Re:High-dimensional spheres are "spikey"
by ColinWright on Sunday May 20, 2012 @04:11AM
(#3998265)
> So 'pointy' cubes reasons out the 'paradoxes'
> without resorting to 'spikey' spheres.
In particular, this comment: http://news.ycombinator.com/item?id=3998259
Thanks for the point - but your argument that 'from a surface almost every step leads you out' doesn't seem to be true in particular.
Imagine a sphere in n-dimensions, S = {Sum(x_i^2)<1}. Top of the sphere is p=(1,0,0,...,0). Now for any random direction r, if you take a very small step from p towards r, you have exactly 50% chance to be inside the sphere.
To be mathematically precise, for any r (unit vector) chosen at random, probability that there is e>0 such that p + e*r is within S is 1/2.
So half the time the steps take you out, half the time it takes you in - considering the step is small enough compared to the radius.
Re:High-dimensional spheres are "spikey"
by ColinWright on Wednesday May 23, 2012 @02:36AM
(#4011853)
Answered here: http://news.ycombinator.com/item?id=3998259
It paints the visual picture where corners of a cube are more 'pointy' in higher dimensions. Another way to see this is basically that the corners basically divide the whole angle in many more parts in higher dimensions. For 2D, four corners cover the full 360 degree angle. For 3D, 8 corners. For a million dimensions, you can connect 2^(1mill) corners of different cubes to each other without overlapping each other.
So 'pointy' cubes reasons out the 'paradoxes' without resorting to 'spikey' spheres.