Substitution invariant cutting sequences

Substitution invariant cutting sequences

Substitution Invariant Beatty Sequences

with θ irrational and taken to satisfy 0 < θ < 1; plainly this may be assumed without loss of generality. Evidently (fn) is a sequence of zeros and ones. Denote by w0 and w1 words on the alphabet {0, 1} ; that is, finite strings in the letters 0 and 1. Then the sequence (fn) is said to be invariant under the substitution W given by W : 0 −→ w0, 1 −→ w1, if the infinite strings fθ = f1f2f3 . . ....

Complete Characterization of Substitution Invariant Sturmian Sequences

We provide a complete characterization of substitution invariant inhomogeneous bidirectional pointed Sturmian sequences. The result is analogous to that obtained by Berthé et al. [5] and Yasutomi [21] for one-directional Sturmian words. The proof is constructive, based on the geometric representation of Sturmian words by a cut-and-project scheme.

A Homeomorphism Invariant for Substitution

We derive a homeomorphism invariant for those tiling spaces which are made by rather general substitution rules on polygonal tiles, including those tilings, like the pinwheel, which contain tiles in infinitely many orientations. The invariant is a quotient of Čech cohomology, is easily computed directly from the substitution rule, and distinguishes many examples, including most pinwheel-like ti...

On Guillotine Cutting Sequences

Imagine a wooden plate with a set of non-overlapping geometric objects painted on it. How many of them can a carpenter cut out using a panel saw making guillotine cuts, i.e., only moving forward through the material along a straight line until it is split into two pieces? Already fifteen years ago, Pach and Tardos investigated whether one can always cut out a constant fraction if all objects ar...