On linear ternary Intersection sequences and their properties

TL;DR

This paper classifies linear ternary intersection sequences derived from lines in a 3D grid, explores their complexity and palindromic properties, and establishes that sequences with factor complexity n+2 correspond to such line intersections.

Contribution

It provides a classification of linear ternary intersection sequences and links sequences with factor complexity n+2 to line intersections in 3D space.

Findings

Classification of linear ternary sequences is achieved.

A family of sequences analogous to Fibonacci words is constructed.

Sequences with factor complexity n+2 are exactly those from line intersections.

Abstract

Let $D^+$ be the first octant of the Euclidean space and consider the integral cube grid $G$ in $D^+$. The intersections of each line with $G$ form an infinite sequence of three letters which can be considered as an extension of well-known Sturmian words. A classification of such linear ternary sequences is presented and a family of examples is constructed from a notable sequence $S^M$ which could be viewed as an analogue of the Fibonacci word in the family of Sturmian words. The factor complexity and the palindromic complexity of these linear ternary sequences are also studied. The last result stated is that each ternary sequence with factor complexity $n+2$ is the intersection sequence of a line.