A memory provides reduction in access latency for frequently-accessed values by self-organizing to always move a requested value to a front-most central storage element of a spiral. The occupant of the central location is swapped backward, which continues backward through the spiral until an empty location is swapped-to, or the last displaced value is cast out of the last location in the spiral. The elements in the spiral may be cache memories or single elements. The resulting cache memory is self-organizing and for the one-dimensional implementation has a worst-case access time proportional to N, where N is the number of tiles in the spiral. A k-dimensional spiral cache has a worst-case access time proportional to N1/k. Further, a spiral cache system provides a basis for a non-inclusive system of cache memory, which reduces the amount of space and power consumed by a cache memory of a given size.