A Septagonal Non-Periodic Tiling
A tiling consists of an arrangement of shapes covering the plane without overlap or gaps. Usually we restrict ourselves to a finite number of shapes and fixed sizes (although we can also consider tiles of geometrically decreasing and increasing sizes). The simplest and, in some sense, most natural tilings are periodic, in the sense that they consist of a meta-tiling of duplicated regions under translation. Three common periodic tilings, one consisting of triangles, one of squares, and one of hexagons, are illustrated below.
It's also possible to tile the plane non-periodically. The best-known non-periodic tilings are the Penrose tilings based on the pentagon and the golden ratio. It's interesting to consider other non-periodic tilings. The figure below illustrates a tiling based on the septagon (7-sided regular polygon), although it actually produces a 14-way symmetry.
The 14-sided polygon is tiled with just the three types of tiles shown below:
The key to this tiling is the fact that these three basic tiles can be combined to produce scaled-up copies of themselves. The scale factor is a/b = 2.24697... Alternatively, these tiles can be sub-divided into scaled-down versions of themselves. Consequently, beginning with the 14-sided tiled polygon shown above, we can "inflate" the figure by a factor of 2.24697... This is done by scaling up the entire figure by this factor, and then sub-dividing each expanded tile into tiles of the original size. Obviously we can continue this process indefinitely, so this proves that we can tile the entire plane with this 14-way symmetry using just these three tiles. The result of the first inflation step is shown below.
We will call the tiles with edge lengths (a,a,b) the "G" tiles (for Green), and the tiles with edge lengths (a,b,c) will be called "B" tiles (for Blue), and the tiles with edge lengths (a,a,c) will be called Y tiles (for Yellow). We can start with an initial 14-gon tiling consisting of just 14 G tiles, but since the overall figure has perfect 14-way symmetry, we need consider only one of these slices, consisting of just a single G tile. Thereafter, with each inflation step, each G tile is partitioned into 2 G tiles, 2 B tiles, and 1 Y tile. Likewise, each B tile is partitioned into 1 G tile, 2 B tiles, and 1 Y tile. Also, each Y tile is partitioned into 2 G tiles, 3 B tiles, and 2 Y tiles. Thus if we let Gn, Bn, and Yn denote the numbers of tiles of the respective types contained in the original slice after the nth inflation step, we have the relations
The determinant of the coefficient array is 1, and the characteristic polynomial is
which has the roots
We also have the recurrence relation for the number of G tiles after the nth inflation
and likewise for the numbers of B and Y tiles. We can also give the number of the different kinds of tiles explicitly in terms of the characteristic roots as follows.
where
Interestingly, the coefficients of the expressions for Bn and Yn are just permutations of each other (with opposite signs). The values of the characteristic roots are l1 = 0.307978..., l2 = 0.643104..., and l3 = 5.048917..., so as n increases the terms involving l1 and l2 go to zero, so only the term involving l3 is significant. As a result, the numbers of tiles of the three types are asymptotic to
and the total number of tiles goes to
From this we can determine the asymptotic proportions of the three kinds of tiles. We have
Notice that Bn/Tn equals the ratio of edge lengths b/a, which equals 2sin(p/14). We can also see that Gn/Tn equals 1 + a/b = l1, and Yn/Tn equals a/b - 2 = a/c - 1. In addition, we have Yn/Gn = c/a. Since the characteristic polynomial is irreducible, we know the roots are irrational, so the asymptotic proportions of the tiles are also irrational, which implies that the tiling is non-periodic. (For a periodic tiling, the entire plane is covered with copies of some finite region, in which the proportions of different tile types is obviously rational, and this is the asymptotic proportion as well.)
Incidentally, although the characteristic polynomial (1) is irreducible over the rationals, if we put l = s2 we have the factorization
In terms of the edge lengths a,b,c of our tiles, we notice that the roots of the right hand factor are c/a, -a/b, and b/a - 1, and the roots of the left hand factor are a/b, -c/a, and c/a + 1. is a root of the left hand factor, and c/a is a root of the right hand factor. Thus it's not surprising that there are numerous relations between the edge lengths and the characteristic roots, such as
We also have several relations between the edge lengths themselves, such as
The left hand relation implies that c/a is the sum of the infinite geometric series in b/a, whereas the right-most relation signifies that b is half the harmonic mean of a and c.
The tiling is self-similar at each inflation stage, in the sense that the pattern on the existing region is replicated once the tiling has been expanded and the larger tiles have been partitioned back into tiles of the original size. Therefore, it is possible in principle to construct the entire tiling incrementally by adding tiles of a fixed size to the outer edge, one at a time. However, the correct pattern that must be followed in order to continuing replicating the tiling on larger and larger scales becomes increasingly complex.
One notable feature of this tiling is that the vertices of the adjoining tiles don't appear to always be aligned. In other words, it appears that the vertex of one tile contacts another tile at some intermediate point on its edge. However, the distinction between edge points and vertices is somewhat conventional, because any point on an edge can be regarded as a vertex between two edges making an angle of p.