Primitive Two-Dimensional Words and Iterated Pedal Triangles via Symbolic Coding
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911633188585472 |
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| author | Smith, Taylor J. |
| author_facet | Smith, Taylor J. |
| contents | The notion of a two-dimensional word arises naturally in the study of combinatorics on words, while the iterative construction of pedal triangles results in a rich dynamical system in the study of geometry. At first, these two classes of objects seem to be unrelated. However, it is known that for all $n \geq 1$, the number of primitive two-dimensional words of dimension $2 \times n$ over a binary alphabet agrees with the number of triangles whose first similar pedal triangle is their $n$th pedal triangle. We construct a finite four-symbol coding of the sorted pedal map and use the resulting branch itineraries to give a bijection between these two classes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_26832 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Primitive Two-Dimensional Words and Iterated Pedal Triangles via Symbolic Coding Smith, Taylor J. Dynamical Systems Formal Languages and Automata Theory Metric Geometry 37B10 (primary), 51M15, 68Q45 (secondary) The notion of a two-dimensional word arises naturally in the study of combinatorics on words, while the iterative construction of pedal triangles results in a rich dynamical system in the study of geometry. At first, these two classes of objects seem to be unrelated. However, it is known that for all $n \geq 1$, the number of primitive two-dimensional words of dimension $2 \times n$ over a binary alphabet agrees with the number of triangles whose first similar pedal triangle is their $n$th pedal triangle. We construct a finite four-symbol coding of the sorted pedal map and use the resulting branch itineraries to give a bijection between these two classes. |
| title | Primitive Two-Dimensional Words and Iterated Pedal Triangles via Symbolic Coding |
| topic | Dynamical Systems Formal Languages and Automata Theory Metric Geometry 37B10 (primary), 51M15, 68Q45 (secondary) |
| url | https://arxiv.org/abs/2604.26832 |