Looking beyond the Lovecraftian mysticism, this is actually pretty fascinating in and of itself. I'm not a true mathematician (only a lowly electrical engineer with some training and study in mathematics), but I would not have expected that equality to hold for complex values, let alone quaternions, etc.
The gossipy narrative style of the article is kind of jarring for an article on a topic like this. It took several paragraphs before it touched on the matter.
I dunno about gossipy, but the narrative style is standard at Quanta. It's written for the subscriber who is reading for leisure, and wants a good story as well as some amount of technical depth, not for the HN reader who wants to quickly judge whether figuring this thing out is worth their time, and will abandon it if not.
I always wonder what a popular science/math magazine would look like if it were oriented towards hackers. In this I mean people who have little background in the field but also the type of person who is used to bluntness and knows to RTFM.
I would subscribe to one. Journal articles are often opaque to people who aren't already in the field, and popular science falls too often into the storytelling trap seen here.
For this particular article, I'm not sure a hacker version could be much better. I'm slightly familiar with this area of research, I'm not sure a more "true" explanation couldn't be done in much less than 20 pages of fairly hard maths, and I don't imagine anyone would want to chew through that.
You could trim this down, but I personally find the background as interesting as the result.
The gap is very hard to close because there is a chasm between the two. RTFM here means years (typically 5+ for research math) of very focused study to follow what it says. While the popular science tries to convey something that is meaningful to the majority of people without any background exposure. The gap between the two is huge.
> I always wonder what a popular science/math magazine would look like if it were oriented towards hackers. In this I mean people who have little background in the field but also the type of person who is used to bluntness and knows to RTFM.
OK, assuming that there are enough people like you to make that a going concern now we just have to solve the problem of getting this level quality distributed through the population that wouldn't care for it enough to send it on to their friends, that is to say through the global network of humans with 1 in 10000 being one of the people willing to pay 1 dollar and the other 9999 people saying "what the hell, who cares"
I always wonder about what something in between popular science/math and academic journals would look like. Something oriented towards people that know a nontrivial amount but are not researchers. E.g., you can assume they know calculus and have a conceptual understanding of what things like manifolds and homotopy but are fuzzy on the details.
For a topic like in the article, you need a lot of words to give a very vague understanding if you are aiming for a general audience. Not clear it is worth it for the reader. For that more targeted audience it could be a lot shorter and give a little more detail.
Probably too small a market, but I would definitely enjoy that type of content a lot.
I hate reading quanta mag because of the narrative, and didn't have the patience to weed through this to learn something conceptual. But I'm glad there are people who enjoy it.
I hate slide decks like these, where every page in the PDF contains one more bullet point than the last one. Maybe I'm particularly bad at this, but I spend way too long scanning each page for the new information. Is it impossible to configure LaTeX to only produce the final animation step as a completed page and skip the intermediate ones?
The ideal way to read a deck like this, is to download the pdf and enter present mode on your pdf viewer so that you can click through each slide and immediately see where the new material has been added.
The first sentence should have been the ball-is-equal-to-egg explanation with mention of topology. Before that I had no idea what they were talking about.
P.s. I have to assume the rules forbid shapes with surfaces of zero thickness. Otherwise I can just smash a ball into an inner-tube. If the shapes have thickness mandated, what is it? Are the thickness of the surfaces a consideration when morphing from one shape to another? Is the surface thickness negative or positive from zero? All of these questions stem from my experience in 3D modeling where these parameters must be defined.
Have you heard the quip that in physics a cow and a point are equivalent? This is because the physicist cares only about the motion of the thing.
In topology, a doughnut and a coffee mug are equivalent (a mug has exactly one hole, in the handle where your fingers grab). Because the mathematician doesn't care about how hard, thick, or breakable it is; they only care about how complex the shape is. So throw out thickness, size, elasticity, etc.
No, you can't. If you'd like an analogy from 3D modeling to see why not:
Any polyhedral mesh has an integer called its "Euler characteristic", which is simply calculated by taking the number of vertices, subtracting the number of edges, and adding the number of faces. (V-E+F)
Obviously, smoothly deforming a surface by moving vertices around doesn't change its Euler characteristic. A bit less obviously, any sequence of local refinements to "patches" of the mesh can't change its Euler characteristic either. (For example, splitting one face into smaller regions that are still connected to their surroundings in the same way.) Anything that you might reasonably call a "smooth" transformation will keep the Euler characteristic unchanged. You can convince yourself of this by experimentation with whatever 3D modeling software you like.
But a spherical mesh has Euler characteristic 2, and a torus mesh has Euler characteristic 0. So no smooth deformation can transform one into the other.
The only way to change the Euler characteristic would be to change the mesh topology itself, which would mean there's at least one pair of faces that are connected by an edge in one mesh and not connected in the other, which means the mesh has been "torn" along that edge.
With a lot of math, you can extend this argument to arbitrary continuous surfaces, not just polygons. If two surfaces have different Euler characteristic, then you cannot find a bidirectional continuous mapping between them. Any such bijection must be discontinuous somewhere, which roughly means that arbitrarily close points are "torn apart" from each other.
You can’t. The informal proof may not be very convincing but it’s that the torus has two circles that remain distinct no matter how you deform the space: the smaller and larger circles in this picture [1].
But on a sphere, every circle can be deformed to any other circle. If the torus were itself the deformation of a sphere, you’d be able to deform it the same way as the sphere to get one circle to the other.
Again though, the version of these objects that mathematicians study is formalized such that this is unambiguous.
Mathematical objects are only loosely analogous to physical objects.
A 2D disk has zero thickness, any movement orthogonal to the plane of the disk will take you off the disk. But the disk can't be distorted into a circle in a continuous way.
To add on to what dullcrisp said, which is all correct, even spheres with thickness are “the same as” spheres of zero thickness from the perspective of homotopy theory. “Sameness” here means homotopy equivalence [1]. In fact the thin sphere is a deformation retract [2] of the thick one. The deformation pushes each point of the thick sphere along radial lines towards the thin sphere. Being a deformation retract implies the two spaces are homotopy equivalent.
>Infinitely more maps from spheres to telescopes means infinitely more maps between spheres themselves. The number of such maps is finite for any difference in dimension, but the new proof shows that the number grows quickly and inexorably.
Lost me at the end, but, don't inner-tubes have 2 holes (genus 2), topologically: one for inflation and one for the wheel to fit in. This makes them distinct from a torus (genus 1) and no homotopy exists between them.
You probably are aware, but just to make it clear for others: topologically a torus has one hole (genus 1). The inner void is not considered a hole.
Similarly, a 2-sphere (surface of solid sphere in 3d) has 0 holes (the inner void is not considered).
You are right, but I am pretty sure they just meant the inner tube ignoring the puncture for inflation.
Usually people just use donut/bagel for an illustration of a torus. Not sure why they used inner tube here - maybe to make it clear it is just the surface?
My understanding is they're non-homeomorphic. You can thread a string in through the inflation hole (valve hole) on an inner tube, around the great circle inside the tube, and then out the valve hole. You can also, of course go through the centre as with a donut/bagel/torus. But you can't do the former threading with a donut, so the torus and whatever the inner-tube are not typologically equivalent.
Another way to consider it is if you have a circle around the valve hole, you can't stretch that circle to be one of the two possible circles you could draw on a torus (around the 'trunk' and around centred on the donut-hole).
Anyway, I think they just chose a bad example, this was just where my mind went.
Eight, for me. Once you can fit packed spheres between each other in the norma "pyramid" configuration, my intuition breaks down. It should probably be true about four dimensions as well but I'm overconfident
do they call them all spheres just to pretend that their work is relevant? I've heard from captain beyond that everything's a circle, but this is one step too far. a 100-dimensional non-uniform egg is not a sphere in any possible way. why is it not called an n-manifold or something like that
I'm not sure what you're talking about. These are, in fact, n-dimensional spheres -- the set of points at unit distance from the origin in n+1 dimensions. (It's n+1 because, e.g., a sphere in 3 dimensions is intrinsically 2-dimensional.)
An n-manifold would just mean any n-dimensional manifold. These are very particular n-dimensional manifolds, namely, spheres.
Now of course, this is topology, so our equivalences are broad; but the thing these are all equivalent (homeomorphic) to is a sphere. Sure, you can take a more complicated shape that's equivalent to a sphere, but that complexity is incidental; the broad equivalences of topology let us ignore them. (Although, alternatively, they also let us turn the sphere into, say, a cube, if that's easier to think about, which often it is.)
Other than the article using the word sphere incessantly, I don't see how any of this is limited to spheres. I don't see even once how uniform distance plays into this except that the sphere is the simplest version of the sorts of things they're talking about. Your failure to banish my suspicions despite effort makes me that much more confident in my original conclusion.
also a hypercube is not a cube--it's an n-cube. otherwise this is just lazy pop science rhetoric to get the kids excited about their field (and eventually suppress wages in mathematics with their newly-supplied labor, degree in hand). except not even science, so even less important
I understand that these objects are topologically equivalent to n-spheres, but that doesn't make them n-spheres, let alone spheres proper. In fact, you point out that cubes and spheres are topologically equivalent despite zero spheres being cubes and zero cubes being spheres.
Mathematicians commonly refer to two objects by the same name if they are equivalent in the given context. In this context, topology, any space that is homeomorphic to a sphere might be referred to as “a sphere” even if literally speaking it’s not a sphere. For instance a topologist would might point at a cube and call it a sphere. In their domain there is no important difference between them so why not?
Also, n-spheres are commonly just called spheres for brevity. So when I say “the fundamental problem of homotopy theory is to compute the homotopy groups of spheres,” I am referring to all homotopy groups of all (n-)spheres simultaneously.
> I don’t see how any of this is limited to spheres.
In fact you’re right, homotopy theory is not just limited to spheres! However, if we could readily compute the homotopy groups of spheres, then we would be able to compute the homotopy groups of any “reasonable space.” Here I’m referring to CW complexes [1] which are a very broad class of spaces that, up to homotopy equivalence, probably includes any space you care to think of. It is for this reason that the problem of computing the homotopy groups of spheres is so fundamental to homotopy theory more broadly.
Looking at all the continuous functions from all dimensions of spheres into a particular topological space ends up giving rich algebraic information about the space. This is a cornerstone of algebraic topology. Turns out calculating this stuff for even just spheres can be subtle and mysterious.
Uniform distance matters not at all for any of this, but it does matter that your family of "spheres" be topologically equivalent to the round spheres.
Another model is you take the iterated suspensions starting with a pair of points (the zero sphere).
Yet another is to take boundaries of simplicies, or even cubes.
Topologists are those who are perfectly happy to call a paper towel tube an annulus.
It is of great comfort to learn that lazy writing did not exist in 1930. And no mathematician would have ever misrepresented his work to the public back then. They wore top hats then. Far too proper to stoop to lowly deception to achieve recognition.
>The n-dimensional unit sphere — called the n-sphere
LOL nevermind—I was right the first time. Thank you for confirming.
Ok, I see you are trolling. I tip my top hat to you, good day.
> Your failure to banish my suspicions despite effort makes me that much more confident in my original conclusion.
Side note: I've never considered this phenomenon in my life, and suddenly other people digging in in the face of evidence makes sense. A dubious "thank you" to you.
it's fine to call them spheres in a freewheeling discussion with a fellow traveler but it's a deliberately misleading shortcut here. at the barest of minimums it could say topological n-spheres
And if you don't become more confident in your idea after a well-orchestrated yet entirely failed attempt to destroy it, you are not a rational person lol. it's called trial by fire and it's older than i am
the sensationalist nature of the writing has generated a lot of discussion so I guess it has does its job
Metallic hydrogen? yeah, i took astrophysics in college. pretty sure that just because something starts conducting doesn't mean it's a metal. but sure, professor
An n-sphere is an n-dimensional manifold that can be translated, via various means such as diffeomorphisms, to other shapes like an n-dimensional egg. Since it's the "base" shape, that's what it's called. Note that an n-sphere has no holes.
There's an applications section in the Wikipedia article, but it's all to other parts of pure math. It's hard to summarize, but they've got to do with obstructions to untangling, unwrapping, or otherwise solving things to do with spaces.
Mathematics has a habit throughout history of coming up with useless (to layman, at first) looking ideas that become insanely valuable. Such as zero, negative numbers, imaginary numbers, group theory and so on. Modern life
and progress needs this.
Proving other theorems, which may themselves either prove further theorems or lead to direct applications. That's how the questions of "what to prove" often materialise.
