Let's learn about [[Axel Thue]], the Norwegian mathematician who made significant contributions to the field of combinatorics on words, including the study of Abelian square-free words. **Who was Axel Thue?** - **Nationality:** Norwegian - **Lived:** 1863 – 1922 - **Areas of Focus:** Number theory, Diophantine equations, and combinatorics on words. **Thue's Work on Word Structure** Thue was a pioneer in investigating the structure and repetitions within words (sequences of symbols). His key contributions include: - **Square-free Words:** Thue investigated words that don't contain any consecutive repeated subwords (squares). For example, "abac" is [[Square-free Word|square-free]], while "b**anan**a" contains the square "**an**" twice. - **Infinite Square-free Words:** Thue proved the existence of infinite square-free words over a **[[three-letter alphabet]]**. This was a groundbreaking result with implications for pattern avoidance in strings. - **Thue-Morse Sequence:** While not directly about [[Abelian Square|Abelian squares]], Thue contributed to the study of the [[Prouhet-Thue-Morse sequence]], a famous infinite binary sequence with fascinating properties and connections to repetition avoidance. **Abelian Square-Free Words** - **Definition:** An Abelian square-free word is a word where no two consecutive subwords are anagrams (permutations) of each other. For example, "abcabc" contains an Abelian square ("abcabc"), whereas "abacb" is Abelian square-free. - **Thue's Connection:** Although Thue didn't explicitly introduce the concept, his fundamental work on constructing infinite square-free words laid the foundation for further investigation of Abelian square-free words and related properties. **Legacy** Axel Thue's investigations into word patterns had a lasting impact on combinatorics on words and theoretical computer science. His findings sparked research into: - **String and Sequence Analysis** - **Formal Languages** - **Applications:** Areas like data compression and DNA analysis sometimes draw upon concepts related to repetition avoidance in sequences. # References ```dataview Table title as Title, authors as Authors where contains(subject, "Axel Thue") or contains(authors, "Axel Thue") sort modified desc, authors, title ```