5 ms·

But there is another way to think of the high-dimensional balls where "spikey" is the right visualisation. Consider the volume of a cap. Take a plane that is

But there is another way to think of the high-dimensional balls where "spikey" is the right visualisation. Consider the volume of a cap. Take a plane that is 90% of the distance from the centre to the edge, and look at what percentage of the volume is "outside" that plane. When the dimension is high, that volume is negligible. And when the dimension is really high you can get quite close to the centre, and still the volume you cut off is very small. In our 3D world the closest thing that has this property is a spike. You can cut off quite close to the centre, and the volume excised is small. The sense in which a high-dimensional ball is not a spikey thing is in the symmetry, and the smoothness. So when you want to develop an intuition for a high dimensional ball you need to think of it as simultaneously symmetrical, smooth, and spikey. Then think of another five impossible things, and you can have breakfast.

Hugsun 2y ago
That's a good point. High dimensional objects can obtain very unintuitive properties, like you describe. This to me feels similar in many ways to how a corner in a high dimensional n-cube, although 90 degrees, no matter how you measure it, seems extremely spiky. As the shape does not increase in width, but the corners extend arbitrarily far away from the center. A property reserved for spiky things in 3D.
dawnofdusk 2y ago
I think our geometric intuition could never be good for a high dimensional object. Consider a sphere in 3D. It's represented by the points which satisfy x^2 + y^2 + z^2 = 1. Because there are only three coordinates, knowing the value of one of them greatly reduces the possibilities for the other two. For example, if I know z is close to 1, then my point is close to the north pole (x and y are both close to zero). However, if I have the n-sphere x_1^2 + x_2^2 + ... + x_n^2 = 1, knowing the value of x_1 gives me very little information about all the other coordinates. And humans' interaction with the geometry is reality is somewhat limited to manipulating one coordinate at a time, i.e., our intuition is for built on things like moving our body linearly through space, not dilating the volume or surface area of our bodies.
- saagarjha 2y ago
  If x_1 is close to one, then the more x_n values there are the closer they are to zero.

Hugsun 2y ago

That's a good point. High dimensional objects can obtain very unintuitive properties, like you describe. This to me feels similar in many ways to how a corner in a high dimensional n-cube, although 90 degrees, no matter how you measure it, seems extremely spiky. As the shape does not increase in width, but the corners extend arbitrarily far away from the center. A property reserved for spiky things in 3D.

dawnofdusk 2y ago

I think our geometric intuition could never be good for a high dimensional object. Consider a sphere in 3D. It's represented by the points which satisfy x^2 + y^2 + z^2 = 1. Because there are only three coordinates, knowing the value of one of them greatly reduces the possibilities for the other two. For example, if I know z is close to 1, then my point is close to the north pole (x and y are both close to zero). However, if I have the n-sphere x_1^2 + x_2^2 + ... + x_n^2 = 1, knowing the value of x_1 gives me very little information about all the other coordinates. And humans' interaction with the geometry is reality is somewhat limited to manipulating one coordinate at a time, i.e., our intuition is for built on things like moving our body linearly through space, not dilating the volume or surface area of our bodies.

saagarjha 2y ago
If x_1 is close to one, then the more x_n values there are the closer they are to zero.