Encoding and Visualization in the Collatz Conjecture

2018/10/30 by George M. Georgiou, Georgiou, George M.
Mathematics · #FOS: Mathematics #History and Overview (math.HO) #math.HO

paper · pdf · doi:10.48550/arxiv.1811.00384

12 pages, 16 figures. Added reference to A304715, added new sections on reverse curves and miracle curves. Minor corrections

arxiv created 2019/01/03 · arxiv updated 2019/01/04

Abstract

The Collatz conjecture is one of the easiest mathematical problems to state and yet it remains unsolved. For each n≥ 2 the Collatz iteration is mapped to a binary sequence and a corresponding unique integer which can recreate the iteration. The binary sequence is used to produce the Collatz curve, a 2-D visualization of the iteration on a grid, which, besides the aesthetics, provides a qualitative way for comparing iterations. Two variants of the curves are explored, the r-curves and on-change-turn-right curves. There is a scarcity of acyclic r-curves and only three r-curves were found having a cycle of minimum length greater than 4.

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