The geometric primitives of APL

Abstract

The roles of φ, θ, and ⦰ in describing the symmetries of the square are investigated. This leads to relationships between the results of these functions and the cyclic and dihedral groups of orders 2 and 4. Numerical modelling of these groups is accomplished through the primitives ⊥ and ⊤ which are thus seen to play an important auxiliary role. Next the behavior of the functions @@@@ and @@@@ applied to permutation vectors is investigated, and parallels are observed between these primitives and the primitives φ, θ, and ⦰. These similarities are heightened by considering the partitioning of all permutations of a given order according to their internal symmetries.