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Basic results on bases of the free abelian group Z2 are proved by elementary and geometric methods. Bases are characterized by determinants of matrices. Primitive triangles and primitive parallelograms are introduced and characterized by tiling of the plane. Christoffel words are introduced by discrete geometric means: they code paths which discretize finite segments with integral endpoints. Their inner maximal factor is a palindrome, which is the cutting word associated with the segment. Their standard factorization is studied; uniqueness is proved (Borel and Laubie). The tree of Christoffel pairs is studied using Sturmian morphisms: these are the substitutions that preserve conjugates of Christoffel words.
Title: Words
Description:
Basic results on bases of the free abelian group Z2 are proved by elementary and geometric methods.
Bases are characterized by determinants of matrices.
Primitive triangles and primitive parallelograms are introduced and characterized by tiling of the plane.
Christoffel words are introduced by discrete geometric means: they code paths which discretize finite segments with integral endpoints.
Their inner maximal factor is a palindrome, which is the cutting word associated with the segment.
Their standard factorization is studied; uniqueness is proved (Borel and Laubie).
The tree of Christoffel pairs is studied using Sturmian morphisms: these are the substitutions that preserve conjugates of Christoffel words.
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