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An Old Conjecture Falls, Making Spheres a Lot More Complicated

quantamagazine.org

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Re: An Old Conjecture Falls, Making Spheres a Lot More Complicated

#11

Post by Sniffnoy »

iraqmtpizza wrote: post #10

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.)

Re: An Old Conjecture Falls, Making Spheres a Lot More Complicated

#12

Post by myownpetard »

iraqmtpizza wrote: post #10

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

Wait until you hear about hypercubes.

Re: An Old Conjecture Falls, Making Spheres a Lot More Complicated

#13

Post by iraqmtpizza »

Sniffnoy wrote: post #11
iraqmtpizza wrote: post #10

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…

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.

Re: An Old Conjecture Falls, Making Spheres a Lot More Complicated

#15

Post by datavirtue »

dullcrisp wrote: post #5
datavirtue wrote: post #4

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…

There is no thickness (or it’s zero if you like). The deformations have to be continuous mathematical functions, so punching a hole isn’t possible. 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.

If it's zero I can make a doughnut from a ball without tearing.

Re: An Old Conjecture Falls, Making Spheres a Lot More Complicated

#17

Post by dullcrisp »

datavirtue wrote: post #15
dullcrisp wrote: post #5

Earlier quoted context omitted.

There is no thickness (or it’s zero if you like). The deformations have to be continuous mathematical functions, so punching a hole isn’t possible. 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.

If it's zero I can make a doughnut from a ball without tearing.

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.

[1] https://en.m.wikipedia.org/wiki/File:Tesseract_torus.png

Re: An Old Conjecture Falls, Making Spheres a Lot More Complicated

#18

Post by agrounds »

datavirtue wrote: post #4

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…

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.

[1] https://en.wikipedia.org/wiki/Homotopy#Homotopy_equivalence

[2] https://en.wikipedia.org/wiki/Retraction_(topology)

Re: An Old Conjecture Falls, Making Spheres a Lot More Complicated

#19

Post by kmill »

iraqmtpizza wrote: post #13
Sniffnoy wrote: post #11

Earlier quoted context omitted.

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…

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…

Higher homotopy groups of spheres goes way back to the 30s. This isn't lazy writing. https://en.m.wikipedia.org/wiki/Homotopy_groups_of_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.

Re: An Old Conjecture Falls, Making Spheres a Lot More Complicated

#20

Post by iraqmtpizza »

kmill wrote: post #19
iraqmtpizza wrote: post #13

Earlier quoted context omitted.

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…

Higher homotopy groups of spheres goes way back to the 30s. This isn't lazy writing. https://en.m.wikipedia.org/wiki/Homotopy_groups_of_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 sub…

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.

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