Reflections on grades

Abstract

The APL function REVERSE and a certain defined function generate a group of order four on permutation vectors. The group is extended with the GRADE-UP function to produce GRADE-DOWN and the order eight group known as D4, the fourth dihedral group. The symmetries of a square are generated by REVERSE and TRANSPOSE on square matrices. The resulting isomorphism maps GRADE-DOWN and its inverse to the ninety degree rotations. APL's powerful formalism easily reveals the isomorphisms between the domains of algebra and geometry.