DYADIC POLYGONS

Abstract

Dyadic rationals are rationals whose denominator is a power of 2. Dyadic triangles and dyadic polygons are, respectively, defined as the intersections with the dyadic plane of a triangle or polygon in the real plane whose vertices lie in the dyadic plane. The one-dimensional analogs are dyadic intervals. Algebraically, dyadic polygons carry the structure of a commutative, entropic and idempotent algebra under the binary operation of arithmetic mean. In this paper, dyadic intervals and triangles are classified to within affine or algebraic isomorphism, and dyadic polygons are shown to be finitely generated as algebras. The auxiliary results include a form of Pythagoras' theorem for dyadic affine geometry.