s p h e r e

Comments

• ignotum@lemmy.world ⁨2⁩ ⁨weeks⁩ ago

  How on earth can an infinite dimentional sphere be able to agree to a contract, let alone signing one?

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  • Chronographs@lemmy.zip ⁨2⁩ ⁨weeks⁩ ago

    As the tip of an infinite dimensional ballpoint pen

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• lauha@lemmy.world ⁨2⁩ ⁨weeks⁩ ago

  Explanation?

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  • someacnt@sh.itjust.works ⁨2⁩ ⁨weeks⁩ ago

    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 - just shrink along each line to the center. 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 isn’t - infinite dimensional sphere is . Somehow, it loses the “hollow space” inside.

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    • zipsglacier@lemmy.world ⁨2⁩ ⁨weeks⁩ ago

      For each finite dimension n (1, or 2, or 3, etc…), the sphere in dimension n can’t be contracted because of that empty n-dimensional space it surrounds. But that same sphere is the “equator” of the sphere in the next higher dimension, n+1. There, the n-dimensional equator can contract along one of the hemispheres, to a pole. But then that whole (n+1)-dimensional sphere still isn’t contractible, because of the (n+1)-dimensional space it surrounds.

      BUT the (n+1)-dimensional sphere can contract along one of the hemispheres in the (n+2)-dimensional sphere. And so on.

      For any particular finite dimension n, there is an n-dimensional obstruction to contracting the sphere in that dimension. But if you go all the way to infinitely-many dimensions, there is no obstruction that ever stops contractibility of the infinite-dimensional sphere.

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    • ComicalMayhem@lemmy.world ⁨2⁩ ⁨weeks⁩ ago

      Sphere refers to the surface area, correct? Ball would be the volume inside?

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  • dont@lemmy.world ⁨2⁩ ⁨weeks⁩ ago

    On what level? A proof? Or just the meaning of the words?

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    • lauha@lemmy.world ⁨2⁩ ⁨weeks⁩ ago

      What is contractible in mathematics? Wikipedia had article on contractible spaces but that was way beyond be.

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• AnarchoEngineer@lemmy.dbzer0.com ⁨2⁩ ⁨weeks⁩ ago

  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.)

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  • pankuleczkapl@lemmy.dbzer0.com ⁨2⁩ ⁨weeks⁩ ago

    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 R^infinity, which is basically what most people would imagine is the infinite equivalent of our dimensional space.

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    • ComicalMayhem@lemmy.world ⁨2⁩ ⁨weeks⁩ ago

      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

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    • phoenixz@lemmy.ca ⁨2⁩ ⁨weeks⁩ ago

      most people would imagine

      I imagine that most people don’t have the faintest clue what you’re talking about, though

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• ivanafterall@lemmy.world ⁨2⁩ ⁨weeks⁩ ago

  Image

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• Etterra@discuss.online ⁨2⁩ ⁨weeks⁩ ago

  How much do you need to contract an infinite sphere before it becomes finite?

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  • dont@lemmy.world ⁨2⁩ ⁨weeks⁩ ago

    Until it becomes a point.

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• Crackhappy@lemmy.world ⁨2⁩ ⁨weeks⁩ ago

  There is no sphere but s p h e r e itself.

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