Let $T$ be a tiling of the plane. Fix an origin and shoot a ray $r$ from the origin. Mark off points $p_i$ along $r$ separated by unit distance. Compute from $r$ a binary number $0 < b(r) < 1$ that alternates $0$'s and $1$'s for each marked point $p_i$ as the ray enters a new tile of $T$. For example, the square tiling and illustrated ray below lead to $$ .00011011100010011011100 \ldots $$

^{ Square tiles side length $=7/3$. Ray slope $= 1/\sqrt{2}$. }

To avoid thin tiles, assume every tile includes a disk of diameter $> 1$ so that more than one $p_i$ could land in a tile. One needs a rule when $p_i$ is on the boundary of a tile to make $b(r)$ well-defined, but I think that detail is not relevant to my question.

It is not difficult to find tilings and rays where $b(r)$ is rational, irrational, or transcendental, for example, by selecting the slope appropriately in the above example.

. Is there a tiling $T$ such that every $b(r)$, for all origins and rays $r$, is transcendental?Q1

If the answer to ** Q1** is

*No*, the following two questions are superfluous:

. What is the class of all suchQ2transcendental tilings(if I may coin a term)?

. How does this class relate to the aperiodic tilings?Q3