4 ms·

> 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 ba

> 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.

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.