For the more visual folk, I would recommend drawing this out and you can see this in action. More clearly, if a sphere became 'spikey' then the distance on the surface of the spike should be further than a neighboring point, which is NOT the case. Not trying to attack you, I just see this same point over and over and think that this warrants more thoughtby carltg_
1/17/2025 at 9:08:26 AM
Yes that is true, but there are other ways to see spiky-like behavior.First, the volume of spheres (or balls rather) in higher dimensions goes to zero as the dimension grows. Said another way, to keep unit volume on a ball you need to grow the radius more and more (which I interpret as spiky).
Second, the volume of spherical caps grows like ~exp(- d h^2 /2), in particular the caps lose volume fast in higher dimensions. To interpret this as "spikyness" I like to visualize it as two balls intersecting (which is just 2x the cap volume). If they are of the same radius, but their centers are just slightly off their intersection volume goes to zero quickly!
by iNic
1/16/2025 at 8:34:16 PM
Yes, but that can be better understood as the hypercube becoming more pointy, not the sphere. And it's true; the cube's vertices get arbitrarily far from the origin, while the centers of its faces stay at ±1.There are other ways in which a hypersphere can be considered "pointy", though; for example, consider a point lying on the surface being moved some epsilon distance to a random direction. As the dimension increases, the probability that the point ends up inside the sphere approaches zero – the sphere spans a smaller and smaller fraction of the "sky".
by Sharlin
1/16/2025 at 8:44:03 PM
Specifically, of course, d = sqrt(N), where N is dimension and d is distance of a vertex of the unit hypercube from the origin.by senderista
1/16/2025 at 2:27:41 PM
https://news.ycombinator.com/item?id=3995615 (both article and comments) describe various ways of looking at this - and there are many implications for machine learning e.g. https://news.ycombinator.com/item?id=3995964 !by btown
1/16/2025 at 6:55:49 PM
I remember learning about the probability of returning to the origin in a 2D random walk versus a 3D random walk when I took stochastic processes. After we proved with probability 1 you return to the origin in a 2D walk (and with probability 0 you return in 3D) my professor said "that's why you hand a drunk man the keys to a car and not an airplane when he leaves the bar". After checking wikipedia it looks like he riffed off this quote from Shizuo Kakutani: "A drunk man will find his way home, but a drunk bird may get lost forever".by jochi427
1/16/2025 at 8:31:06 PM
That's interesting, about the probability being zero in 3D. Is this on an integer lattice? The source that cannot be cited on HN without loss of karma says that the probability of returning to the origin in Z^3 is approximately 0.34.I don't see how it could possibly be zero, even for reals, unless you're relying on the idea that the probability of any given real emerging from a uniform RNG is zero. That would seem to apply in 2D as well.
by CamperBob2
1/17/2025 at 2:27:39 AM
Here's how to formulate the question in continuous space/time:Random walks can be defined on continuous space and time as a probability distribution on functions R -> R^n (Brownian motion in n dimensions).
We can then ask whether Brownian motion beginning at the origin will ever revisit it i.e.
Given 2D Brownian motion X such that X(0)=(0,0), the probability that there exists t>0 such that X(t)=(0,0) is 1.
Given 3D Brownian motion X such that X(0)=(0,0,0), the probability that there exists t>0 such that X(t)=(0,0,0) is 0. (This is more clearly true when it doesn't begin at the origin, but it's almost certainly not at the origin at t=1, and you can divide the half open interval (0,1] into a countable number of intervals, each of which have 0 probability of passing through the origin.)
Random walks in 2D are space filling curves; random walks in 3D are not.
by penteract
1/16/2025 at 9:09:30 PM
I'm sure I am just misremembering -- it was definitely on Z^3 so I guess its actually 34%. Thanks for letting me knowby jochi427
1/16/2025 at 11:58:13 PM
They definitely don't visualize 24 dimensional spheres. When I did my PhD in pure math I found that gradually I just became comfortable working without any visual or spatial intuition and instead relying on the (algebraic and topological) machinery that had been put in place before me. Terence Tao had a nice essay [1] talking about how the final stage in becoming a professional mathematician is developing the intuition to know what is likely true or not in these very abstract spaces.I also never used a computer for anything other than latex.
[1] https://terrytao.wordpress.com/career-advice/theres-more-to-...
by jebarker
1/16/2025 at 11:54:22 AM
Likewise!In higher dimensions, are the spheres just a visual metaphor based on the 3-dimensional problem, or are mathematicians really visualising spheres with physical space between them?
Is that even a valid question, or does it just betray my inability to perceive higher dimensions?
This is fascinating and I'm in awe of the people that do this work.
by davethedevguy
1/16/2025 at 12:01:26 PM
> just a visual metaphorIt's not really a metaphor.
An n-sphere is the set of all points that are the same distance away from the same centre, in (n+1)-dimensional space. That generalises perfectly well to any number of dimensions.
In 1 dimension you get 2 points (0-sphere), in 2 dimensions you get a circle (1-sphere), in 3 dimensions you get a sphere (2-sphere), etc.
EDIT: Also, if you slice a plane through a sphere, you get a circle. If you slice a line through a circle, you get 2 points. If you slice a 3d space through a hypersphere in 4d space, do you get a normal sphere? Probably.
by jstanley
1/16/2025 at 10:43:45 PM
Note that the solid set, all points within a certain distance of the center, is called a ball: https://en.wikipedia.org/wiki/Ball_(mathematics)If the boundary is included, it's a closed ball, otherwise it's an open ball.
So the sphere is the "skin", the ball is the whole thing.
A bit different than common usage.
by JJMcJ
1/16/2025 at 12:25:56 PM
> etc.That's handwaving the answer just as you were getting to the crux of the matter. "Are mathematicians really visualising spheres with physical space between them" in higher dimensions than 3 (or maybe 4)?
From the experience of some of the bigger minds in mathematics I met during my PhD, they don't actually visualize a practical representation of the sphere in this case since that would be untenable especially in much higher dimensions, like 24 (!). They all "visualized" the equations but in ways that gave them much more insight than you or I might imagine just by looking at the text.
by close04
1/16/2025 at 1:22:41 PM
Reportedly, Geoffrey Hinton said: “To deal with a 14-dimensional space, visualize a 3-D space and say 'fourteen' to yourself very loudly. Everyone does it.”by Majromax
1/16/2025 at 2:12:20 PM
My sister is a mathematican and she used to say that if you want to understand a 24-dimensional space, you start from a generalized n-dimensional space and then set n=24.This wasn't atypical of her. She would also say that if your house is on fire then you call the firefighters, but if it is not on fire then you set it on fire, thereby reducing the problem to something that you have already solved.
by david-gpu
1/16/2025 at 8:36:13 PM
> Reportedly, Geoffrey Hinton said: “To deal with a 14-dimensional space, visualize a 3-D space and say 'fourteen' to yourself very loudly. Everyone does it.”He did. You can see / hear that line in this video from his old Coursera course.
https://youtu.be/TNhgCkYDc8M?list=PLLssT5z_DsK_gyrQ_biidwvPY...
Exactly how seriously he intended this to be taken is a matter of debate, but he definitely said it.
by mindcrime
1/16/2025 at 12:55:15 PM
I have dyscalculia so I'm always studying how people who have "math minds" work, especially because I have an strong spacial visual thinking style, i thought i should be good at thinking about physical math. When I found out they're not visualizing the stuff but instead "visualized the equations together and imaging them into new ones" - I gave up my journey into math.by neom
1/16/2025 at 2:45:36 PM
My two cents on this: I've done a lot of math, up to graduate courses in weird stuff like operator algebra. I've also read quite a bit of maths pedagogy.I've come to understand that the key thing that determines success in math is ability to compress concepts.
When young children learn arithmetic, some are able to compress addition such that it takes almost zero effort, and then they can play around with the concept in their minds. For them, taking the next step to multiplication is almost trivial.
When a college math student learns the triangle inequality, >99.99% understand it on a superficial level. But <0.01% compress it and play around with it in their minds, and can subsequently wield it like an elegant tool in surprising contexts. These are the people with "math minds".
by semi-extrinsic
1/16/2025 at 2:51:18 PM
wow.I have been posting on hackernews "I have dyscalculia" for years in hopes for a comment like this, basically praying someone like you would reply with the right "thinking framework" for me - THANK YOU! This is the first time I've heard this, thought about this, and I sort of understand what you mean, if you're able to expand on it in any way, that concept, maybe I can think how I do it in other areas I can map it? I also have dyslexia, and have not found a good strategy for phonics yet, and I'm now 40, so I'm not sure I ever will hehe :))
I even struggle with times tables because the lifting is really hard for me for some reason, it always amazes me people can do 8x12 in their heads.
by neom
1/16/2025 at 7:01:43 PM
Just a tangent, but there's a nice trick for 8 x 12.In algebra, you learn that (a - b)(a + b) = a^2 - b^2. It's not too hard to spot this when it's all variables with a little practice but it's easy to overlook that you can apply this to arithmetic too anywhere that you can rewrite a problem as (a-b)(a+b). This happens when the difference between the two numbers you're trying to multiply is even.
For a, take the halfway point between the two numbers, and for b, take half the difference between the numbers. So a = (8 + 12) / 2 = 10. b = (12 - 8) / 2 = 2.
Here, 8 = 10 - 2 and 12 = 10 + 2. So you can do something like (10 - 2)(10 + 2) = 10^2 - 2^2 = 100 - 4 = 96.
It's kind of a tossup if it's more useful on these smaller problems but it can be pretty fun to apply it to something like 17 x 23 which looks daunting on its own but 17 x 23 = (20-3)(20+3) = 20^2 - 3^2 = 400 - 9 = 391
by Cyber_Mobius
1/16/2025 at 3:31:39 PM
You're welcome :)The foundations for these concepts were laid by Piaget and Brissiaud, but most of their work is in french. In English, "Young children reinvent arithmetic" by Kamii is an excellent and practically oriented book based on Piaget's theories, that you may find useful. Although it is 250 pages.
This approach has become mainstream in maths teaching today, but unfortunately often misunderstood by teachers. The point of using different strategies to arrive at the same answer in arithmetics is NOT that children should memorize different strategies, but that they should be given as many tools as possible to increase the chance that they are able to play around with and compress the concept being learned.
The clearest expression of the concept of compression is maybe in this paper, I don't know if it helps or if it's too academic.
by semi-extrinsic
1/16/2025 at 3:37:58 PM
I should be able to chat with an llm about this paper, but my gut says you've given me the glimmer of where I need to go. This is something I've been deeply deeply frustrated about for 30 years now, I had really given up hope of ever being able to process mathematics (whatever they are) properly, it's a real task to figure out how to get someone to see how your brain work and then have them understand how to provide you with some framework to grasp what they know.Once again I wanted to thank you for slowing down and taking the time to leave this thoughtful comment, if everyone took 5 minutes to try to understand what the other person is saying to see if they can help, the world would be a considerably better place. Thank you.
by neom
1/16/2025 at 5:56:47 PM
Calculating 8x12 in my head relies on a trick / technique - they call it "chunking", I believe, in the Common Core maths curriculum that US parents get so angry about - that (I'm also in my 40s) was never demonstrated in schools when we were kids. (They tried to make me memorize the 12x table, which I couldn't, so I calculated it my way instead; took a little longer, but not so much that anyone caught on that I wasn't doing what the teacher said.) I'd like to think I was smart enough to work it out for myself, but I suspect my dad showed it to me.I'll show it to you, but first: are you able to add 80 + 16 in your head? (There's another trick to learn for that.)
by eszed
1/16/2025 at 7:56:51 PM
96, easy. Lets go, real time math tutoring in the hackernews comments, 2025 baby! :Dby neom
1/17/2025 at 3:38:11 AM
:-)12 is made up of a 10 and a 2.
What's 8 x 10? 80.
What's 8 x 2? 16.
Add 'em up? 96, baby!
They teach you to do math on paper from right to left (ones column -> tens column, etc), I find chunking works best if you approach from left to right. Like, multiply the hundreds, then the tens (and add the extra digit to the hundreds-total you already derived), then the ones place (ditto).
It's limited by your short-term memory. I can do a single-digit times anything up to maybe five digits. Two-digits by two digits, mostly. Three-digits times three digits I don't have the working memory for.
by eszed
1/17/2025 at 3:43:53 AM
Seems my math teachers in school...er..didn't. That makes sense, I know how to write math out on paper and solve it, but then my instinct has always been to reach for that method mentally, so I literally draw a pen and paper in my imagination, and look at it and do the math and it takes way too long so I just give up, this seems like I can just learn more rules and then apply them, as long as I have the rules.Thank you kindly for taking the time to teach me this! This thread has been one of the most useful things in a long ass time that's for sure. If I can ever be helpful to you, email is in the bio. :)
by neom
1/17/2025 at 6:27:24 AM
My pleasure! I'm no one's idea of a mathematician, but I enjoy employing arithmetic tricks and shortcuts like this one.A few years ago I had an in-depth conversation with a (then) sixth-grader of my acquaintance, and came away impressed with the "Common Core" way of teaching maths. His parents were frustrated with it, because it didn't match the paper-based methods of calculation they (and you and I) had been taught, but he'd learned a bunch of these sorts of tricks, and from them had derived a good (probably, if I'm honest, better than mine) intuition for arithmetic relationships.
by eszed
1/16/2025 at 9:35:20 PM
Shortly after graduating as an engineer, I remember receiving much help regarding mathematical thinking from a book by Keith Devlin titled "The Language of Mathematics: Making the Invisible Visible".What stuck with me (written from memory, so might differ somewhat from the text):
In the introductory chapter, he describes mathematics as the science of patterns. E.g. number theory deals with patterns of numbers, calculus with patterns of change, statistics with patterns of uncertainty, and geometry with patterns of shapes and spaces..
Mathematical thinking involves abstraction: you identify the salient structures & quantities and describe their relationships, discarding irrelevant details. This is kind of like how, when playing chess, you can play with physical pieces or with a board on a computer screen - the pieces themselves don't matter, it's what each piece represents and the rules of the game that matters.
Now, these relationships and quantities need to be represented somehow: this could be a diagram or formulas using some notation. There are usually different options here. Different notations can highlight or obscure structures and relationships by emphasizing certain properties and de-emphasizing others. With a good notation, certain proofs that would otherwise be cumbersome might be very short. (Note also that notations typically have rules associated with them that govern how expressions can be manipulated - these rules typically correspond in some way to the things being represented and their properties.)
Now, roughly speaking, mathematicians may study various abstract structures and relationships without caring about how these correspond to the real world. They develop frameworks, notations and tools useful in dealing with these kinds of patterns. Physicists care about which patterns describe the world we live in, using the above mathematical tools to express theories that can make predictions that correspond to things we observe in the real world. As an engineer, I take a real-world problem and identify the salient features and physical theories that apply. I then convert the problem into an abstract representation, apply the mathematical tools (informed by the relevant physical theories), and develop a solution. I then translate the mathematical solution back into real-world terms.
One example of the above in action is how Riemann geometry, the geometry of curved surfaces, was created by developing a geometry where parallel lines can cross. Later, this geometry became integral in expressing the ideas of relativity.
This maps back to the idea of "making the invisible visible": Using the language of mathematics we can describe the invisible forces of aerodynamics that cause a 400 ton aircraft suspended in the air. For the latter, we can "run the numbers" on computers to visualize airflow and the subsequent forces acting on the airframe. At various stages of design, the level of abstraction might be very course (napkin calculations, discarding a lot of detail) or very fine (taking into account many different effects).
Lastly, regarding your post of 'When I found out they're not visualizing the stuff but instead "visualized the equations together and imaging them into new ones"':
Sometimes when studying relationships between physical things you notice that there are recurring patterns in the relationships themselves. For example, the same equations crop up in certain mechanical systems than does in certain electrical ones. (In the past there were mechanical computers that have now been replaced with the familiar electronic ones). With these higher order patterns, you don't necessarily care about physical things in the real world anymore. You apply the abstraction recursively: what are the salient parts of the relationships and how do they relate. This is roughly how you can generalize things from 2 dimensions to 3 and eventually n. Like learning a language, you begin to "see" the patterns as you immerse yourself in them.
by neodimium
1/17/2025 at 3:50:00 AM
I have to wonder why your last paragraph made me feel quite uncomfortable, I can't even tell why except by the time I was done an uncomfortable feeling was inside. I appreciate your time to provide me with this much context, it seems between all the comments in this thread, I'm truly out of reasons to avoid learning math, and I suppose maybe that's why your last paragraph made me uncomfortable... I wonder what I'll see... :)Thank you again.
by neom
1/16/2025 at 1:28:33 PM
> If you slice a 3d space through a hypersphere in 4d space, do you get a normal sphere? Probably.Yep — and this will generally be the case, as the equation looks like: x1^2 + x2^2 + … + xn^2 = r^2. If you fix one dimension, you have a hyperplane perpendicular to that axis — and a sphere of one dimension lower in that hyperplane.
For four dimensions, you can sort of visualize that as x^2 + y^2 + z^2 + t^2 = r^2, where xyz are your normal 3D and t is time. From t=-r to t=r, you have it start as a point then spheres of growing size until you hit t=0, then the spheres shrink back to a point.
by zmgsabst
1/16/2025 at 11:59:58 AM
> In higher dimensions, are the spheres just a visual metaphor based on the 3-dimensional problem, or are mathematicians really visualising spheres with physical space between them?For such discrete geometry problems, high-dimensional spaces often behave "weirdly" - your geometric intuition from R^3 will often barely help you.
You thus typically rather rely on ideas such as symmetry, or calculations whether "there is still space inbetween that you can fill", or sometimes stochastic/averaging arguments to show the existence of some configuration.
by aleph_minus_one
1/16/2025 at 2:09:28 PM
In my PhD I did study systems in higher dimensions (including fractal dimensions) and it is not a metaphor and no, I did not visualize them, it was more like defining a mathematical representation of the system geometry and working on top of it.by evandrofisico
1/16/2025 at 11:59:50 AM
I have a hard time visualizing even 3 dimension, but 4 dimensions and up, I just think of it as a spreadsheet where each thing has 4 or more columns of data rather than 3. Whether a 4th column is time, spin, color, smell or yet another coordinate.by bux93
1/16/2025 at 12:22:22 PM
It sort of like the visualizable 3D "kissing spheres" is the story that makes it interesting, captivating and accessible and therefore competitive/social which makes it interesting even more, but basically at higher dims it's a bunch of equations as it is impossible to visualise on human wetware.You could do kissing starfish but no one cares as there is no lore. A bit like 125m world record doesn't matter. 100m is the thing.
This is not a knock ... it is interesting how social / tradition based maths is.
Another example is Fermat's Last Theorem. It had legendary status.
by nejsjsjsbsb
1/16/2025 at 12:26:53 PM
However, the use of spheres means that it is applicable to error correcting codes, whereas "kissing starfish" wouldn't be useful.by ndsipa_pomu
1/16/2025 at 1:24:13 PM
A circle from a flat 2d manifold can be from a 3d sphere, cylinder, or other cross section.Our mental models don't extend well beyond 3, possibly 4, dimensions, hence _all_ of our intuition starts to be doubtful after 3 dimensions.
by tomrod
1/16/2025 at 12:10:17 PM
In many cases you are "translating" the higher-dimensional geometry into something that is not geometric or which is much lower dimensional. You don't generally visualize 24 dimensions. You can get a decent intuition for 4 with practice but at some point this breaks down.For example, the 24-dimensional packing corresponds to the Leech lattice which itself corresponds to the Golay code:
by scythe
1/16/2025 at 12:06:28 PM
I suspect that you have plenty of company...but from a journalism PoV, those kind of things are where it gets tricky. Explaining in detail, and at length, is a lot more work than this short article. Then there are the decisions - "just how much detail?", "just how long?", (worse) "how much mathematical background should we assume, in our readers?", and (worst) "how willing will our readers be, to slog through serious mathematics?".(I'm assuming you've already searched for math bloggers, and similar "labor of love" coverage of the topic.)
by bell-cot