6 ms·

What trips me up (but the article glosses over) is the concept of "the size of a dimension". How are dimensions larger and smaller? To me, a dimension is usua

What trips me up (but the article glosses over) is the concept of "the size of a dimension". How are dimensions larger and smaller? To me, a dimension is usually something you measure in. My height is my extent in the vertical diredction (if standing up). What would it mean for that dimension to be small? Our 3 spatial dimensions I assume are "large", meaning what? The axes extend to some apparent infinity without repeating? Does a "small" dimension behave like a longitude, that after a certain while it repeats itself? So the two dimensions of a sphere surface are "smaller" than the 2 dimensions of an infinite plane, is that how to interpret the small/large? So [the surface of] an infinitely long but finitely thick cylinder would have one large (along) and one small (around) dimension? Is "small" then always finite, but large can be large-but-finite or inifinte?

codethief 6y ago
> So an infinitely long but finitely thick cylinder would have one large (along) and one small (around) dimension? Exactly. The mathematical concept behind this is the idea of compactness[0, 1]. A compact manifold always has finite volume. In the same way, a compact dimension always has finite length and this is what is meant here.[2] Note that the converse does not hold: Not every dimension of finite length is necessarily compact. (The mathematical reason being that the metric tensor could become smaller and smaller ("fade out") towards the infinite ends.) But for the purposes of talking about string theory, you can usually equate the terms "finite length/volume" and "compact". > Is "small" then always finite, but large can be large-but-finite or inifinte? Yes, "small" is always finite (compact). As for "large" dimensions, I would say that's a matter of terminology and physicists tend to not be very precise with their terminology. It can both be large-but-finite (compact) or infinite (non-compact), depending on the context. [0]: https://en.wikipedia.org/wiki/Compact_space https://en.wikipedia.org/wiki/Compact_space [1]: https://mathworld.wolfram.com/CompactManifold.html https://mathworld.wolfram.com/CompactManifold.html [2]: This is also what string theorists mean when they "compactify a dimension": They take this infinite dimension and "wrap it around" a circle (of a given circumference) and end up with a cylinder or some more complicated object (depending on what they started with).
- chartpath 6y ago
  We live in a numpy array.
  - nine_k 6y ago
    If spacetime is indeed quantized, it's even a finite-precision array.
    - fmax30 6y ago
      > spacetime is quantized. Doesn't that just adds more evidence for us living in a simulation, and not the nice kind either. Would be interesting to see when the hardware is upgraded, physcists would suddenly discover that spacetime isn't as quantized as they initially thought.
      - codethief 6y ago
        > physcists would suddenly discover that spacetime isn't as quantized as they initially thought I don't think the majority of physicists think that spacetime is quantized. Sure, something is going to happen at the Planck scale but whether that's quantization of time and space is very much an open question.
      - simiones 6y ago
        Quantizing space and time would not necessarily make simulations simpler - just like quantizing forces actually turns out to probably make the simulation exponentially harder on a classical computer - quantum computers can be simulated by classical computers in exponential time with the best algorithms that we know of, though it is still possible that Polynomial algorithms could exist. Of course, nothing at all can actually prove/disprove that we live in a simulation in the scientific sense - it's just a transcendental model for people who don't like theistic transcendental models, but no more "scientific" than "creation science" or "the gods did it" theories.
      - cheaprentalyeti 6y ago
        It wasn't until Columbus came along that they finally bothered to move us from the "flat" default starter server to the spherical one.
pfortuny 6y ago
Think of a cylinder whose radius is of atomic size. More or less that is what they mean. That space is “felt” as 1-dimensional because there is no way to tell one point on a circumference from another on the same one.
- nine_k 6y ago
  If it's atomic size, it can be relatively easily detected, I suppose, by setting things in a spiral motion. The tricky variant is when the compact dimension is seriously sub-atomic, so you have trouble moving anything easily detectable along it.
  - caf 6y ago
    If two (neutral) objects were only slightly separated along that dimension, no matter how compact it is, wouldn't they be able to move through each other as seen from our perspective (as in, occupy the same position in the usual three dimensions at the same time, because they are separated along that additional spatial dimension)?
    - simiones 6y ago
      Only if the objects themselves are really small along that extra dimension relative to that extra dimension - if they occupy most of the dimension, they would "bump into each other" anyway.
      - cutemonster 6y ago
        Makes me think of quantum tunneling
    - pas 6y ago
      Quantum stuff can already move through each other. (Bosons easily, fermions have the Pauli exclusion principle.)
    - amelius 6y ago
      Not if the strong nuclear force extends significantly beyond the small dimension.
jerf 6y ago
Look at a game of Asteroids: https://www.youtube.com/watch?v=_TKiRvGfw3Q https://www.youtube.com/watch?v=_TKiRvGfw3Q It takes place on what is topologically a torus, because the top of the screen and the bottom are the same, and the left and right are the same. From the point of view of the player, the space never ends; if there were no asteroids and you simply went left forever, you'd never find the end. Nevertheless, the X and Y dimensions are finite, because that is the shape of space in that game. But note you could grow or shrink those all you like, and all you'd change is the size of the playspace; nothing else about such shrinking or growing would be impossible or fundamentally change anything. There is more than one way to hook such structures together: https://www.youtube.com/watch?v=jj5lDmaQTuo&t=0s https://www.youtube.com/watch?v=jj5lDmaQTuo&t=0s In that video the author plays with discrete cases of topologies like the asteroids case where otherwise flat space is just glued together in various configurations but there are continuous analogues to at least most, if not all of them, along with options on how to handle the curvature. The video also deal with macroscopically-sized things, using a model of Earth as its example, but if you can imagine it being shrunk down arbitrarily small in one dimension you can get an idea. Obviously, if you shrink one of them down arbitrarily small you would get a "3D space" that is effectively only 2D, because one of the numbers would seem to be irrelevant since it is always very small. In the real universe this would be some set of dimensions beyond the usual 3 spatial dimensions.

codethief 6y ago

> So an infinitely long but finitely thick cylinder would have one large (along) and one small (around) dimension? Exactly. The mathematical concept behind this is the idea of compactness[0, 1]. A compact manifold always has finite volume. In the same way, a compact dimension always has finite length and this is what is meant here.[2] Note that the converse does not hold: Not every dimension of finite length is necessarily compact. (The mathematical reason being that the metric tensor could become smaller and smaller ("fade out") towards the infinite ends.) But for the purposes of talking about string theory, you can usually equate the terms "finite length/volume" and "compact". > Is "small" then always finite, but large can be large-but-finite or inifinte? Yes, "small" is always finite (compact). As for "large" dimensions, I would say that's a matter of terminology and physicists tend to not be very precise with their terminology. It can both be large-but-finite (compact) or infinite (non-compact), depending on the context. [0]: https://en.wikipedia.org/wiki/Compact_space https://en.wikipedia.org/wiki/Compact_space [1]: https://mathworld.wolfram.com/CompactManifold.html https://mathworld.wolfram.com/CompactManifold.html [2]: This is also what string theorists mean when they "compactify a dimension": They take this infinite dimension and "wrap it around" a circle (of a given circumference) and end up with a cylinder or some more complicated object (depending on what they started with).

chartpath 6y ago
We live in a numpy array.
- nine_k 6y ago
  If spacetime is indeed quantized, it's even a finite-precision array.
  - fmax30 6y ago
    > spacetime is quantized. Doesn't that just adds more evidence for us living in a simulation, and not the nice kind either. Would be interesting to see when the hardware is upgraded, physcists would suddenly discover that spacetime isn't as quantized as they initially thought.
    - codethief 6y ago
      > physcists would suddenly discover that spacetime isn't as quantized as they initially thought I don't think the majority of physicists think that spacetime is quantized. Sure, something is going to happen at the Planck scale but whether that's quantization of time and space is very much an open question.
    - simiones 6y ago
      Quantizing space and time would not necessarily make simulations simpler - just like quantizing forces actually turns out to probably make the simulation exponentially harder on a classical computer - quantum computers can be simulated by classical computers in exponential time with the best algorithms that we know of, though it is still possible that Polynomial algorithms could exist. Of course, nothing at all can actually prove/disprove that we live in a simulation in the scientific sense - it's just a transcendental model for people who don't like theistic transcendental models, but no more "scientific" than "creation science" or "the gods did it" theories.
    - cheaprentalyeti 6y ago
      It wasn't until Columbus came along that they finally bothered to move us from the "flat" default starter server to the spherical one.

pfortuny 6y ago

Think of a cylinder whose radius is of atomic size. More or less that is what they mean. That space is “felt” as 1-dimensional because there is no way to tell one point on a circumference from another on the same one.

nine_k 6y ago
If it's atomic size, it can be relatively easily detected, I suppose, by setting things in a spiral motion. The tricky variant is when the compact dimension is seriously sub-atomic, so you have trouble moving anything easily detectable along it.
- caf 6y ago
  If two (neutral) objects were only slightly separated along that dimension, no matter how compact it is, wouldn't they be able to move through each other as seen from our perspective (as in, occupy the same position in the usual three dimensions at the same time, because they are separated along that additional spatial dimension)?
  - simiones 6y ago
    Only if the objects themselves are really small along that extra dimension relative to that extra dimension - if they occupy most of the dimension, they would "bump into each other" anyway.
    - cutemonster 6y ago
      Makes me think of quantum tunneling
  - pas 6y ago
    Quantum stuff can already move through each other. (Bosons easily, fermions have the Pauli exclusion principle.)
  - amelius 6y ago
    Not if the strong nuclear force extends significantly beyond the small dimension.

jerf 6y ago

Look at a game of Asteroids: https://www.youtube.com/watch?v=_TKiRvGfw3Q https://www.youtube.com/watch?v=_TKiRvGfw3Q It takes place on what is topologically a torus, because the top of the screen and the bottom are the same, and the left and right are the same. From the point of view of the player, the space never ends; if there were no asteroids and you simply went left forever, you'd never find the end. Nevertheless, the X and Y dimensions are finite, because that is the shape of space in that game. But note you could grow or shrink those all you like, and all you'd change is the size of the playspace; nothing else about such shrinking or growing would be impossible or fundamentally change anything. There is more than one way to hook such structures together: https://www.youtube.com/watch?v=jj5lDmaQTuo&t=0s https://www.youtube.com/watch?v=jj5lDmaQTuo&t=0s In that video the author plays with discrete cases of topologies like the asteroids case where otherwise flat space is just glued together in various configurations but there are continuous analogues to at least most, if not all of them, along with options on how to handle the curvature. The video also deal with macroscopically-sized things, using a model of Earth as its example, but if you can imagine it being shrunk down arbitrarily small in one dimension you can get an idea. Obviously, if you shrink one of them down arbitrarily small you would get a "3D space" that is effectively only 2D, because one of the numbers would seem to be irrelevant since it is always very small. In the real universe this would be some set of dimensions beyond the usual 3 spatial dimensions.