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It depends on how you define "concaveness". A cube is concaver than a square in a sense. The travel from the center of a unit square to its side takes 0.5, and
by wruza 5y ago
It depends on how you define "concaveness". A cube is concaver than a square in a sense. The travel from the center of a unit square to its side takes 0.5, and to its vertex ~0.7. For a cube, 0.5 and ~0.87. For a 100d cube, 0.5 and 5. (For completeness, for 1d cube it's 0.5, 0.5). Of course that is not a true concaveness, but it gives a nice sense of inside distances, especially with spheres, which are defined as equidistant.
We are just used to project a cube in a way that prevents to see its linearly-spatial configuration (a projection messes up lengths), but if you preserve these lengths and "flatten" them instead, a cube will flatten out to a sort of a shuriken.
- function_seven 5y ago> It depends on how you define "concaveness". A cube is concaver than a square in a sense. Yup. Even a regular 3D cube (and 2D square) has concave faces if you're viewing it from a polar perspective. As I stand in the center of the cube, measuring distances from me to the surface, I'll see that measurement follow a concave pattern. Yeah, I know that's not the definition of concavity or whatever, but when relating a sphere to a cube, and trying to get an intuition of higher-dimensional spaces, I think it helps to look at it from the sphere's perspective rather than a Cartesian one.
- brummm 5y agoThere is a very clear definition of convexity, as can be seen here [1] on Wikipedia. Nothing to discuss about with regards to definition. [1] https://en.wikipedia.org/wiki/Convex_set https://en.wikipedia.org/wiki/Convex_set
- wruza 5y agoMaybe there is a better word to describe this idea?
- vnorilo 5y agoThank you! This comment solidified for me that the increasing ratio of distances from origin to vertex/side together with the fact that the hypercube surface is still not curved in any way whatsoever is the major thing my spatial intuition (unsurprisingly) struggles with.