Primitive roots of bi-periodic infinite pictures

Nicolas Bacquey (1)

1 Equipe AMACC - Laboratoire GREYC - UMR6072

This paper defines and studies the notion of primitive root of a bi-periodic infinite picture, that is a rectangular pattern that tiles the bi-periodic picture and contains exactly one representative of each equivalence class of its pixels. This notion extends to dimension 2 the notion of primitive root of a bi-infinite periodic word. We prove that, for each bi-periodic infinite picture P , – there exists at least one primitive root of P ; – there are at most two ordered pairs of positive integers (m, n) such that every primitive root of P has size m × n; – for each such pair (m, n), every rectangular pattern of size m × n extracted from P is a primitive root of P. We also discuss some additional properties of primitive roots.

Mots clés: biperiodic pictures

Domaines: Logique en informatique [cs.LO]; Informatique [cs]; Mathématique discrète [cs.DM]

Nicolas Bacquey. Primitive roots of bi-periodic infinite pictures. Words 2015, Sep 2015, Kiel, Germany. ⟨hal-01178256⟩