Farey Structure in Modulo Krinkle Tilings: Mediant Splicing and Generation of Prototiles from a Single Edge

Source 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 $t\ge 2$. 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 $k_1=m^{-1}\bmod k$, 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 $tk_2$.