A space (shape) is contractible if you can "contract" (shrink) it to a point without cutting, pinching or punching through holes. For example, a mattress is contractible, since you can shrink it to the center - each point can follow the line to the center, continuously. Meanwhile, a doughnut, a circle or a hollow sphere are not contractible, you can never remove the inner "hole" to shrink to a point without cutting.
In general, any dimensional sphere is not contractible... Until it is - infinite dimensional sphere is contractible. Somehow, it loses the "hollow space" inside.
Also isn’t an infinite dimensional sphere practically hollow?
(If you were to integrate the sphere to calculate volume like you do for lower dimensional ones, you would sum the volume of shells—which is just their surface area times a thickness—making it up. With infinite dimensions, each shell becomes infinitely larger than the preceding shell no matter how fine you make the slices. This means the largest shell contains basically all the volume.)
This reasoning is pretty weird, but the conclusion is basically right. That is, there is absolutely no way to extend the conventional notion of volume to Rinfinity, which is basically what most people would imagine is the infinite equivalent of our dimensional space. Edit: what I mean by Rinfinity is a bit ambiguous, but let's say for the purpose of a hypersphere we want something like l^2 hilbert space to ensure no vectors with infinite length appear, then we have a separable space and the proof is complete.
Wait, I thought the volume of a sphere approaches 0 as dimensions go to infinity, no? Thr general formula for the volume of any nth dimensional sphere has the gamma function in the denominator, which rises faster than whatever is in the numerator. At some point (5 dimensions, iirc) the volume starts decreasing
21 Comments
ignotum@lemmy.world · 53 pts · 106d
How on earth can an infinite dimensional sphere be able to agree to a contract, how would it even sign it?
Lodespawn@aussie.zone · 11 pts · 106d
Bet that sphere is just fantastic at plastering though ..
derek@infosec.pub · 3 pts · 106d
Yet I never seem to catch 'em at the local pub.
Chronographs@lemmy.zip · 5 pts · 106d
As the tip of an infinite dimensional ballpoint pen
lauha@lemmy.world · 36 pts · 106d
Explanation?
someacnt@sh.itjust.works · 19 pts · 105d
I am not a topologist, but I can try..
A space (shape) is contractible if you can "contract" (shrink) it to a point without cutting, pinching or punching through holes. For example, a mattress is contractible, since you can shrink it to the center - each point can follow the line to the center, continuously. Meanwhile, a doughnut, a circle or a hollow sphere are not contractible, you can never remove the inner "hole" to shrink to a point without cutting.
In general, any dimensional sphere is not contractible... Until it is - infinite dimensional sphere is contractible. Somehow, it loses the "hollow space" inside.
zipsglacier@lemmy.world · 17 pts · 105d
DeadDigger@lemmy.zip · 2 pts · 103d
Ty
ComicalMayhem@lemmy.world · 4 pts · 105d
Sphere refers to the surface area, correct? Ball would be the volume inside?
someacnt@sh.itjust.works · 4 pts · 105d
Indeed.
dont@lemmy.world · 4 pts · 105d
On what level? A proof? Or just the meaning of the words?
lauha@lemmy.world · 12 pts · 105d
What is contractible in mathematics? Wikipedia had article on contractible spaces but that was way beyond be.
kogasa@programming.dev · 2 pts · 105d
https://en.wikipedia.org/wiki/Contractibility_of_unit_sphere_in_Hilbert_space
I can offer no ELI5 but here's the context
AnarchoEngineer@lemmy.dbzer0.com · 8 pts · 106d
Also isn’t an infinite dimensional sphere practically hollow?
(If you were to integrate the sphere to calculate volume like you do for lower dimensional ones, you would sum the volume of shells—which is just their surface area times a thickness—making it up. With infinite dimensions, each shell becomes infinitely larger than the preceding shell no matter how fine you make the slices. This means the largest shell contains basically all the volume.)
pankuleczkapl@lemmy.dbzer0.com · 7 pts · 106d
This reasoning is pretty weird, but the conclusion is basically right. That is, there is absolutely no way to extend the conventional notion of volume to Rinfinity, which is basically what most people would imagine is the infinite equivalent of our dimensional space. Edit: what I mean by Rinfinity is a bit ambiguous, but let's say for the purpose of a hypersphere we want something like l^2 hilbert space to ensure no vectors with infinite length appear, then we have a separable space and the proof is complete.
ComicalMayhem@lemmy.world · 2 pts · 105d
Wait, I thought the volume of a sphere approaches 0 as dimensions go to infinity, no? Thr general formula for the volume of any nth dimensional sphere has the gamma function in the denominator, which rises faster than whatever is in the numerator. At some point (5 dimensions, iirc) the volume starts decreasing
pankuleczkapl@lemmy.dbzer0.com · 2 pts · 104d
Yes, but we aren't talking about the limit of the volume. We are talking about volume in actually infinite dimensional space.
phoenixz@lemmy.ca · 2 pts · 105d
I imagine that most people don't have the faintest clue what you're talking about, though
ivanafterall@lemmy.world · 7 pts · 105d
Etterra@discuss.online · 4 pts · 105d
How much do you need to contract an infinite sphere before it becomes finite?
dont@lemmy.world · 6 pts · 105d
Until it becomes a point.
Crackhappy@lemmy.world · 3 pts · 106d
There is no sphere but s p h e r e itself.