Illustration of how the projection of high-dimensional odor representations onto lower-dimensional surfaces yields fragmented maps. A. Three-dimensional matrix of numbers representing stimulus qualities, where each element is similar to its neighbors in proportion to their Euclidean distance. For clarity, each digit identifies the location of an element in one of these dimensions; hence, the element 333 is at the center of the 5 × 5 × 5 cube depicted. B. The same three-dimensional matrix as in A, projected onto one dimension and only partially depicted. The six nearest neighbors (distance = 1 in Figure 2A) of element 224, highlighted in A, are now clustered at distances of 1 to 3 from its location, retaining their nearest-neighbor relationships to the maximum extent possible. Consequently, the five remaining nearest neighbors of element 324 (for example) can at best be clustered at distances of 3 to 7 from 324, with the interposition of non-neighboring elements 225 and 234; subsequently, the four other nearest neighbors of element 225 can at best be clustered at distances of 7 to 10 from its location, with an increasing number of non-neighboring interpositions (namely, the nearest neighbors of element 324). Various optimized retentions of local similarity relationships can be obtained using a self-organizing map algorithm [29, 30] to generate a fragmented remapping of the three-dimensional matrix onto one dimension, but the fundamental problem is unavoidable: the distance relationships among elements in 3-space (Figure 2A) cannot be replicated in 1-space (Figure 2B).