I've dealt with the need to explore the set of all directions (say, points on the unit sphere) in high-dimensional Euclidean space, and it is certainly true that it's *much* harder to get close to all possible directions. This is known as the "curse of dimensionality" (a phrase that probably originated with John Tukey). For the case in point, Mathematica tells me that a spherical cap in R^n of radius = pi/4 has a relative solid angle of 2.22 x 10^(-273). ----------------------------------------------------------------- In[1]:= alpha[n_] := 2.0 * Pi^(n/2) / Gamma[n/2] In[2]:= scap[theta_,n_] := alpha[n-1] * NIntegrate[Sin[u]^(n-2), {u,0,theta}] In[3]:= fraccap[theta_,n_] := scap[theta,n] / scap[Pi,n] In[4]:= fraccap[Pi/4,1800] -273 Out[4]= 2.223961284661308 10 ----------------------------------------------------------------- But despite the interesting geometrical phenomenon that, starting in 9 dimensions, the sphere tangent to the 2^n "corner" spheres of an n-cube (each of diameter = half the side of the cube) has diameter >= 1 (with strict inequality for dimensions > 9), I would *not* call high-dimensional spheres "spiky". This is imo misleading and not helpful to one's intuition. Rather, I'd call them "roomy". --Dan << http://www.penzba.co.uk/cgi-bin/PvsNP.py?SpikeySpheres " ... I've recently been working on an optimisation problem, and I've come to realise that I can consider it as wandering around on a smooth landscape in 1800 dimensions. Strange number, I know, but that's the way it's worked out. The problem is that the usual visualisation of hill-climbing, or hill-descending, is that you're on a gently undulating vista, and that some directions are up, some are down, and it's easy to decide which is which. You write the code, set off, and somehow the system never finds a good solution. Part of the problem is that there is simply a lot of space to explore. If you discretise space and have 1000 places to be in each dimension, 2 dimensions gives you a million places to be. That's not so bad. 1800 dimensions gives you 10^5400 places to be. That's not good. You definitely need to move in moderately large strides, and then hone your solution by using binary chop or similar techniques. But it's worse than that. The problem is that the error function may be "smooth," but your intuition of what this means is wrong. ... "
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