I took a class taught by David Huffman (of Huffman coding) called Cybernetics (IIUC it was the UCSC equivalent of a class Wiener taught at MIT.
The very first day, he started out by talking about kissing spheres and concluded the lecture with "and that's why kissing spheres are easy in 7 dimensions" (or something like that).
Every lecture of his was like being placed in front of a window looking upon a wonderful new world, incomprehensible at first, but slowly becoming more and more clear as he explained. Sometimes I wish I could play in the garden of math.
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!
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"
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).
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".
> For example high dimensional spheres are "pointy" and most of their volume are near their surface
I 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?
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).
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.
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.
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.
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.”
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.
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.
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".
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.
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. 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
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.)
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.
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.
> 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.
>
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.
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.
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.
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.
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:
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.)
It's strange the article doesn't even mention just trying to simulate the problem computationally.
Surely it's not too difficult to repeatedly place spheres around a central sphere in 17 dimensions, maximizing how many kiss for each new sphere added, until you get a number for how many fit? And add some randomness to the choices to get a range of answers Monte Carlo-style, to then get some idea of the lower bound? [Edit: I meant upper bound, whoops.]
Obviously ideally you want to discover a mathematically regular approach if you can. But surely computation must also play a role here in narrowing down reasonable bounds for the problem?
And computation will of course be essential if the answer turns out to be chaotic for certain numbers of dimensions, if the optimal solution is just a jumble without any kind of symmetry at all.
Maybe the author thought it was so obvious that you could get some lower bounds that way, that it didn't seem worth mentioning! :) Wikipedia has a list, where I presume the lower bounds are mostly demonstrated constructively. Upper bounds must be demonstrated non-constructively, so I presume computers don't really help there.
> In two dimensions, the answer is clearly six: Put a penny on a table, and you’ll find that when you arrange another six pennies around it, they fit snugly into a daisylike pattern.
Is there an intuitive reason for why 6 fits so perfectly? Like, it could be a small gap somewhere, like in 3d when it's 12, but it isn't. Something to do with tessellation and hexagons, perhaps?
> They look for ways to arrange spheres as symmetrically as possible. But there’s still a possibility that the best arrangements might look a lot weirder.
Three pennies form an equilateral triangle with (of course) 60 degree angles.
Six of those equilateral triangles will perfectly add to 360 degrees. Intuitive enough? (I'm being a little hand-wavey by skipping over the part where each penny triangle shares two pennies with a neighbor — why the answer is not 18 for example.)
For my mind though, the intuitiveness ends in dimension 2 though. ;-)
It would be fun to make that square packing for 11 from wood and give it to puzzle enthusiasts with this task: Rearrange the squares so you can add an additional 12th square. And then watch them struggle putting even those 11 squares back in.
> Had she been one of his graduate students, he would have tried harder to convince her to work on something else. “If they work on something hopeless, it’ll be bad for their career,” he said.
A small anecdote: my dad is a mathematician. For a significant portion of his postdoc/early career (in the 80's/90's) he worked on proving a particular conjecture. Eventually he abandoned it and went to be much more successful in other areas.
A few years ago someone found a counterexample. He was quite depressed for a few weeks at the thought of how much of his strongest research years had been devoted to something impossible.
Choosing a "good first problem" in math is quite difficult. It needs to be "novel," somewhat accessible, and possible to solve (which is an unknown when you're starting out)!
> “There may be structures without any symmetry at all,” said Gabriele Nebe (opens a new tab) of RWTH Aachen University in Germany. “And no good way to find them.”
She taught Lineare Algebra II when I took it! It was one of the toughest lectures I took during university. I remember looking to the person next to me and one of us asked "do you understand anything?" and the other said "no! I haven't understood anything for like 20 minutes" and we burst out laughing and couldn't get it together until we were asked to quiet down. Wadim if you hang out here, send me a mail or something!
> Mathematicians often visualize this problem in terms of spheres. You can think of each code word as a high-dimensional point at the center of a sphere. If an error-filled message (when represented as a high-dimensional point) lives inside a given sphere, you know that the code word at the sphere’s center was the intended message. You don’t want these spheres to overlap — otherwise, a received message might be interpreted in more than one way. But the spheres shouldn’t be too far apart, either. Packing the spheres tightly means you can communicate more efficiently.
I went to math prep school for 2 years, attended 12 hours of math class in agebra and analysis per week, which I think proves I've done more math than most people in the general population, and this makes no sense to me. It either lacks introduction required to understand the analogy, or I've become really dumb. I want to understand this based on what the article says, but I can't. I can't represent error-filled messages as high-dimensional points. It's easier for me to imagine what the intersection between 4D spheres would look like in geometry.
I'm not sure if this helps makes things clearer, but see this diagram for symbols in Quadrature Amplitude Modulation [1]. The valid symbols are mapped to certain points in the vector space. Now, imagine non-overlapping circles around each symbol. If a received signal falls within a circle, it would be mapped to that symbol in the centre of the circle.
This can be extended to 3-D or higher dimension spaces.
> I want to understand this based on what the article says, but I can't. I can't represent error-filled messages as high-dimensional points.
Well, start with an analogy. Let's say you and I want to communicate a message, which comes from a set of let's say 4 possible messages: "YES", "NO", "GOOD", and "BYE". Let's further suppose that the medium for this message (the "data channel") is going to be a single point selected from a 2D square. We'll agree beforehand on four points in the square that will represent our four possible messages. Then, you're going to position a dot at one of those points, and I'm going to observe that dot and infer your message from its position.
If the "data channel" is "error-free" (a.k.a. "lossless"), then it really doesn't matter which points we agree on: you could say that the exact center of the square is "YES", the point one millimeter to the left is "NO", the point two millimeters to the left is "GOOD", and so on. But if the data channel is "lossy," then the dot might get shaken around before I observe it. Or equivalently, I might observe its position slightly incorrectly. So we should choose our "code" so as to minimize the effect of this "error."
The best way to do that, on a square, is to place our four "code points" all the way at the four corners of the square, as far away from each other as possible. By "as far away from each other as possible," I mean in the sense of https://en.wikipedia.org/wiki/Pole_of_inaccessibility — I mean we want to maximize the minimum distance between any two points. A mathematician would notice that this is the same thing as maximizing the radius R such that we can draw a circle of radius R around each of our code points without any of the circles intersecting. (R in this case is half of the square's side length.)
If we add a fifth code point, this same reasoning would lead us to place that fifth point right smack in the center of the square. And the sixth point... well, I feel like that gets tricky.
BUT! In actual communications, we don't send messages encoded as real points in 2D squares. We send messages as discrete bit-strings, i.e., strings of zeros-and-ones of length N, which you can see as discrete points at the corners of an N-dimensional hypercube. Then, if we want to send K different messages robust against errors(+), we should pick as our code points some K corners of the hypercube so as to maximize the minimum Manhattan distance along the hypercube's edges between any two code points. This is the basic idea behind error-correcting codes.
A digital error-correcting code is "K code points in a bounded region of N-dimensional hyperspace (namely the discrete set of corners of a unit hypercube), selected so as to maximize the minimum distance between any two of them." The kissing-hyperspheres problem is "K sphere-centers in a bounded region of N-dimensional hyperspace (namely the continuous set of points at unit distance from the origin), selected so as to maximize the minimum distance between any two of them (and then, if that minimum distance is still >=1, increase K and try again)."
If all you meant is "Those two problem statements don't seem 100% equivalent," I think I agree with you. But if you meant you didn't see the similarity at all... well, I hope this helped.
(+) — edited to add: Robust against the traditional model of error, i.e., our "threat model" is that any given bit has a constant small probability of getting flipped, so that our observed point may be some random Manhattan walk away from the code point you actually sent. You could instead use a different threat model — e.g. supposing that the bits sent in the actual digital message's "low-order" bits would flip more often than the high-order bits — in which case the optimal selection of code points wouldn't be as simple as "just maximize Manhattan distance."
This helped a lot, thanks! I now see a similiarity where I was missing the bridges between geometry and lossy information channels. It's really interesting, though it's a really complex problem.
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