Geometry of dyadic polygons I: the structure of dyadic triangles

TL;DR

The paper addresses the classification of dyadic triangles in the dyadic plane $\\mathbb{D}^2$ within the commutative binary mode framework. It shows that every pointed dyadic triangle is isomorphic to a representative hat $T_{i,j,m}$ with odd $i,j,m$ and introduces encoding triples to index pointed-orthogonal isomorphism classes, providing a corrected and streamlined characterization that supersedes earlier results. A key outcome is that two pointed dyadic triangles are isomorphic (with the same orientation) precisely when their encoding triples match, enabling a complete, explicit classification. The results lay groundwork for a broader study of dyadic polygons and their connections to lattice polygons and digital geometry.

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.

Figures (2)

• Figure 1: The triangle $T_{i,j,m,n}$
• Figure 2: The isomorphic hats $T_{i,j,m,0}$ and $T_{m-i,j,m,0}$.

Theorems & Definitions (39)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Example 3.1
• Lemma 3.2
• Lemma 3.3
• proof
• Remark 3.4
• Corollary 3.5
• proof