Tue, 30 Sep 2025, 1:25 pm MT
In this talk, we will use an algebraic approach to study convex subsets of affine spaces over the ring of dyadic rationals.
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 dyadic space. Dyadic convex sets appear 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.
We will examine algebraic concepts and properties of dyadic polytopes (especially polygons), and contrast their behavior with that of polytopes in real affine space. On the basis of older results about the generation and structure of dyadic polytopes, we will present recent results (obtained with A. Mucka) on the representation, characterization and classification of dyadic triangles.