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In my recent question about polygonal tilings where tiles can occur in infinitely many positions, both constructions given as solutions are of self-similar nature. This means in particular that there is no periodicity and thus no fundamental unit.

What if we exclude the possibility of self-similarity by requiring the tiling to be periodic?
The two tilings I used as initial examples are both periodic with $p(\mathcal T)=12$.

  • Can we do better than $12$?

EDIT: Yes, of course we can, as per Wlodek Kuperberg's comment. This excludes the possibility of self-similarity only partially, so to speak. So I will squeeze this out by sharpening the conditions as follows:

  • If we require moreover that the tiling has a fundamental domain which contains only one tile of each position, how big can such a fundamental domain be?

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  • Same question for periodic tilings of $\mathbb R^3$, where an easy construction takes a truncated octahedron (which tiles $\mathbb R^3$, left picture) as fundamental unit and subdivides it into 48 identical tiles.

enter image description here $\qquad$ enter image description here

The right picture shows one of those pyramid-like tiles (more precisely, a triangular and a quadrangular pyramid with a common face). There are two for each vertex of the truncated octahedron, and they have obviously all different orientations. They are cut off by the $6+3$ perpendicular bisector planes of the edges and the 3 planes of the octahedron's squares. This realizes $p(T)=48$ and generalizes quite directly the plane construction with a regular hexagon as fundamental unit, subdivided by the 3 main diagonals and the 3 "heights" into 12 triangles with angles $30°,60°,90°$.

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  • $\begingroup$ The answer to your previous question posted by Jan Kyncl also answers this one: At a finite step of the construction described there, a large square Q is tiled by copies of the 3-4-5 triangle T in a large number of positions; the larger T, the greater the number of positions. Since Q tiles the plane by a lattice, you get a periodic tiling with copies of T with an arbitrarily large number of positions. The question in 3 dimensions is answered automatically, by taking a prism over T, then by iteration of the prism operation, an analogous example is produced in higher dimensions. $\endgroup$
    – Wlodek Kuperberg
    Commented Dec 22, 2017 at 17:39
  • $\begingroup$ @WlodekKuperberg Sure enough... I was so impressed by the self-similarity that it didn't occur to me that just stopping at some finite step yields arbitrarily large numbers of positions! What I had in mind initially would be patterns with no "potential to be iterated in a self-similar way"... but I guess such a restriction is hard to formulate... $\endgroup$
    – Wolfgang
    Commented Dec 22, 2017 at 21:07
  • $\begingroup$ Well, one very sharp restriction would work: to require that there is a fundamental domain which only contains one tile of each position. $\endgroup$
    – Wolfgang
    Commented Dec 22, 2017 at 21:09

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