"Transcendental tilings": Do they exist? 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.

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

If the answer to Q1 is No, the following two questions are superfluous:

Q2. What is the class of all such transcendental tilings
  (if I may coin a term)?
Q3. How does this class relate to the 
  aperiodic tilings?

 A: Answering Q1, I believe there is a transcendental tiling. Let us begin with this tiling with congruent convex pentagons:
          

Notice that one can rearrange any of the triples of pentagons that form the regular hexagon, by rotating any triple we want by 180 degrees. Thus we can have two kinds of triples: pointing up or down. Now, the entire tiling can be viewed as formed by triples, each filling a hexagon, and we can view the tiling by hexagons as the union of concentric "annuli" with disjoint interiors. Make it so that the annuli's thickness (not just diameter) is strictly increasing, if needed, increasing exponentially. Then arrange all triples in the same annulus to be pointing in the same direction, alternating the direction when passing from an annulus to the adjacent one.




(Image added by J.O'Rourke.)


Then, although I do not have a complete proof, I believe the tiling will be transcendental. My feeling is based on the example of a binary transcendental number formed by alternating blocks of zeros and ones of strictly increasing length.
