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.
