Stochastic Rhombic Tiling

Using the de Bruijn grid method (or the equivalent projection method) we can construct quasi-periodic tilings using (n−1)/2 distinct types of rhombic tiles for any odd integer n greater than 3. The rhombs have internal angles of [jϕ, (n−j)ϕ] with j = 1, 2, (n−1)/2, where ϕ = π/n. For fairly small values of n, such tilings contain occasional complete 2n-gons, and these always are comprised of exactly n copies of each of the (n−1)/2 types of tiles. With n = 5, the figure below shows the only two distinct (up to rotation) “tiled circles” that can appear in a rhombic Penrose tiling:

It’s worth noting that not all possible tiled circles can appear in such a tiling. For example, using the same set of tile we can construct the decagon

and yet this pattern cannot occur in a quasi-periodic tiling, as can be seen from the fact that it doesn’t satisfy the “matching conditions” for Penrose tilings.

For larger values of n, the number of possible 2n-gons appearing in a quasi-periodic rhombic tiling increases rapidly. For example, the figure below shows just some of the 14-gons that appear in septa-tilings (i.e., tilings based on n = 7).

For any n, the regions of the de Bruijn grid corresponding to a 2n-gon consist of n lines, one in each of the n principal directions, and each tile corresponds to one of the n(n−1)/2 intersections of two of these lines. This leads (again) to the fact that each of the (n−1)/2 types of tiles appears exactly n times. Of course, the most obvious and symmetrical such arrangement of grid lines is based on the regular n-gon as illustrated below for the case n = 7.

This grid corresponds to the left-most circular region of tiling in the figure below, and the similar grids for n = 9 and n = 11 lead to the other two “circular” regions shown below.

The standard form of the 2n-gon for n = 19 is shown below.

In every case (for an odd integer n) the 2n-gon consists of the (n−1)/2 types of tiles described previously, as is obvious from considering the sequence of angles as shown in the figure below for the case n = 11.

We can also see that this standard form of the 2n-gon tiling contains several subsets whose outlines are radially symmetrical, even though the tiles within those outlines are not radially symmetrical. Examples of these subsets are highlighted in the figured below (again for n = 11).

We can re-arrange the tiles by radially inverting any of these subsets. The same can be done for the tiled circles for any other n, and this leads to multiple possible arrangements, such as those depicted at the beginning of this article for the case n = 7. This raises some interesting questions. First, what is the number of distinct arrangements (up to rotations and reflections) for any given n, and how many of these can appear in a quasi-periodic tiling? Second, can all of these arrangements be reached from the “standard form” purely by means of inversions? Regarding the second question, note that whenever a vertex joins precisely three edges, the three tiles can be inverted, so we could ask the restricted question, i.e., whether all possible arrangements can be reached by three-tile inversions. (Of course, we’ve already seen that not all of the arrangements that can be reached by inversion can occur in a quasi-periodic tiling.) We could also ask if it is possible to tile one of these circles with some other multi-set of these tiles.

Since each of the n types of tiles occur equally often in a tiled circle of this kind, such circles must become progressively more rare as n increases, because the overall numbers of the different types of tiles must be in proportion to the sine of the minor interior angle. In terms of the grid lines, the outer radius of the “star” of intersections as depicted above must become progressively larger – in proportion to the radius of the inner polygon of intersections – as n increases. This also applies to non-standard intersections, i.e., those with different arrangements of intersections, while still being contiguous in just a single set of n grid lines. The absolute radius of the “star” is limited by the unit spacing between grid lines, so the radius of the inner polygon must become progressively smaller (relative to the grid spacing). This implies that a complete tiled circle requires a near simultaneous intersection of all n grid lines. Clearly this becomes less frequent as n increases. Nevertheless, there seems to be nothing to prevent their occurrence in an infinite grid for any n, so every quasi-periodic rhomb tiling should possess infinitely many complete tiled circles, although they are exceedingly rare for large values of n.

It’s interesting to consider the tiling the results from an arbitrarily large value of n, and then taking this to the limit as n increases to infinity. We then get a continuous density distribution for the continuous spectrum of tile types. Letting α denote the smaller of the two interior angles in a rhomb tile, the range of tiles extends from α = 0 to α = π/2, and the asymptotic density is simply

Thus the “thinnest” tiles occur least frequently, and the “squarest” occur most frequently. A region consisting of any fixed number of tiles becomes smaller and smaller relative to the distance between the grid lines, so the quasi-periodicity becomes less noticeable. In the limit as n increases to infinity, we simply have a region constructed with grid lines, no two of which are exactly parallel, because the spacing between parallel grid lines becomes infinite in proportion to the size of the tiles. In a sense, the quasi-regularity has disappeared completely, and the tiling is entirely stochastic, based entirely on the arbitrary offsets of the individual grid lines. To illustrate, the figure below shows a typical region of a tiling with n = 41.

Even though n is still finite, this already shows the general character of a completely stochastic tiling. The greater frequency of “squarish” tiles and the rarity of extremely thin tiles is apparent. Some limited portions of this tiling almost look like a regular grid on a curved surface, but then imperfection appear. Notice that, as always, any vertex where precisely three edges meet can be inverted, so we could re-arrange the tiling in a step-wise manner. Some tiles can be moved a great distance in this way.

Strictly speaking, in the limit as n goes to infinity, no two of the tiles appearing in any finite region would be exactly alike. The tiles would be drawn from a continuous spectrum, with the density distribution noted above. So, in a sense, this is at the opposite extreme from efforts to tile the plane with the minimum number of distinct shapes. Here we have no two shapes exactly the same, and yet there is still a very visible lawfulness in the arrangement.

Just as in the case of small n, we find that even with very large n there are some clusters or more than three tiles that can be inverted. Two such clusters are highlighted in yellow in the figure below.

In each case the equality of opposite angles can easily be seen, due to the parallel sides of each rhomb. Every pair of edges meeting on the boundary can be traced, by parallel steps, to a pair of adjoining edges on the opposite side of the boundary, so the angles are equal. Each of these regions has eight edges. Another region of the n = 41 tiling, with some of the convex clusters highlighted, is shown below.

Of course, each cluster of three tiles meeting at a vertex has a boundary with six edges. It seems that every convex cluster has a symmetrical boundary – like an anti-podal sphere – because opposite edges can be identified with each other.

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