Aperiodic approximants bridging quasicrystals and modulated structures

Aperiodic crystals constitute a class of materials that includes incommensurate (IC) modulated structures and quasicrystals (QCs). Although these two categories share a common foundation in the concept of superspace, the relationship between them has remained enigmatic and largely unexplored. Here, we show “any metallic-mean” QCs, surpassing the confines of Penrose-like structures, and explore their connection with IC modulated structures. In contrast to periodic approximants of QCs, our work introduces the pivotal role of “aperiodic approximants”, articulated through a series of k-th metallic-mean tilings serving as aperiodic approximants for the honeycomb crystal, while simultaneously redefining this tiling as a metallic-mean IC modulated structure, highlighting the intricate interplay between these crystallographic phenomena. We extend our findings to real-world applications, discovering these tiles in a terpolymer/homopolymer blend and applying our QC theory to a colloidal simulation displaying planar IC structures. In these structures, domain walls are viewed as essential components of a quasicrystal, introducing additional dimensions in superspace. Our research provides a fresh perspective on the intricate world of aperiodic crystals, shedding light on their broader implications for domain wall structures across various fields.

Prior to the discovery of quasicrystals (QCs) as the advent of aperiodicity in materials science, incommensurate (IC) modulated structures and IC composite structures were investigated, wherein IC spatial modulations were added to the background crystalline structures [1,2].Then, the concept of superspace and additional degrees of freedom known as phasons were introduced.After Shechtman's discovery [3], aperiodic crystals, including IC modulated structures and QCs, emerged as an important class of materials [4][5][6][7][8].Aperiodicity is characterized by irrational numbers, thereby making a distinction between the two.In QCs, the irrational numbers are locked by two-length scales [9][10][11]20] in geometry, whereas in IC modulated structures, these numbers remain unlocked.QCs typically consist of concentric shell clusters arranged quasiperiodically, locking in the golden mean in icosahedral QCs.In certain alloys, such as Au-Al-Yb [12], periodic approximants are synthesized, where the clusters are arranged periodically.Consequently, "periodic approximants" have been extensively studied to gain a better understanding of QCs.A crucial aspect of periodic approximants is that they exhibit local quasiperiodicity (resembling QCs), but globally display periodicity.Moreover, as the degree of approximation increases, these periodic approximants converge towards QCs [13].
A complementary treatment has been explored where a quasiperiodic structure approaches the periodic one by varying the characteristic irrational.Such treatments are known as "aperiodic approximants" [14].An elementary example of aperiodic approximants is the generalized Fibonacci sequence, which comprises two letters, A and B. The sequence is generated by the substitution rules: where k is a natural number.The numbers of the letters A and B at iteration n (N where the maximum eigenvalue of the matrix is given by the metallic-mean: τ k = (k + √ k 2 + 4)/2.When k = 1, the sequence is the conventional Fibonacci one with the golden mean.The eigenvector of the matrix is given by (τ k , 1) T , indicating → τ k as n → ∞, where the sequence is filled with the letter A for large values of k.In the limit k → ∞, the sequence converges to a crystal consisting of consecutive "A"s.Hence, the generalized Fibonacci sequence with the metallic-mean can be considered as the aperiodic approximants of the onedimensional crystal AAA • • • .
Similarly, aperiodic approximants of triangular lattices were proposed.These metallic-mean quasiperiodic tilings start from the bronze-mean tilings [10].Majority tiles increase with increasing k, and eventually, the systems converge to the triangular lattices in the limit k → ∞.A crucial aspect of these is that they are locally periodic, but globally quasiperiodic, in other words, they are considered as planar IC modulated structures.
Here we present hexagonal metallic-mean approximants of the honeycomb lattice, which bridge the gap between QCs and IC modulated structures.Schematic of our view is presented in Fig. 1.As the metallic-mean increases, the size of honeycomb domains bounded by the parallelograms also increases, and the whole tiling converges to the honeycomb lattice.Conversely, the metallic-mean IC modulation is introduced to the honeycomb crystals in terms of the metallic-mean tilings.The domain walls composed of parallelograms in the honey-Fig.1. Aperiodic approximants.Schematic showing the role of aperiodic approximants as a link between quasicrystals and periodic crystals.The link is a series of k-th metallicmean tilings as an aperiodic approximant of the honeycomb crystal (top arrow), which is simultaneously regarded as a metallic-mean (incommensurate) modulated honeycomb crystal (bottom arrow).
comb crystal are regarded as ingredients of a quasicrystal adding superspace dimensions.Significantly, we show that the metallic-mean tiling scheme is applicable to a polymer system [15] and colloidal systems [16,17] in softmatter self-assemblies.

Metallic-mean tilings
We construct the metallic-mean approximants of the honeycomb lattice, which are composed of large hexagons (L), parallelograms (P) and small hexagons (S) shown in Fig. 2a.The ratio between the long (ℓ) and short (s) lengths is given by the metallic-mean τ k (= ℓ/s).Consequently, the ratio of areas for the three tiles is given by 3τ 2 k : τ k : 3. We elaborate the substitution rules for these tiles as a natural extension of those for the hexagonal golden-mean tiling [11] (Fig. 2b).The substitution rules for k = 2 and k = 3 are illustrated in Figs.2c and 2d, respectively.Notice that the matching rule of the tilings are introduced by solid and open circles.When the deflation rule is applied to an L tile, an S tile is generated at the center of the original L tile, thereby, six zig-zag chains of P tiles emanate from the central S tile, which is clearly found in the case with k = 3.The rest region is filled by L tiles.Upon one deflation process, a P tile is changed to one P tile and L tiles, and an S tile is changed to one L tile.Hence, one can construct the substitution rules of three tiles for any k, which are subjected to the substitution rule for the generalized Fibonacci sequence: ℓ → ℓ k s and s → ℓ.See also Supplementary Fig. 1 showing how these rules are extended to the cases of k = 4 and k = 5.
Two-dimensional space is covered without gaps after iterative deflation processes, as shown in Figs.2e-g and Supplementary Fig. 2 for k = 1 − 5.Because of the deflation process, self-similarity is an inherent property of the metallic-mean tilings: Supplementary Fig. 3 exem-plifies exact self-similarity for k = 2 and k = 3.We find that a finite number of adjacent L tiles are bounded by the P tiles, which can be regarded as an isolated "honeycomb domain".For examples, in the case with k = 2, the domains are composed of one, three, or six L tiles, as shown in Fig. 2f.We confirm that each honeycomb domain bounded by the P tiles is composed of a k−1 , a k , or a k+1 adjacent L tiles in the kth metallic-mean tiling, where a k = k(k+1)/2, see Supplementary Note 3. Therefore, increasing k, the number of the L tiles in each honeycomb domain quadratically increases.On the other hand, the S and P tiles are located around the corners and edges of the honeycomb domains, and thereby their numbers should be O(1) and O(k), respectively.These suggest that the L tiles become majority in the large k case and the single honeycomb domain is realized in the limit k → ∞, as shown in Fig. 2h.Using a deflation matrix described in Method, it is easy to evaluate the frequencies of tiles (f L , f P , and f S ) and the ratio of the corresponding areas (S L , S P , and S S ) rendered in Fig. 2i and its inset.For k = 5, more than ninety percent of the two-dimensional space is occupied by L tiles.See Method in details.
The tiling has eight unique types of vertices as shown in Fig. 3a classified by their coordination numbers and their circumstances.The frequency of each type can be exactly computed, and the explicit formulae for any k are presented in Supplementary Note 2. Figure 3b shows the frequencies of the vertex types as a function of k.As expected, the frequencies of the C 0 and C 1 vertices monotonically increase and approach 1/2 implying the convergence to the honeycomb lattice.
The metallic-mean tilings are bipartite since they are composed of hexagons and parallelograms.As shown in Fig. 3a, the vertex types C 1 -C 3 belong to the A sublattice and the others belong to the B sublattice, as depicted by open and solid circles, respectively.We find that the sublattice imbalance in the system given as ∆ are the fractions of the A and B sublattices, respectively.This distinct property is in contrast to those for the bipartite Penrose, Ammann-Beenker, and Socolar dodecagonal tilings where each type of vertices equally belongs to both sublattices [21].
As shown in Fig. 3c, we can distinguish two kinds of L tiles denoted by L △ and L ▽ , introducing up and down triangles located at their centers so that three corners of each triangle point the filled circles on the vertices of the L tile.In Fig. 3d, we find the following properties: plementary Fig. 6.

Superspace representation
To provide a theoretical basis for the metallic-mean tilings, we construct their higher-dimensional description.In this powerful method, the superspace is divided into the physical space and its complement, known as the perpendicular space.A tiling is viewed as a projection of a hypercubic crystal in the superspace onto the two-dimensional physical space.The projections onto the perpendicular space are densely filled in specific areas, as illustrated in Fig. 3f, which are referred to as windows.These windows are derived from sections perpendicular to the threefold axis of a rhombohedron (octahedron) that is the projection of the hypercubic unit cell, showcasing hexagonal and triangular shapes in Extended Data Fig. 1.The figure also highlights the regions associated with the eight vertex types, as detailed in the Method section and Supplementary Fig. 8.

Application to soft matter
The metallic-mean tilings are physical entities in two softmatter systems.We consider self-assembled crystalline structures obtained in soft materials with the P31m plane group, as illustrated in Fig. 4a, which belongs to the twodimensional hexagonal Bravais lattice but lacks hexago-nal rotational axes.Further crystallographic description is given in Supplementary Notes 8 and 9 for colloidal particles and polymer blends, respectively.
The first application of the metallic-mean tiling is a polymer system reported by Izumi et.al., who found a complex ordered structure in an ABC triblock terpolymer/homopolymer blend system [15].Sample preparation is provided in Method.Figure 4b illustrates the decoration of L, P, and S tiles by three kinds of polymers.In the previous study, regular large domains consisting of only L tiles were observed.It is noticed that the triangle inside a hexagon has two directions, up and down.In the present study, we searched samples again and found P tiles in a TEM picture rendered in Fig. 4c.In Fig. 4c, a regular region of an extended L ▽ area in the center and a domain wall represented by a row of zigzag P tiles on the left-hand side.We can interpret the rows of P tiles within the L sea as twin boundaries, which mark a transition between different crystal orientations, L △ and L ▽ .It's worth emphasizing that a row of P tiles physically changes the crystal orientations, demonstrating the tangible properties of P tiles beyond mathematical concepts.We note that the decoration of S tile (Fig. 4b) is hypothetical and it has not been observed in the samples.
The second application of the metallic-mean tiling is a colloidal particle simulation in two dimensions conducted by Engel [16].It utilizes a Lennard-Jones-Gauss (LJG) potential [22] that has two distinct length scales.In Method, we have reproduced his result.We find that the LJG particles occupy the same positions as the dark gray circles in Figs.4a-c.Moreover, in Fig. 2 of the Engel's paper and his Supplementary Figure S1 in particular, it was shown to form twin-boundary superstructures on a scale much larger than the potential range: the size of superstructures depends on the temperature reversibly; the lower the temperature, the larger the size.One finds that regular L △ or L ▽ domains form triangle shapes of several sizes, which property is also characteristic of the metallic-mean tiling.Additionally, it was observed that twin boundaries only intersect at triple junctions, which situation mimics the metallic-mean tilings, where triple rows consisting of the P tiles meet at the location of an S hexagon, though the correspondence between the P tiles and domain walls in the LJG system is not always exact.Nonetheless, the metallic-mean scheme mimics Engel's modulated superstructures with changing scale ratios or k values.
In Fig. 4d, an ideal decoration model for the particle system generated by the higher-dimensional quasicrystal theory with the 5-th metallic-mean modulation (Supplementary Note 12).In these cases, as shown in Figs.4e  and 4f, the structure factor S(q) = 1 N i e iq•ri 2 theoretically calculated in terms of the superspace represen-tation dramatically reproduces the numerical FFT for the diffraction images shown in Fig. 3 of the Engel's paper.As clearly shown in the magnified views (Fig. 4f), the prominent peaks appear at almost the same positions, while the aperiodic modulation of the metallicmean tiling yields the satellite peaks in the vicinity of the main peaks, which is the characteristic property of IC structures.

Discussion
Our previous study has covered the multiples-of-3 metallic-means, through the hexagonal aperiodic approximants of the triangular lattice [14].The present work broadens the scope of aperiodic approximants.Firstly, our tiling serves as the approximant of the honeycomb lattice.Secondly, it enables an inflation ratio of any metallic-mean, thereby enhancing the applicability.
In fact, we have applied the tiling concept to explore real materials, such as polymer and colloidal systems.Our analysis successfully identifies large hexagons as regular structures and parallelograms as twin boundaries.It is noted that similar IC triangular domain structures were discovered in quartz and aluminum phosphate long time ago [18,19], known as Dauphiné twins in trigonal quartz.We surmise that there is a similar mechanism behind the formation.
We emphasize that the decorated perpendicular space windows in 6D generate the 2D IC structures, whose method has been developed in the field of QC studies.It is striking that the satellite peaks can be calculated not by direct real-space Fourier transform, but by perpendicular-space Fourier transform of the windows.By comparing these peaks with those observed in twodimensionally IC modulated structures, we establish a foundation for analysis of IC structures in terms of the QC methodology.
One of the origins of the P31m plane group demonstrated here is the aggregation tendency of pentagons.Regular pentagons cannot tile the entire plane without gaps, as shown by Dürer-Kepler-Penrose, however, there are pentagon-related tilings if we abort five-fold symmetry.In Supplementary Note 10, we demonstrate the accommodation of pentagons within both a square and a hexagon.Using 4-fold symmetry, the Cairo pentagonal tiling and its dual, i.e., the 3 2 .4.3.4Archimedean tiling with the P4gm have been considered [23].The latter Archimedean tiling is associated with the σ phase found in complex metallic and soft-matter phases, which is recognized as a periodic approximant of dodecagonal QCs [24][25][26][27][28][29].It is noteworthy that P31m plane group structure is a 3-fold variant of the Cairo tiling and the σ phase.
Our study highlights the effectiveness of aperiodic approximants in inducing modulations within selfassembled soft-matter systems employing the P31m plane group.Specifically, we utilized the rows of P tiles as domain boundaries in the honeycomb lattice, thereby bridging metallic-mean hexagonal QCs and IC modu- lated honeycomb lattices.The dynamic movement of domain walls while maintaining triple junctions can be explained by the phason flips of L, S, and P tiles, as illustrated in Extended Data Fig. 3 and Supplementary Note 4. In this context, the colloidal system appears to be a phason-random tiling version of the metallic-mean tiling system.Lastly, applying the deterministic growth rules, known as OSDS rules [30], reveals that dead surfaces consist of these domain walls.Overall, our research offers a fresh perspective, providing novel insights into the realm of both aperiodic crystals and their broader implications for domain wall structures across various fields.

Methods
Deflation matrix of the metallic-mean tiling The metallic-mean tilings are regarded as the aperiodic approximants of the honeycomb lattice.To discuss quantitatively how the metallic-mean tilings approach the honeycomb lattice with increasing k, we construct the deflation matrix.At each deflation process, the increase of the numbers of L, P, and S tiles is explicitly given by where is the number of the tile α, which stands for L, P, or S at iteration n.The maximum eigenvalue of the matrix M is τ 2 k , and the corresponding eigenvector is given as (τ 2 k 6τ k 1) T .We evaluate the frequencies for these tiles in the large k limit approach The kdependent frequencies for three tiles are shown in Fig. 2i.Increasing k, the frequency of the L tiles monotonically increases and approaches unity.

Domain boundaries
The domain boundaries composed of consecutive zig-zag P tiles intersect at small hexagons and pass through the opposite edge of the small hexagons with keeping alternating directions of P tiles.If we ignore these slithering configuration of P tiles, there are three sets of parallel domain walls, as displayed in Fig. 3d.Focusing on a set of parallel domain walls, we observe two types of intervals between the domain walls denoted by S S and S L , as shown in Supplementary Fig. 6 for silver-and bronzemean tilings.There are intriguing properties for the intervals.First, for the k-th metallic-mean tilings, the interval S S and S L consists of k and k + 1 consecutive L tiles.Second, upon the deflation, we find the substitution rules: L , and where the maximum eigenvalue of the matrix is given by the metallic-mean τ k .The eigenvector of the matrix is given by , where the sequence is filled with S S intervals for large k values.Therefore, we conclude that the intervals between domain walls are metallic-mean modulated.

Superspace representation
We outline main steps of the construction of the metallicmean tiling by projection of a higher-dimensional hyperlattice onto the physical space.Let ℓ and s be the lengths of the long and short edges of the tiling.We here assume that the ratio η = s/ℓ is a variable to apply the tiling to soft-matter systems, while the ratio in the perpendicular space is set to be 1/τ k to keep the arrangement of the metallic-mean tiling.When η = 1/τ k , the tiling is the exact self-similar metallic-mean tiling generated by the deflation rules.
(1, 0) Extended Data Fig. 1.Superspace perspective.a, the perpendicular space for the bronze-mean tiling.b, Four windows on the right-hand side are obtained from a regular octahedron (middle part of a rhombohedron) of edge length √ 3(1 + τ −1 k ).The top (1, 0) and bottom (0, 1) windows are equilateral triangular faces of the solid, and hexagonal windows indicated by (0, 0) and (1, 1) are the sections of the octahedron.In the solid, blue and red colors correspond to honeycomb domains with L △ and L▽, respectively.In each window, each color corresponds to the vertex type rendered in Fig. 3b.
Each vertex site in the tiling is described by a sixdimensional lattice point ⃗ n = (n 0 , n 1 , • • • , n 5 ) T , labeled with integers n m .Let the six-dimensional lattice point ⃗ r h in the six-dimensional space S h as ⃗ r h = R⃗ n: where R is the mapping matrix and c6 = cos(π/6), s6 = sin(π/6).Namely, the matrix is represented by the sixdimensional basis vectors ⃗ e h i (i = 0, 1, • • • , 5): (⃗ e h i )j = Rji.The vertex site r in the physical space S is given by the first two components of the vector: r = (⃗ r h )0, (⃗ r h )1 = 5 m=0 nmem, where the projected vectors of the form em = (R0m, R1m) with lengths ℓ and s are displayed in Fig. 3e.The remaining four-dimensional perpendicular space is split into two-dimensional spaces S and S ⊥ , and the corresponding coordinates r and r ⊥ are given as r = (⃗ r h )2, (⃗ r h )3 = Note that r points are densely filled on four planes with } denoted by {(1,0), (0,0), (1,1), (0,1)}, having polygonal windows shown in Extended Data Fig. 1.Notice that the windows are faces and sections for a regular octahedron.This octahedron is the middle part of a rhombohedron of edge length √ 3(1 + τ −1 k ), which is the projection of the hypercubic unit cell.In Extended Data Figure 1, r is plotted in the (x, y)directions, while for r ⊥ both (⃗ r h )4 and (⃗ r h )5 are projected onto the z component.We find that in the limit k → ∞, the upper and lower hexagons get closer to the top and bottom faces, respectively, and finally they become the equilateral triangles.The explicit sizes of hexagonal windows are presented in Supplementary Fig. 8.
The six-dimensional reciprocal lattice vectors ⃗ q h i are defined to have the following property ⃗ e h i • ⃗ q h j = 2πδij with δij is the Kronecker delta.It is easy to find (⃗ q h j )i = Qij, where RQ T = 2πδij and with nmqm, where the projected vectors qm = (Q0m, Q1m) with lengths 1/ℓ and 1/(ℓτ k ) are displayed in Fig. 4g.The remaining four-dimensional perpendicular space is split into two-dimensional reciprocal spaces and the corresponding reciprocal vectors q and q ⊥ are given as q = (⃗ q h i )2, (⃗ q h i )3 = 5 m=0 nm qm, q ⊥ = (⃗ q h i )4, (⃗ q h i )5 = 5 m=0 nmq ⊥ m , where qm = (Q2m, Q3m) and q ⊥ m = (Q4m, Q5m).The detailed procedure is given in Supplementary Note 5.
When computing the Fourier transforms, we rely on the following identity for any pair of vectors in the superspace lattice ⃗ r h and in the corresponding reciprocal lattice ⃗ q h : 1 = exp(i⃗ q h • ⃗ r h ) = exp(iq • x) exp(iq • x) exp(iq ⊥ • x ⊥ ).If particle's positions are described by δ-functions so that the density reads f (r) = N j=1 δ(r − rj), then the Fourier transform of the density is calculated as in the last step, we resorted to the above identity.
To construct decorated tilings (Fig. 4d) for soft-matter systems, we set η = s/ℓ = 0.6249.In this case, we employ sections of a rhombohedron as extensional windows.Detailed procedures are presented in Supplementary Notes 8 and 12.

Polymer details
An ISP (I: polyisoprene, S: polystyrene, P: poly(2vinylpyridine)) triblock terpolymer sample was prepared by a sequential monomer addition technique of an anionic polymerization from cumyl-potassium as an initiator in tetrahydrofuran (THF), while styrene homopolymer was synthesized anionically with sec-butyllithium in benzene.The average molecular weight of the terpolymer is 161k and the composition is ϕI/ϕS/ϕP = 0.25/0.53/0.22,whereas that of the styrene homopolymer is 9k.The overall composition of the blend sample is ϕI/ϕS/ϕP = 0.17/0.68/0.15,where polystyrene block/styrene homopolymer ratio of wS(b)/wS(h) = 1.4.The sample film was obtained by casting for two weeks from a dilute solution of THF followed by heating at 150 • C for two days.The specimens for morphological observation were cut by an ultramicrotome of Leica model Ultracut UCT into ultrathin sections of about 100 nm thickness and stained with OsO4 for the TEM observation.Further details are provided in the previous reference [15].
Colloidal simulation.Monte Carlo simulation for the Lennard-Jones-Gauss potential at T = 0.270.

Simulations of colloidal particles
We used NPT Monte Carlo simulations of N = 10000 colloidal particles interacting with the Lennard-Jones-Gauss potential [16,22] given by with parameters σ 2 = 0.042, ϵ = 1.8, r0 = 1.42 at T = 0.270, P = 0.0.There are slight differences between simulations (Extended Data Fig. 2) and the metallic-mean tiling model (Fig. 4): (1) Dynamically, P tiles are not always perfect.(2) There are five particles in an S tile in simulations, while six particles in the latter.The effect of these is negligible in the structure factor.Further data including diffraction images is provided in Supplementary Note 11.

Phasons
Domain walls dynamically move with keeping triple junctions can be explained by the phason flips of L, S and P tiles, as shown in Extended Data Fig. 3.In this sense, the colloidal system appears to be a phason-random tiling version of the metallic-mean tiling system.The existence and the conservation of S tiles in the phason flips is the key of triple junctions of domain walls at moderate thermal excitations.See also Supplementary Note 4.

Supplementary Note 2 Fractions of the vertices
We derive the fractions of vertices for k ̸ = 1 (the fractions for k = 1 have been given in Ref 1 ).In the case of k ̸ = 1, we could not find the F vertex shared by six P tiles (f F = 0).This is because vertices shared by two adjacent P tiles are always shared by the L or S tile according to the substitution rules for k ̸ = 1, as shown in Supplementary Figs.1c-1f.When one evaluates the fractions for certain graphs such as vertices and domains, it is convenient to consider the ratio between numbers of tiles and vertices for the hexagonal metallic-mean tiling in the thermodynamic limit.Supplementary Figure 1a clearly shows that the net numbers of sites in L, P, and S tiles are two, one, and two, respectively.Therefore, we obtain the ratio r k as where We first focus on the bipartite structure.The sublattice structures for the L, P, and S tiles are shown as the open and solid circles in Supplementary Fig. 1a, where these are referred to as A and B sublattices.By counting the net numbers of the site belonging to each sublattice in L, P, and S tiles, we obtain its fractions as, This naturally leads to the sublattice imbalance in the hexagonal metallic-mean tilings, Since the sublattice A (B) is composed of C 1 , C 2 , and C 3 (C 0 , D 0 , D 1 , and E) vertices, we obtain the following equations, In the tilings, two adjacent tiles share the edge, which is connected between the neighboring sites in A and B sublattices.Therefore, we obtain the equations for the total number of longer and shorter edges, where the left (right) hand side of the equations represents the total number of edges, which is expressed by the numbers of vertices belonging to the A (B) sublattice.According to the substitution rule, the C 3 , D 1 , and E vertices always appear around the S tile for k ̸ = 1.Therefore, these fractions are then given as From these equations, we obtain the exact fractions of vertices in the hexagonal metallic-mean tilings as The average of the coordination number is given by In the hexagonal metallic-mean tilings, the average of the coordination number depends on k. z k → 3 when the system approaches the honeycomb lattice k → ∞.
Supplementary Note 3 Honeycomb domain Supplementary Fig. 4. Upper panels: α, β0, β1, β2, and γ domains in the hexagonal silver-mean tiling, which are bounded by some P and S tiles.Here, we focus on the honeycomb domain in the hexagonal metallic-mean tilings with k ̸ = 1, which is composed of finite number of the L tiles and is bounded by the P and S tiles.As seen in Supplementary Fig. 2, in the hexagonal metallic-mean tiling with k ̸ = 1, there exist three kinds of domains composed of a k−1 , a k or a k+1 L tiles, where a k = k(k + 1)/2.These are referred to as α, β and γ domains.Supplementary Figure 4 shows α, β, and γ domains in the silver-mean tilings, as an example.We find that in the α domain, the E vertex shared by one L tile and four P tiles, which is shown as the circle, is located at each corner site.In the γ domain, the D 1 vertex shared by one L tile, two P tiles and one S tile, which is shown as the square, is located at each corner site.On the other hand, the β domains can be divided into the β i (i = 0, 1, 2) domains, where i E vertices and (3 − i) D 1 vertices are located at three corner sites, as shown in Supplementary Fig. 4. The absence of the β 3 domains will be proved below.
To examine the fraction of each domain, we consider the substitution rule for the tiles.In the lower panels of Supplementary Fig. 4, we show the tiling structure obtained by the inflation operation as the red lines.We find that the C 1 , C 0 , D 0 , E vertices, and S tiles generated by an inflation operation are located at the center of the α, β 0 , β 1 , β 2 , and γ domains.Therefore, we obtain the following equations as where f X is the ratio of the number of X(= α, β i , γ) domains to the total number of tiles.Since f L = a k−1 f α + a k 2 i=0 f βi + a k+1 f γ , we prove that each L tile belongs to α, β i (i = 0, 1, 2) or γ domain, and β 3 domains never appear in the hexagonal metallic-mean tiling with k ̸ = 1.As for the golden-mean tiling with k = 1, α and β 0 domains do not appear, but β 3 domains appear due to the existence of the F vertices.The fractions of the β 1 , β 2 , and γ domains are given by Eqs. ( 21), (22), and (23), and the fraction of the β 3 domains is given as We show in Supplementary Fig. 5 the fraction of the L tiles which belong to each domain.When k is small, the β 1 , β 2 , and γ domains are majority in the tilings.On the other hand, when the system approaches the honeycomb lattice, α and β 0 domains become dominant in the system.This originates from the fact that, in the large k case, the vertices are almost composed of the C 1 and C 0 vertices, and thereby α and β 0 domains, which are generated by applying the deflation operation to the above vertices, become dominant.Figures 6b and 6d show the deflations of the tilings shown in Supplementary Figs.6a and 6c, respectively.When the deflation operation is applied to the tiling, k and k + 1 domain boundaries are equally-spaced generated in the original S S and S L spaces, respectively.Namely, k −1 and k S S are generated.On the other hand, under one deflation operation, one S L is generated at each domain boundary.Therefore, the numbers of S S and S L at iteration n (N S S and N S L ) satisfy where the maximum eigenvalue is given by τ k and the corresponding eigenvector (kτ k , 1 + τ k ) T .This means that the self-similarity inherent in the metallic ratio τ k appears in the tiles, honeycomb domains, and spaces between adjacent domain boundaries.
where ϕ = 2π/3 and θ is constant.ℓ and s are the lengths for longer and shorter edges of the tiles.The vectors e m are schematically shown in Supplementary Fig. 7.
In Eq. ( 26), the vertex site r can be regarded as the projection from a six-dimensional lattice point, where the vectors e m are the projections from the six-dimensional basis vectors.Thereby one can define the projections onto the other four-dimensional space (perpendicular space).For a unified understanding of the projection, it is convenient to introduce the six-dimensional space S h including the physical and perpendicular spaces.Then, ⃗ n is mapped to the six-dimensional lattice point ⃗ r h in S h as, where M is the mapping matrix.Then, one can discuss vertex properties in both physical and perpendicular space.Namely, in the physical space S, the vertex site r is given by the first two components of the vector as The four-dimensional perpendicular space is split into two-dimensional spaces S and S ⊥ , and the corresponding coordinates r and r ⊥ are given as where ẽm = (M 2m , M 3m ) and e ⊥ m = (M 4m , M 5m ).We find that r ⊥ = (x ⊥ , y ⊥ ) takes four values x ⊥ = 0, √ 2τ −1 k and y ⊥ = − √ 2, 0 in the hexagonal metallic-mean tilings.In each r ⊥ plane, the r points densely cover a certain window.We find that the window in planes r ⊥ = (0, 0), ( ] has a hexagonal (triangular) structure.Supplementary Figure 8 shows the perpendicular spaces for the hexagonal metallic-mean tilings with , and (0, − √ 2)] of the hexagonal metallicmean tilings for k = 1, 2, 3, 4, and ∞.Each area bounded by the solid lines is the region of one of eight types of vertices with △ and ▽. λ1 and λ2 (λ3) are the characteristic lengths of the windows with r ⊥ = (0, 0) and ( k = 1, 2, 3, 4, and ∞.The characteristic lengths λ 1 , λ 2 and λ 3 will be obtained in Supplementary Note 6.We find that eight types of vertices are mapped into specific regions.This implies that the perpendicular spaces reflect the local environments for the lattice sites.Namely, the areas of each vertex region in perpendicular spaces are proportional to its fraction in the physical space.The region of the F (C 0 ) vertices appears only in the case of k = 1 (k ̸ = 1).This means the absence of C 0 (F) vertices in the hexagonal metallic-mean tilings with k = 1 (k ̸ = 1), which is consistent with the results discussed in the main text and Supplementary Note 2. The areas of the C 0 , C 1 vertices (the others) monotonically increase (decrease) with increasing k.The planes r ⊥ = (0, 0) and ( )] are fully occupied only by the C 0 and C 1 vertices in the limit k → ∞ since f C0 , f C1 → 1/2.We also find that the vertices in the A (B) sublattice are mapped to the planes with (0, 0) and ( This can be explained by the following.The sublattice index for each vertex is uniquely determined, as discussed above.Since upon moving from one site to its neighbor only one of the n m 's changes by ±1, the site The six-dimensional lattice points ⃗ r h with indices ⃗ n relevant for the generalized hexagonal metallic-mean tiling satisfies the condition that their projections onto the perpendicular space are located inside the windows, which are shown in Supplementary Fig. 8.Then, projecting these six-dimensional coordinates onto the physical space, we can obtain the generalized hexagonal tilings.Golden-mean, silver-mean, and bronze-mean tilings with ℓ/s = 1/2 and ℓ/s = 2 generated by means of the cut-and-project scheme are shown in Supplementary Fig. 10.We wish to note that the self-similarity in the tilings is inherent in the case ℓ/s = τ k and the substitution rule cannot be defined in generic case with arbitrary ℓ and s.Supplementary Fig. 10.Hexagonal golden-, silver-and bronze-mean tilings generated by the cut-and-project scheme with a ℓ/s = 1/2 and b ℓ/s = 2.

Reciprocal vectors
Each vertex site in the tilings is described by the six integer indices ⃗ n and that mapped to the six-dimensional space S h is represented as where ⃗ e h i (i = 0, 1, • • • , 5) are the six kinds of six-dimensional basis vectors with (⃗ e h i ) j [= M ij ].The six kinds of six-dimensional reciprocal vectors ⃗ q h i are obtained by imposing the condition ⃗ e h i • ⃗ q h j = 2πδ ij with δ ij is Kronecker delta.These are explicitly given as, where The reciprocal vectors projected onto the physical space are given as, The six reciprocal vectors are composed of three long vectors q 0 , q 1 , q 2 and three short vectors q 3 , q 4 , q 5 .The ratio of their lengths is given by τ k .The longer length monotonically increases with increasing k and approaches the constant 4π/(3ℓ) in larger k.On the other hand, the shorter length ] monotonically decreases and vanishes in the limit k → ∞.This is consistent with the fact that, in the limit k → ∞, three of the reciprocal vectors are reduced to those for the honeycomb lattice and the metallic-mean tiling can be regarded as the aperiodic approximant of the honeycomb lattice.We wish to note that the net number of the reciprocal vectors is two due to satisfying q 0 = −(q 1 + q 2 ) in the limit k → ∞.This is consistent with the fact that the honeycomb lattice is periodic.Supplementary Fig. 11.Reciprocal lattice basis in the real space q0, • • • , q5 and those in the perpendicular space q0, • • • , q5.

Supplementary Note 6 Phason flips
We examine phason flips in the hexagonal metallic-mean tilings.As seen in Supplementary Note 5, the vertices in the metallic-mean tilings can be mapped inside of the windows in the perpendicular space.When the windows in the perpendicular space slightly slide, positions for some vertices become located outside of the windows and those for some vertices become located inside.This slightly changes the vertex structure in the physical space, that is, some vertices can be regarded to move, so called, phason flips.To discuss how the phason flips occur in the metallic-mean tiling, we show the bronze-mean tiling and the tiling with the slightly shifted windows in Supplementary Figs.12a  and 12b.For clarify, the difference between these two tilings is shown in color.We find the phason flip along a certain domain boundary.Furthermore, we find that this change can be described by three kinds of local flips.Note that a single local flip never appears due to the matching rule of the tilings.One of local flips is the position change around the C 2 vertex sharing one L tile and two P tiles, as shown in Supplementary Fig. 12c, where the C 2 vertex and the D 0 , D 1 and E vertices connected to it by the longer edges change their positions.The local flip around the C 3 vertex also occurs at the same time, as shown in Supplementary Fig. 12d, where the C 3 vertex and the D 1 vertices connected to it by the shorter edges change their positions.The other flip appears at the intersection of two domain boundaries, as shown in Supplementary Fig. 12e.In the case, some D 0 , D 1 and E vertices change their positions.When one focuses on the E vertex, the change in the physical space is characterized by e 2 −e 0 , as shown in Supplementary Fig. 13a.The corresponding move in the perpendicular space appears between the corners of the triangular window for E vertices, which is characterized by ẽ2 − ẽ0 schematically shown in Supplementary Fig. 13b.Therefore, an edge length of the window of E vertex in the perpendicular space (if k = 1, a longer edge length of the trapezoid) is |ẽ 2 − ẽ0 | = √ 3τ −1 k .We immediately obtain the characteristic lengths of the windows in the perpendicular space as, We note that the above phason flip also changes the areas of the honeycomb domains.This should be observed as thermal fluctuations of Monte Carlo and MD simulations 2 (See the main text for the details).

( 1 )Fig. 2 .
Fig.2.Metallic-mean tilings.a, Large hexagons (L), parallelograms (P) and small hexagons (S) with edge lengths ℓ and s.Vertices are decorated with open and solid circles alternatively to introduce the matching rule of the tiling.b-d, Substitution rules for the golden-mean tiling (k = 1)(b), silver-mean tiling (k = 2)(c), and bronze-mean tiling (k = 3)(d).e, Golden-mean tiling.f, Silver-mean tiling.g, Bronze-mean tiling.h, Honeycomb lattice.i, Frequencies of L, P, and S tiles as a function of k, and the corresponding fraction for each area (inset).The convergence to the periodic honeycomb lattice is assessed.

Fig. 3 .
Fig. 3. Vertex types.Vertices are alternatively decorated with the open and solid circles to define A and B sublattices, respectively.a, Eight types of vertices.b, Frequencies of the vertex types.Dashed line represents the sublattice imbalance ∆. c, Two kinds of the L tiles, L △ and L▽.d, Honeycomb domain structures for the hexagonal bronzemean tilings.L △ tiles form up-triangular domains, and L▽ tiles form down-triangular domains.e, Projected basis vectors ei (i = 0, • • • , 5) from fundamental translation vectors in six dimensions.f, Windows in the perpendicular space.Each area shows the vertex types.

Fig. 4 .
Fig. 4. Application to soft matter.a, Diagram of the P31m plane group.b, Schematic decoration of L, P, and S tiles by ABC triblock terpolymer/homopolymer blend ISP-III/S.Dark gray circles indicate polyisoprene (PI), light gray circles indicate poly(2-vinylpyridine) (PVP), and the other matrix region is polystyrene (PS).c, TEM image from the ABC triblock terpolymer/homopolymer ISP-III/S.d, Ideal particle decoration for a colloidal system generated by the 5-th metallic-mean tiling.Up and down triangles form blue and yellow triangular domains reproducing colloidal simulations.e, Each sector shows the structure factor for the decorated k-th metallic-mean tiling when k = 1, 3, 5, 7, 9, and ∞. f, Magnified views of slices of the structure factor indicated by a dashed rectangle in e.In the vicinity of main peaks, the aperiodic modulation yields satellite peaks characterizing IC structures.g, Six reciprocal vectors qi (i = 0, 1, • • • , 5) for k = 1, 3, 5, 7, and 9.

. 3 .Contents 1 .Supplementary Fig. 2 .
Phason flips.a, Schematic phason move in self-assembled pattern from ABC triblock terpolymer/homopolymer blend.b, Two types of phason flips in the metallic-mean tiling.c, Move of a twin boundary by a row of phason flips.Substitution rules for the hexagonal metallic-mean tilings 2Hexagonal metallic-mean tilings.a Golden-mean tiling (k = 1) 1 , b silver-mean tiling (k = 2) and c bronze-mean tiling (k = 3).d and e represent metallic-mean tilings with k = 4 and k = 5. f represents the honeycomb lattice with k → ∞. a b Supplementary Fig. 3. Self-similarity of hexagonal metallic-mean tilings.a Silver-mean tiling.b Bronze-mean tiling.The colored tilings are obtained by applying the deflation operation to the tilings shown as the black lines.
Circles and squares indicate E and D1 vertices at the corners of the honeycomb domains.Lower panels: the red lines represent the results of the inflation operation applied to the tiles around the honeycomb domains shown in the upper panels.Red open (solid) circles at the vertices indicate the A (B) sublattice in the inflated tiling.

α
Supplementary Fig.5.a Fractions of α, βi (i = 0, 1, 2, 3), and γ domains for the total number of the domains are shown as the cumulative bar chart.b Fraction of the L tiles belonging to each domain are shown as the cumulative bar chart.

Supplementary Fig. 12 .
Schematic pictures for the phason flips in the hexagonal bronze-mean tiling.a and b show the bronzemean tiling and the tiling with the slightly shifted windows, respectively.The difference is explicitly shown in color.c (d) The local flips around the C2 (C3) vertex, e The local flip at the intersection of two domain boundaries.In c,d and e, circles represent the vertices which move due to the phason flip.

E
Supplementary Fig. 13. a The position change of the E vertex due to the phason flip.The black (red) circle indicates the position of the E vertex before (after) the flip.b shows the corresponding position change in the perpendicular space r ⊥ = (0, − √ 2).