Metallic-mean quasicrystals as aperiodic approximants of periodic crystals

Ever since the discovery of quasicrystals, periodic approximants of these aperiodic structures constitute a very useful experimental and theoretical device. Characterized by packing motifs typical for quasicrystals arranged in large unit cells, these approximants bridge the gap between periodic and aperiodic positional order. Here we propose a class of sequences of 2-D quasicrystals that consist of increasingly larger periodic domains and are marked by an ever more pronounced periodicity, thereby representing aperiodic approximants of a periodic crystal. Consisting of small and large triangles and rectangles, these tilings are based on the metallic means of multiples of 3, have a 6-fold rotational symmetry, and can be viewed as a projection of a non-cubic 4-D superspace lattice. Together with the non-metallic-mean three-tile hexagonal tilings, they provide a comprehensive theoretical framework for the complex structures seen, e.g., in some binary nanoparticles, oxide films, and intermetallic alloys.


Shape of fundamental motifs
Type IA tilings are parametrized by two integers defined by the subdivision patterns of the ST, LT, and R tile shown in Fig. 1k of the main text. To obtain the second-generation tiling from the first-generation tiling, we may either apply the subdivision rules in Fig. 1k or place the second-generation fundamental motifs at the vertices of the first-generation tiling. Generally, the thus placed second-generation fundamental motifs at the neighboring vertices partly overlap. If we choose the type IA fundamental motif such that its six LT domains are triangular as shown in Supplementary Fig. 1a, then the overlap of two such motifs along the short edges of the first-generation tiling and thus the m parameter of the subdivision rule in Fig. 1k of the main text are not unambiguously defined. In the sketch shown in Supplementary Fig. 1a, they overlap by 2 R tiles (so that m = 4) but they could also be pushed 1 R tile closer to each other (so that m would be 3) or 1 and 2 R tiles farther apart (so that m would be 5 and 6, respectively).

a b
Supplementary Figure 1: Two choices for the fundamental motifs for k = 6 type IA tiling. Panel a shows a smaller motif which, when placed at the vertices of the ST, LT, and R tiles, does not define the value of m that characterizes the subdivision rule in Fig. 1k of the main text. The motif in panel b removes the ambiguity when packed along the short edges of the first-generation tiling such that the overlap is as large as possible. In bottom row, the fundamental motifs are semitransparent so as to highlight the overlap.
To fix the value of m, we add ST tiles at the outer edge of the triangular ST wedges ( Supplementary Fig. 1b); in the case shown, we add a single row containing 5 ST tiles. These fundamental motifs are then packed such that along the short edges of the second-generation tiling the overlap is maximal, which defines m. The uncovered area in the centers of the first-generation R and LT tiles can be filled out unambiguously using second-generation tiles.
The thus defined fundamental motif offers a clear interpretation of the two parameters of type IA tiling, n and m: The former gives the number of R tiles in the 6 spokes radiating from the central rosette containing 6 LT tiles and the latter gives the number of rows in the diamond-shaped wedges containing ST tiles measured in the radial direction ( Supplementary Fig. 2a). For m = n, the wedges reduce to equilateral triangles whereas for m = 2n they reduce to 60 • − 120 • rhombi. These rhombi also represent the limiting case; tilings with m larger than 2n, which are also possible, cannot be represented by fundamental motifs. In Supplementary Fig. 2b we show the fundamental motif of type IB tiling with indicated n and m. The central rosette of this motif is a hexagonal LT domain with n LT tiles along each edge, and each of the 6 trapezoid-shaped domains filling the gaps between the 6 radial spokes containing R tiles contains m rows of LT tiles.

Second-generation patterns
To complement the second-generation k = 6 type IA tiling in Fig. 1m of the main text, we display the secondgeneration k = 9 type IA, k = 6 type IB, and k = 9 type IB tilings in Supplementary Figs. 3, 4, and 5, respectively. In the k = 6 and k = 9 figures, the first-generation patterns magnified by factors β 6 ≈ 6.16228 and β 9 ≈ 9.10977, respectively, are plotted using dark blue lines.
Supplementary Figure 3: Second-generation k = 9 type IA pattern; here the parameters of the subdivision rule from Fig. 1k in the main text are n = 5, m = 7. Overlaid is the first-generation pattern magnified by β9 ≈ 9.10977 (dark blue lines).
In Supplementary Fig. 3, we note the dominant triangular domains arranged in six diamond-shapes wedges stitched together along six radial seams. The patterns in Supplementary Figs. 4 and 5 can be viewed as a network consisting of three kinds of hexagonal domains containing LT tiles. The contour of the largest domains at the vertices of the first-generation pattern is a regular hexagon, whereas the contours of the other two domains seen in the center of the first-generation R and LT tile are not.
Apart from the symmetric flower-like patterns shown in Supplementary Figs. 3, 4, and 5 which illustrate the selfsimilar nature of our tilings, it is also instructive to examine their zoomed-in parts. Supplementary Figs. 6 and 7 present two such examples. The former figure contains the k = 9 type IA R tile after two subdivisions; the firstgeneration structure can still be easily recognized. This patch is dominated by (i) triangular domains consisting of the LT tiles. The other two types of domains-(ii) trapezoidal and (iii) diamond-shaped domains-constitute the roughly X-shaped cut across the rectangle, dotted by the single LT tiles plotted in blue.
In Supplementary Fig. 7, we zoom in on portions of the third-generation k = 9 type IB tiling. Here too there exist three types of LT-tile domains: (i) regular-hexagonal domains, (ii) domains with 3-fold symmetry and 3 short and 3 long sides, and (iii) domains with 2-fold symmetry and 2 short and 4 long side. The domain boundaries along the short and the long sides of the 2-fold and the 3-fold symmetric domains consist of 2 and 3 R tiles, respectively. Yet their structure is somewhat more intricate and not as immediately appreciable than that of type IA tilings; it is perhaps best to consider it by examining the arrangement of the regular-hexagonal domains, which are generally separated from each other by the 2-fold and the 3-fold domains but also appear in pairs and in three-way triplets. Also instructive is viewing these patterns at an angle along one of the preferred directions and observing the zig-zagging of the domain boundaries along a given direction.

Supplementary Note 3
Five k = 6 type I tilings For k = 6, the identification of the different type I tilings described in the Methods section of the main text leads to the patterns shown in Supplementary Fig. 8. We first note that as shown in Supplementary Table I, the self-similar length ratios φ of type IC and ID tilings are smaller rather than larger than 1. This means that the physically longer length obeys the subdivision rule formally associated with the second row of the transformation matrix in Eq. (17) of the main text whereas the physically shorter length transforms according to the first row of the matrix. In Supplementary Fig. 8, we colored the ST and the LT tiles red and blue like in type IA, IB, and IE tilings where φ > 1 for consistency. Secondly, when we construct the subdivision pattern for type IF tiling we recover the same pattern as in type ID tiling, which can be explained by the fact that type IF length ratio is reciprocal to that of type ID tiling. As a result, the type IF ↔ type ID transformation involves a mere relabeling of the long and short edges without any physical difference.
The reader will notice that the inflation factors for type IA and IC tilings are also reciprocals so that a similar symmetry should also exist in these two patterns. Indeed, if one disregards the six rows of five ST tiles arranged around the very perimeter of type IA pattern, then type IA and IC fundamental motifs differ only in the arrangement of the very center and in the first annulus around it. In type IA, the center consists of six LT tiles and the first annulus contains six R tiles in the azimuthal orientation and six ST tiles, whereas in type IC the center is formed by six ST tiles and the first annulus consists of six R tiles in radial orientation separated by six LT tiles. This leads to two distinct sets of subdivision rules at the same length ratio. We thus conclude that there exist a total of 5 different k = 6 type I tilings.

Physical-space and reciprocal-space basis vectors
The physical-space basis vectors for type IA and type IB tilings with k = 3, 6, 9, 12, and ∞ are shown in Supplementary Fig. 9. At k = 3 corresponding to the bronze-mean tiling, the four basis vectors are evidently all independent which is witnessed by the non-integer ratio of their lengths given by Eq. (12) of the main text; at k = 3, Supplementary Figure 9: Physical-space basis vectors a j (j = 1, 2, 3, 4) and reciprocal-space basis vectors q j (j = 1, 2, 3, 4) for k = 3, 6, 9, 12 and ∞ type IA and IB tilings. As k is increased, the four physical-space basis vectors become increasingly less linearly independent and for k → ∞, only two of them are. Concomitantly, the magnitudes of vectors q 1 and q 3 in type IA tilings and the magnitudes of vectors q 2 and q 4 in type IB tilings decrease; in the limit k → ∞, these four vectors vanish.
a odd a even = √ 39 + √ 3 6 ≈ 1.330. (1) As k is increased, this ratio in the type IA tilings approaches where we used Eqs. (13) and (14) of the main text. As the basis vectors are 30 • apart, when k → ∞ then a 1 +a 3 = 2a 2 and a 1 + 2a 4 = 2a 3 etc. so that the four basis vectors a 1 , a 2 , a 3 , and a 4 are no longer independent from each other. As a result, only two basis vectors are needed to describe the tiling, which indicates translational periodicity; the rotational symmetry is evidently six-fold. In type IB tilings at k → ∞, a 2 + a 4 = a 3 etc., which leads to the same conclusion. Supplementary Fig. 9 also shows the reciprocal-space basis vectors, which too reflect the convergence towards the hexagonal lattice as k is increased. This is witnessed by the increasingly smaller magnitude of q 1 and q 3 in type IA tiling and q 2 and q 4 in type IB tiling.
Supplementary Note 5

Self-similarity in diffraction patterns
To find the metallic means in diffraction images in experiments, it is helpful to use the following identities pertaining to the ratio of long and short edges: √ 3/ψ +2 = β k (type IA) and √ 3ψ +1 = β k (type IB). These identities imply that for example in the k = 6 type IB pattern in Fig. 3b in the main text or in the right panel of Supplementary Fig. 10 the intensities of the small six satellite peaks forming a hexagon around the pronounced peak with indices (2, 1, 1, 0) are scaled by a factor of 1/β 6 .
Extended areas of k = 6 Fourier transforms of type IA and IB tilings are shown in Supplementary Fig. 10. To demonstrate the presence of the sixth metallic mean, we drew a few hexagons characterizing the pattern of three different sizes (yellow lines); on going form the smallest to the middle and from the middle to the largest hexagon, the linear dimension is increased by a factor of the sixth metallic mean 3 + √ 10 ≈ 6.162.

Supplementary Note 6 Inflation factors
In Supplementary Tables IV and V, we list the inflation factors of type IB and type IIA tilings λ IB + = m + 1 + 2 n + ( m − 1) 2 + 4 n(2 + m + n) 2 , and λ IIA + = √ 3(n + 1) + 3(n − 1) 2 + 8m 2 , respectively, for the physically meaningful combinations of n and m or n and m where n and n ≤ 5. These tables complement Table I of the main text. Interestingly, if n = m then the inflation factor of the type IB tiling is an integer. These special cases correspond to periodic-crystalline rather than quasicrystalline tilings as illustrated by the n = m = 2 type IB tiling encoded by the subdivision patterns in Supplementary Fig. 11. Supplementary Fig. 12 shows the second-generation n = m = 2 type IB tiling both without and with the outline of the first-generation tiling; the former more evidently shows the periodic-crystalline nature of the pattern. In this type IB tiling, the ratio of long and short lengths φ is √ 3 and the inflation factor is 7.
The periodic-crystalline nature of the n = m type IB tiling is exact as the tiling consists of a single hexagonal domain evident in the left panel of Supplementary Fig. 12. Yet the existence of the periodic-crystalline n = m type IB