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I’m a little bit interested in intuitive properties that high dimensional space retains as well, just to get a handle on how weird these shapes get. Like, you c

by fwlr 3y ago

I’m a little bit interested in intuitive properties that high dimensional space retains as well, just to get a handle on how weird these shapes get. Like, you can inscribe a circle inside a square and have it touch all sides without extending outside the square, and you can similarly “inscribe” a sphere inside a cube such that the sphere touches all sides of the cube without extending outside the cube. Does this hold true in higher dimensions? The picture near the end of the article showing hyper-cubes with jutting out vertices and hyper-spheres being pushed inwards sort of indicates maybe they do, but obviously that isn’t an actual mathematical answer.

sillymath 3y ago
> Like, you can inscribe a circle inside a square and have it touch all sides without extending outside the square, and you can similarly “inscribe” a sphere inside a cube such that the sphere touches all sides of the cube without extending outside the cube. Does this hold true in higher dimensions? Of course, points with only one nonzero coordinates xi in $\{1,-1}$ are in both the sphere and the sides of the cube, and those are the minimal distance points from the sides of the cube to the center of the circle (since they are orthogonal to the hyperplane xi=+1, or xi=-1 that contains it). That also shows that no side of the cube extends outside of the circle. Some more math: A well known result in math is that the minimum distance from the origin O to a linear subspace of R^n is attained at points P such that OP is orthogonal to the direction of the subspace. In this case, since the linear space is a hyperplane there is just one orthogonal direction to that hyperplane, and there is just one point in the intersection of the hyperplane and the line defined by the point O and the orthogonal direction to the hyperplane, and that intersection is just the points we alluded before.
layer8 3y ago
It does. For an n-dimensional cube of size 2, all surface points are at least distance 1 away from the center of the cube (the surfaces are where one of the dimensions is exactly 1 from the center, and the remaining dimensions can only increase that distance), and so a unit sphere around the center point is always contained within the cube.
meindnoch 3y ago
I. Hypersphere: length(v) <= 1 II. Hypercube: abs(max(v)) <= 1 Clearly II implies I, therefore the hypersphere is inside the hypercube. The points where they touch are: length(v) = abs(max(v)) = 1, the solutions of which are the points with exactly one ±1 coordinate.
jstx1 3y ago
You can and as the dimensions go up the volume of the inscribed ball as a percentage of the volume of the hypercube shrinks down a lot. There's a nice video about this from Numberphile and 3blue1brown - https://www.youtube.com/watch?v=6_yU9eJ0NxA https://www.youtube.com/watch?v=6_yU9eJ0NxA

by fwlr 3y ago

sillymath 3y ago

> Like, you can inscribe a circle inside a square and have it touch all sides without extending outside the square, and you can similarly “inscribe” a sphere inside a cube such that the sphere touches all sides of the cube without extending outside the cube. Does this hold true in higher dimensions? Of course, points with only one nonzero coordinates xi in $\{1,-1}$ are in both the sphere and the sides of the cube, and those are the minimal distance points from the sides of the cube to the center of the circle (since they are orthogonal to the hyperplane xi=+1, or xi=-1 that contains it). That also shows that no side of the cube extends outside of the circle. Some more math: A well known result in math is that the minimum distance from the origin O to a linear subspace of R^n is attained at points P such that OP is orthogonal to the direction of the subspace. In this case, since the linear space is a hyperplane there is just one orthogonal direction to that hyperplane, and there is just one point in the intersection of the hyperplane and the line defined by the point O and the orthogonal direction to the hyperplane, and that intersection is just the points we alluded before.

layer8 3y ago

It does. For an n-dimensional cube of size 2, all surface points are at least distance 1 away from the center of the cube (the surfaces are where one of the dimensions is exactly 1 from the center, and the remaining dimensions can only increase that distance), and so a unit sphere around the center point is always contained within the cube.

meindnoch 3y ago

I. Hypersphere: length(v) <= 1 II. Hypercube: abs(max(v)) <= 1 Clearly II implies I, therefore the hypersphere is inside the hypercube. The points where they touch are: length(v) = abs(max(v)) = 1, the solutions of which are the points with exactly one ±1 coordinate.

jstx1 3y ago

You can and as the dimensions go up the volume of the inscribed ball as a percentage of the volume of the hypercube shrinks down a lot. There's a nice video about this from Numberphile and 3blue1brown - https://www.youtube.com/watch?v=6_yU9eJ0NxA https://www.youtube.com/watch?v=6_yU9eJ0NxA