Comment by danwills 2 days ago
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!
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).
> 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).
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 thought
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!
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".
Specifically, of course, d = sqrt(N), where N is dimension and d is distance of a vertex of the unit hypercube from the origin.
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 !
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".
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.
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.
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:
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-...
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.)
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.
> just a visual metaphor
It'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.
> 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.
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.
> 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.
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.
> 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.
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.
However, the use of spheres means that it is applicable to error correcting codes, whereas "kissing starfish" wouldn't be useful.
> 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"