Why is space three-dimensional and how may groups be seen? (Q1073398)

This paper contains two applications of tolerance space theory. The first is a speculative application of the result [see \textit{A. B. Sossinskij}, ibid. 5, 137-167 (1986; see the preceding review)] that tolerance dimension can only take the values 0, 1, 2, or 3. This is used to ''justify'' the three dimensionality of the visual world. The second application introduces the notion of the group crystal associated to a set \(\{H_ i\}\), \(i\in I\), of subgroups of a group G. This is a tolerance space T defined on the set of right cosets of the \(H_ i\) in G. Two cosets \(aH_ i\) and \(bH_ i\) are within tolerance of each other if their intersection is nonempty. The idea is illustrated with a discussion of two generator groups.