Geometry of dyadic polygons I: the structure of dyadic triangles

Abstract: Dyadic rationals are rationals whose denominator is a power of 2. A dyadic n-dimensional convex set is defined as the intersection with n-dimensional dyadic space of an n-dimensional real convex set. Such a dyadic convex set is said to be a dyadic n-dimensional polytope if the real convex set is a polytope whose vertices lie in the dyadic space. Dyadic convex sets are described as subalgebras of reducts of faithful affine spaces over the ring of dyadic numbers, or equivalently as commutative, entropic and idempotent groupoids (binars or magmas) under the binary operation of arithmetic mean. This paper investigates the structure of dyadic polygons (two-dimensional polytopes), in particular dyadic triangles, following some earlier results. A new classification of dyadic triangles is provided. In addition, dyadic triangles with a pointed vertex are characterized.