3 ms·
There’s a certain way in which a cube “feels sharper” or “feels spikier” than a square. Trying to formalize that, you can compare the edge of a 3d box where two
by 6gvONxR4sf7o 5y ago
There’s a certain way in which a cube “feels sharper” or “feels spikier” than a square. Trying to formalize that, you can compare the edge of a 3d box where two faces meet to the point where three faces meet. I’d rather step on the two-face edge than the three-face corner, and there’s definitely a sense in which the cube is spikier.
It seems reasonable to extend that same intuition to n-D sharpness/spikiness in an accurate way. Adding an extra “face” just chops more off making those vertices sharper and sharper, at least relative to the high dimensional space around it.
- feoren 5y agoI think that's a great insight; I especially like comparing the sharpness of an two-face edge to a three-face corner; we could expect stepping on a seven-face corner to be slightly worse than stepping on a six-face. I think that "sharpness" idea surely must be related to this phenomenon. However, one should be careful not to let that "increasing sharpness" idea lead to the mental image of a concave shape, especially not a sea urchin. That would be a false resolution to this "paradox" and that image of a N-cube would lead to all sorts of other incorrect ideas, e.g. regarding where the volume of the shape is.
- 6gvONxR4sf7o 5y agoMaybe the right intuition is a sea urchin that is, due to high dimensional unintuitive properties, still convex. It’s almost entirely extremely pointy corners, and yet they’re “magically” connected to each other in an entirely convex manner.