An Old Conjecture Falls, Making Spheres a Lot More Complicated
quantamagazine.org
quantamagazine.org
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.
To be fair, manuals are often pretty opaque without the requisite background knowledge.
Expensive.
… [0] https://news.ycombinator.com/item?id=37171553
(This comment reads faster with tail recursion.)
I think it may not work out.
Any source that filtered their articles so only that quality was released, I would pay $5.
And instead of filtering out their writers work, it would mostly be about telling them to rework, do more homework, get it right.
90% of quality is simply recognizing when you haven’t hit the bar yet and stepping up. Repeatedly if necessary.
The subject is rarely the problem.
Downvote me as much as you like. It's painful to take an honest look into the mirror.
ne supra crepidam
You could trim this down, but I personally find the background as interesting as the result.
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.
What a nice 65th birthday present to finally close off the last dangling piece of your almost-complete research agenda!
Cheeky.
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.
The study is about the properties of (higher dimensional) shapes rather than concrete objects. It’s like asking what’s the thickness of a circle.
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.
[1] https://en.m.wikipedia.org/wiki/File:Tesseract_torus.png
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.
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.
[1] https://en.wikipedia.org/wiki/Homotopy#Homotopy_equivalence
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.
is it actually infinitely - or just a lot?
(There are finitely many maps for each possible dimensional difference, and countably many possible dimensional differences.)
Clearly IANAM.
A torus is like an inner tube - an inner void and a big hole in the middle.
A solid torus just has a big hole in the middle, like a donut.
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?
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.
Sure, no problem, author.
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.)
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.
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.
>The n-dimensional unit sphere — called the n-sphere
LOL nevermind—I was right the first time. Thank you for confirming.
> 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.
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
A "metal" in astronomy is everything other than hydrogen or helium.
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.
please no knee-jerk 'this is pure mathematics, it doesn't need applicability' answers.
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.
The Internet existed when Feynman died in 1988, but Internet porn did not, at least not watchable (video) Internet porn.