Editing Curse of dimensionality (section)

=== Sampling ===
There is an exponential increase in volume associated with adding extra dimensions to a [[Space (mathematics)|mathematical space]]. For example, 10<sup>2</sup>&nbsp;=&thinsp;100 evenly spaced sample points suffice to sample a [[unit interval]] (try to visualize a "1-dimensional" cube, i.e. a line) with no more than 10<sup>−2</sup> = 0.01 distance between points; an equivalent sampling of a 10-dimensional [[unit hypercube]] with a lattice that has a spacing of 10<sup>−2</sup> = 0.01 between adjacent points would require 10<sup>20</sup> = [(10<sup>2</sup>)<sup>10</sup>] sample points. In general, with a spacing distance of 10<sup>−''n''</sup> the 10-dimensional hypercube appears to be a factor of 10<sup>''n''(10−1)</sup> = [(10<sup>''n''</sup>)<sup>10</sup>/(10<sup>''n''</sup>)] "larger" than the 1-dimensional hypercube, which is the unit interval. In the above example ''n'' = 2: when using a sampling distance of 0.01 the 10-dimensional hypercube appears to be 10<sup>18</sup> "larger" than the unit interval. This effect is a combination of the combinatorics problems above and the distance function problems explained below.