300 dimensions seems like way too many to have any kind of spatial intuition about. For me, it's easier to think of a n-tuple where n is 300. I'm not sure reall

300 dimensions seems like way too many to have any kind of spatial intuition about. For me, it's easier to think of a n-tuple where n is 300. I'm not sure really what the cardinality of each dimension in this problem is, but I'll assume its a) the same for every dimension and b) something like a subset of the natural numbers (like, one int per 100k english words or something). The sphere part is a little confusing at first, but it's really just a fancy (and yes, elegant) way of specifying a constraint on the n-tuple. Given some radius R, it defines the constraint sum(n_i^2) = R^2 for i=1..300. It might be useful to normalize R, and it is convenient to select R=300 so that the average contribution of each n_i^2 term is 1. At this point, you just have to pick 3M n-tuples. I would probably use a random number generator to a) fill in the n-tuple, b) adjust a constant factor to fit with real world R, and c) repeat 3M times.