Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. The curse of dimensionality (also known as the Hughes effect) is the problem caused by the exponential increase in volume associated with adding extra dimensions to a (mathematical) space. The term was coined by Richard Bellman. For example, 100 evenly-spaced sample points suffice to sample a unit interval with no more than 0.01 distance between points; an equivalent sampling of a 10- dimensional unit hypercube with a lattice with a spacing of 0.01 between adjacent points would require 1020 sample points: thus, in some sense, the 10-dimensional hypercube can be said to be a factor of 1018 "larger" than the unit interval. Another way to illustrate the "vastness" of high- dimensional Euclidean space is to compare the proportion of a hypersphere with radius r and dimension d, to that of a hypercube with sides of length 2r, and equivalent dimension.

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