Explicit formula for the position of "bent wedge" tiles

Question

Is there an explicit formula to compute the position and the angle of the $n$th tile of a bent wedge tiling ?

Answer 1

Yes. In general, such formulas are long ugly aggregations of not particularly advanced mathematics. It will be much more readable to write them as pseudocode than as pure algebra.

One general approach could be to let $a=n \bmod 48$ determine the orientation of the tile, and then use one of $48$ subsidiary functions to determine its location based on $\left\lfloor \frac{n}{48}\right\rfloor$. Typically many of these subsidiary functions will be the same except for global rotations about the origin, and each of these functions just place the tiles at selected positions in a rectangular grid. The only tricky thing is then to make sure that the right subset of the positions in the rectangular grid get filled.

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Of course, the "bent wedge" tile also mathematically allows tilings of the infinite plane that cannot be expressed by any formula or program. Since each annulus of the tiling can turn left or right independently of the others, there are at uncountably many essentially different tilings, but there are only countably many possible formulas. (This is not a particular property of the bent-wedge tile, though -- it's also true for something as plain as a 2×1 rectangular tile).