Geometry of dyadic polygons II: isomorphisms of dyadic triangles

TL;DR

This work completes the classification of dyadic triangle isomorphisms by leveraging representative hats and their automorphisms within the affine dyadic plane $D^2$. It provides a comprehensive, case-driven framework that ties boundary-type data, hull areas, and gcd relations to precise isomorphism and automorphism criteria, including explicit dyadic-matrix realizations. The results yield a full taxonomy of isomorphism types for dyadic hats and their pointed counterparts, enabling direct decision procedures for isomorphism and shedding light on the symmetry structures of dyadic convex sets and related commutative, entropic, idempotent groupoids. The methods solidify the algebraic-geometric understanding of dyadic polytopes and their automorphism groups, with potential implications for the broader theory of dyadic convexity and affine $D$-spaces.

Abstract

This paper is the second part of a two-part paper investigating the structure and properties of dyadic polygons. A dyadic polygon is the intersection of the dyadic subplane $D^2$ of the real plane $R^2$ and a real convex polygon with vertices in the dyadic plane. Such polygons are described as subreducts (subalgebras of reducts) of the affine dyadic plane $D^2$, or equivalently as commutative, entropic and idempotent groupoids under the binary operation of arithmetic mean. The first part of the paper contained a new classification of dyadic triangles, considered as such groupoids, and a characterization of dyadic triangles with a pointed vertex. This second part investigates isomorphisms of dyadic triangles, and provides a full classification of their isomorphism types.

Figures (2)

• Figure 1:
• Figure 2: The isomorphic hats $T_{1,3,5}$ and $T_{5,15,1}$.

Theorems & Definitions (44)

• Theorem 3.1
• Proposition 3.2
• Theorem 3.3
• Proposition 4.1
• Lemma 5.1
• proof
• Lemma 5.2
• proof
• Lemma 5.3
• proof