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there is an assumption the shuttlecock is the Universe and there is nothing outside to observe the object. It's not an assumption, it's explicitly part of the

there is an assumption the shuttlecock is the Universe and there is nothing outside to observe the object. It's not an assumption, it's explicitly part of the model. Same with "What's south of the south pole." It ignores the stars and galaxies past Earth itself. If I say "the latex in this balloon is like the surface of the earth," I'm not so much ignoring the stars & galaxies, they're just not part of the analogy. Is your point that, for an object to be curved, there must be something for it to curve into? For example, in order for a 2D surface (like the latex in the balloon) to be curved, it necessarily must be in a 3D (or higher) space?

deleted 7y ago
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throwawaymath 7y ago
> Is your point that, for an object to be curved, there must be something for it to curve into? For example, in order for a 2D surface (like the latex in the balloon) to be curved, it necessarily must be in a 3D (or higher) space? Given the context I'm guessing you asked this rhetorically, but if not: it is not true in general that a curved n-dimensional object must be embedded in a n + k-dimensional space. Curvature can be intrinsic.
- pbhjpbhj 7y ago
  Could you expand on that statement a little, please. You can take a curved object in 3D space and project it into 2D space and it remains curved? In the examples of Gaussian curvature here https://www.maths.ox.ac.uk/about-us/departmental-art/theory/differential-geometry https://www.maths.ox.ac.uk/about-us/departmental-art/theory/... they only appear to be curved because they're 3D (an arc-section of a sphere, a paraboloid), none of them is 2D. Are the examples just weak? Wolfram has a curious definition: >"A curvature such as Gaussian curvature which is detectable to the "inhabitants" of a surface and not just outside observers. An extrinsic curvature, on the other hand, is not detectable to someone who can't study the three-dimensional space surrounding the surface on which he resides." (http://mathworld.wolfram.com/IntrinsicCurvature.html http://mathworld.wolfram.com/IntrinsicCurvature.html) // but it doesn't make sense to me. A cylinder has no intrinsic curvature but the curvature is discoverable by travelling in one direction only to return from the other?
  - SAI_Peregrinus 7y ago
    A sphere has positive Gaussian curvature everywhere, in 3D. You don't need to view it from 4D to discover this, you can draw a triangle on the surface and note that the interior angles add up to >180 degrees. You can't project it into 2D space and keep the curvature, which is why all maps have distortion.

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throwawaymath 7y ago

> Is your point that, for an object to be curved, there must be something for it to curve into? For example, in order for a 2D surface (like the latex in the balloon) to be curved, it necessarily must be in a 3D (or higher) space? Given the context I'm guessing you asked this rhetorically, but if not: it is not true in general that a curved n-dimensional object must be embedded in a n + k-dimensional space. Curvature can be intrinsic.

pbhjpbhj 7y ago
Could you expand on that statement a little, please. You can take a curved object in 3D space and project it into 2D space and it remains curved? In the examples of Gaussian curvature here https://www.maths.ox.ac.uk/about-us/departmental-art/theory/differential-geometry https://www.maths.ox.ac.uk/about-us/departmental-art/theory/... they only appear to be curved because they're 3D (an arc-section of a sphere, a paraboloid), none of them is 2D. Are the examples just weak? Wolfram has a curious definition: >"A curvature such as Gaussian curvature which is detectable to the "inhabitants" of a surface and not just outside observers. An extrinsic curvature, on the other hand, is not detectable to someone who can't study the three-dimensional space surrounding the surface on which he resides." (http://mathworld.wolfram.com/IntrinsicCurvature.html http://mathworld.wolfram.com/IntrinsicCurvature.html) // but it doesn't make sense to me. A cylinder has no intrinsic curvature but the curvature is discoverable by travelling in one direction only to return from the other?
- SAI_Peregrinus 7y ago
  A sphere has positive Gaussian curvature everywhere, in 3D. You don't need to view it from 4D to discover this, you can draw a triangle on the surface and note that the interior angles add up to >180 degrees. You can't project it into 2D space and keep the curvature, which is why all maps have distortion.