Farey Structure in Modulo Krinkle Tilings: Mediant Splicing and Generation of Prototiles from a Single Edge
Abstract
The Modulo Krinkle tilings of Imura (arXiv:2506.07638) form a family of non-periodic, spiral monohedral tilings parametrized by a reduced fraction m/k and an integer t≥ 2. We show that the Farey sum (mediant) (m1+m2)/(k1+k2) of two Farey-adjacent parameters is realized by an exact geometric operation on prototiles: the lower boundary path of the (m1+m2,k1+k2)-prototile is obtained by concatenating the parents' lower paths after an edge-length-preserving progressive rotation (fan-twist) of their edges. Conversely, every prototile admits exactly one fan-twist splice decomposition -- no non-adjacent parameters ever splice -- the cut position being k1=m−1bmod k, and the recursion descends the Stern-Brocot tree to a single unit edge. The combinatorial core of the operation is the classical standard factorization of Christoffel words; the contribution here is its exact edge-isometric realization on circular direction systems and the resulting structure theory for the Modulo Krinkle family, including the recently introduced variants: we prove a separation theorem stating that every variant prototile is the common recursively-generated core plus finitely many direction-invariant decoration edges. As a corollary of Imura's spiral-arm count formula, the two Farey parents are visible in the offset-free tiling itself: the numbers of counterclockwise and clockwise spiral arms are tk1 and tk2.
Related
- Parity-dependent sublattice structure of local 2×2 correlations in domino tilings of the Aztec diamond
- The structure and generation of the second maximal subgroups of the almost simple groups with alternating, classical or sporadic socle
- Automata, groups, limit spaces and tilings
- Tile sets consisting of two types of concave polygons derived from periodic tilings corresponding to non-periodic tilings with hat and turtle tiles
- Two examples in amplituhedron tilings
BibTeX
@misc{fujiwara2026farey,
title = {{Farey Structure in Modulo Krinkle Tilings: Mediant Splicing and Generation of Prototiles from a Single Edge}},
author = {Mikihiro Fujiwara},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.01270},
eprint = {2609.01270},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01270},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Farey Structure in Modulo Krinkle Tilings: Mediant Splicing and Generation of Prototiles from a Single Edge
AU - Mikihiro Fujiwara
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2609.01270
UR - https://arxiv.org/abs/2609.01270
AB - The Modulo Krinkle tilings of Imura (arXiv:2506.07638) form a family of non-periodic, spiral monohedral tilings parametrized by a reduced fraction m/k and an integer t≥ 2. We show that the Farey sum (mediant) (m_1+m_2)/(k_1+k_2) of two Farey-adjacent parameters is realized by an exact geometric operation on prototiles: the lower boundary path of the (m_1+m_2,k_1+k_2)-prototile is obtained by concatenating the parents' lower paths after an edge-length-preserving progressive rotation (fan-twist) of their edges. Conversely, every prototile admits exactly one fan-twist splice decomposition -- no non-adjacent parameters ever splice -- the cut position being k_1=m^−1bmod k, and the recursion descends the Stern-Brocot tree to a single unit edge. The combinatorial core of the operation is the classical standard factorization of Christoffel words; the contribution here is its exact edge-isometric realization on circular direction systems and the resulting structure theory for the Modulo Krinkle family, including the recently introduced variants: we prove a separation theorem stating that every variant prototile is the common recursively-generated core plus finitely many direction-invariant decoration edges. As a corollary of Imura's spiral-arm count formula, the two Farey parents are visible in the offset-free tiling itself: the numbers of counterclockwise and clockwise spiral arms are tk_1 and tk_2.
ER -