Local growth of icosahedral quasicrystalline tilings

Connor Hann and Joshua E. S. SocolarPhysics Department, Duke University, Durham, NC 27708

Paul J. SteinhardtPhysics Department, Princeton University, Princeton, NJ 08544

(Dated: June 14, 2016)

Icosahedral quasicrystals (IQCs) with extremely high degrees of translational order have beenproduced in the laboratory and found in naturally occurring minerals, yet questions remain abouthow IQCs form. In particular, the fundamental question of how locally determined additions to agrowing cluster can lead to the intricate long-range correlations in IQCs remains open. In answer tothis question, we have developed an algorithm that is capable of producing a perfectly ordered IQC,yet relies exclusively on local rules for sequential, face-to-face addition of tiles to a cluster. Whenthe algorithm is seeded with a special type of cluster containing a defect, we find that growth isforced to infinity with high probability and that the resultant IQC has a vanishing density of defects.The geometric features underlying this algorithm can inform analyses of experimental systems andnumerical models that generate highly ordered quasicrystals.

I. INTRODUCTION

Icosahedral quasicrystals (IQCs) with extremelyhigh degrees of translational order have been pro-duced in the lab [1] and found in naturally occurringminerals [2]. These materials possess icosahedralpoint group symmetry and quasiperiodic structure.Their diffraction patterns consist of Bragg peaks atall integer linear combinations of a set of six indepen-dent basis vectors pointing to the vertices of a reg-ular icosahedron, a dense set that includes wavevec-tors of arbitrarily small magnitude. The presence ofincommensurate collinear wavevectors gives rise to“phason” symmetries that have no analogue in crys-tals and strongly affect the elasticity and plasticityof the quasicrystal [3].

While the existence of IQCs is well established,the processes by which they form are not well under-stood. It is known that thermal annealing can im-prove the quality of a quasicrystal [4, 5], but highlydeveloped translational order has also been observedin rapidly quenched samples [5], suggesting that nu-cleation and local growth kinetics produce a well-ordered IQC. The kinetics of nucleation and growthfrom the liquid is also thought to play an importantrole in creating a sample that can be successfully an-nealed. (See, for example, Refs. [6, 7].) There are,however, geometric features of quasicrystal structureand of defects associated with the phason degrees offreedom that raise questions about how any kineticprocess can give rise to a well ordered sample.

The atomic structure of a well ordered quasicrys-tal alloy can be described in terms of a space-fillingtiling of two or more types of “unit cells” [1, 8]. Ifone imagines building the tiling one cell at a time, adifficulty is quickly encountered: the proper choice ofwhich tile to add at some surface sites on the grow-

ing cluster can depend on choices that have beenmade in distant locations [9]. Growth of a perfectsample would appear to require interactions of ar-bitrarily long range, without which the growth pro-cess could not avoid the inclusion of a finite densityof certain types of defects representative of phasonfluctuations. The problem can be mitigated to someextent by allowing for annealing in a surface layerduring the growth, but as long as the depth of thelayer is finite, some degree of phason strain wouldappear to be inevitable.

In this paper we address the question of whetherit is possible in principle for nucleation and growthto produce a perfectly ordered IQC. We find that it ispossible to produce with exceedingly high probabil-ity an IQC with a vanishing density of defects, usinga local growth algorithm for sequentially adding tilesof two different shapes to a growing cluster. By “lo-cal,” we mean that the choice of how to add a tileat any selected surface site is based only on informa-tion about the local environment at that site. Theinfinite growth occurs when the algorithm is seededwith a special type of cluster containing a defect.

The apparent requirement of nonlocality isavoided by introducing a distinction between forcedsites and unforced sites on the surface of a growingcluster [10]. At a forced site, the local configura-tion already present uniquely specifies how a tile (orcluster of atoms) can be added. At an unforced site,there are at least two ways of adding tiles that wouldbe consistent with the local environment, thoughpossibly inconsistent with distant parts of the ex-isting cluster. To prevent inconsistent additions, theprobability of adding any tile to a randomly selectedsurface site is taken to be zero at an unforced siteand nonzero at a forced site. In this way, informationabout distant parts of a cluster can be transmitted

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through locally forced additions until a tile is addednear a previously unforced site that resolves any am-biguity, converting it to a forced one. The questionis whether, even in principle, a set of local forcingrules can be found that is sufficient to produce infi-nite growth rather than terminating with a clusterwhose surface consists entirely of unforced sites.

Our results are analogous to previously publishedresults on the 2D Penrose tilings, which are qua-sicrystals with decagonal symmetry. [10–14] Impor-tant new features arise, however, due to the differ-ent topologies of 2D and 3D phason defects. Unlikethe 2D growth algorithm that produces a perfectPenrose tiling from a decapod seed that contains asingle point defect [10], the IQCs generated by our3D algorithm necessarily contain line defects. Thenumber of defects, however, grows only linearly withthe cluster radius, leaving the bulk 3D sample witha vanishingly small density of defects, which occuronly along special planes passing through the seedand correspond only to infinitesimal fluctuations ofthe phason field.

Recent numerical investigations [15–18] and ex-periments [19] strongly suggest that favoring certaingrowth sites near the surface of a growing cluster caninstill a high degree of long-range order. It is notclear, however, how (or whether) these growth pro-cesses manage to avoid the generation of finite pha-son fluctuations or linear phason strain. The presentwork shows that local growth can, in principle, ac-count for the high degree of order in an IQC and elu-cidates mechanisms for generating nearly perfectlyordered, large samples via purely local growth ki-netics.

In Sec. II, we describe the tiling model due toAmmann that we use as the basis for our investiga-tion. Section III presents a local growth algorithm inwhich a tile is added to a surface vertex of a growingcluster in a manner determined completely by the al-ready placed tiles that share that vertex. Section IVpresents an analysis of the growth produced by thealgorithm, showing that certain seeds give rise tonearly perfect growth that proceeds to infinity witha high probability. We conclude with some remarksand discussion in Sec. V.

II. THE AMMANN TILINGS

The tilings considered in this work are formedfrom oblate and prolate rhombohedra decorated asshown in Fig. 1. Matching rules, which may bethought of as indicating energetically favored localconfigurations, specify that dots of the same coloron a face shared by two tiles must coincide. Thesetiles and the rules enforced by the decorations were

FIG. 1. Ammann tiles decorated with matching rulemarkings. The decoration of each tile is chiral, and bothenantiomorphs are needed for each tile shape. Positionsof dots on the faces not visible may be inferred from thevisible dots: for the prolate tiles (left pair), the blackdot on a hidden face is located in the same corner as theblack dot on the corresponding parallel visible face. Thered dot is located at the opposite corner from that of thecorresponding parallel face. The reverse is true for theoblate tiles (right pair).

discovered by Ammann [20], and we refer to theclass of defect-free tilings that can be made fromthem as Ammann tilings [8, 21]. Ammann’s mark-ings of the rhombohedral tiles are known to be atleast weak matching rules that enforce long rangequasicrystalline order. [22, 23] These particular ruleshave not been rigorously proven to be perfect match-ing rules (i.e., to force a single local isomorphismclass of tilings), though closely related rules havebeen shown to do so. [8, 24] We proceed here onthe assumption that the Ammann markings are in-deed perfect matching rules, an assumption that isstrongly supported by our finding that there existrules for forcing growth of space-filling, infinite clus-ters.

The vertices of an Ammann tiling may be ob-tained by direct projection of a subset of latticepoints of a six-dimensional hyper-cubic lattice ontoa three-dimensional subspace called the tiling spaceand denoted by E||. (See Ref. [25? ? ] for a reviewof projection methods for quasicrystal construction.)The projection onto the tiling space is defined as theprojection that takes the six mutually perpendicu-lar basis vectors of the hypercubic lattice into the six“star vectors” pointing to the vertices of an icosahe-dron.

ek =

{1√5

(2 cos

(2πk5

), 2 sin

(2πk5

), 1)

k ≤ 4

(0, 0, 1) k = 5.

(1)The projection of a hypercubic lattice point a =(a0, a1, a2, a3, a4, a5) onto the tiling space is

P‖(a) =

5∑k=0

akek . (2)

The subset of points that is projected is deter-mined by a projection onto the orthogonal comple-ment of the tiling space, generally referred to as

3

“perp-space” and denoted by E⊥. We define a setof perp-space star vectors:

e′k =

{e〈3k〉 k ≤ 4

−ek k = 5. (3)

The projection of a into E⊥ is

P⊥(a) =

5∑k=0

ake′k . (4)

To generate the vertices of an Ammann tiling, onedefines a perp-space volume that is the projection byP⊥ of a unit hypercube, which forms a rhombic tri-acontahedron called the “perp-space window,” des-ignated W. The vertices of the tiling are the pro-jections by P‖ of all hypercubic lattice points a forwhich P⊥(a) lies within W. Note that the locationof W in E⊥ can be chosen arbitrarily, with differentchoices producing globally distinct Ammann tilingsthat are locally isomorphic; i.e., that cannot be dis-tinguished by examination of local configurations ofany size. Note also thatW has the point group sym-metry of a regular icosahedron.

Individual tiles may be constructed from the set ofprojected vertices by connecting each pair of verticeswith unit separation. The above procedure yieldstwo distinct tile shapes: one prolate rhombohedron,with edges parallel to (e0, e1, e5) or any symmetryrelated triple of star vectors; and one oblate rhom-bohedron, with edges parallel to (e0, e1, e2) or anysymmetry related triple.

III. ICOSAHEDRAL GROWTHALGORITHM

Using the above matching rules, and inspired bythe results of Onoda et al. for the two-dimensionalPenrose tilings [10], we consider a growth algorithmfor Ammann tilings that relies exclusively on a localvertex rule to determine where and how additionaltiles should be added. We first compile a catalog ofall vertex configurations appearing in these tilings.A complete specification of the catalog is presentedin Table 1.

To construct the catalog, we first identify the do-mains within W for which the corresponding ver-tex is part of a tile face with edges along two givenstar vectors ±ei and ±ej . There are two types ofsuch vertex-face domains, depending on whether theangle between the two star vectors at the vertex isacute or obtuse. The two types of domain are bothrhombic dodecahedra [24], but are positioned differ-ently within W. An example of each type is shownin Fig. 2(a), which also shows a fundamental domainof W under the full icosahedral group Ih.

The face corresponding to a given vertex-face do-main may be decorated in any of four distinct waysby the matching rules markings; the red dot can beat either acute angle and the black at either obtuseangle. These distinct markings correspond to dis-tinct domains within the vertex-face domain, whichgets divided symmetrically as shown in Fig. 2(b).(The dividing planes are determined by tracing pos-sible paths of edges from the vertex until a vertex isplaced that implies a tile specifying the location ofthe relevant mark. See Katz [24] for a closely relatedanalysis associated with a set of matching rules re-quiring 14 distinct decorations of the prolate rhom-bohedron and 8 distinct decorations of the oblateone.) Each quadrant of a given vertex-face domaincorresponds to a distinctly oriented and marked faceattached to a tiling vertex that projects into thatdomain in W. The number of distinct complete ver-tex configurations, up to Ih symmetry operations,is obtained by examining a single fundamental do-main of W to see how it is subdivided into cells bythe boundaries of all of the quadrants of all of thevertex-face domains. These boundaries, shown inFig. 2(c), form 39 cells.

Each of the 39 cells corresponds to a unique vertexconfiguration specified by a row in Table 1. An entryin the table specifies a particular face as follows. Thetwo numbers ij specify that the edges of the face thatemanate from the vertex of interest are ei and ej ,with x indicating −ex. The order ij indicates thatthere is a matching rule dot at the tip of edge j, andthe arrow indicates the location of the other dot,with “↑” indicating a dot near the vertex of interestand “↓” indicating a dot at the opposite corner ofthe face. Figure 3 illustrates the meaning of thefirst row of the table. Two of the tiles sharing thevertex are not shown so that we can see the vertexof interest. Consider, for example, the face 34 ↑. Ithas a (red) dot at the corner along the e4 direction,and a (black) dot at the vertex of interest.

Each colored band of rows in the table representsa set of cells that lie in the same set of vertex-face do-mains but not in the same quadrants of all of them;i.e., a set of cells specifying the same geometric ver-tex configuration but with different matching ruledecorations. As noted by Katz, there are 24 suchcells [24]. Figure 4 illustrates the difference betweentwo rows in the gray band of three rows at the topof the table. Again, two tiles have been removed tomake the vertex visible. The two vertices shown areidentical except for the location of the black dot onthe 53 face, which shows up in the table as a differ-ence in the arrow directions for the first two rows inthe band.

The existence of 24 distinct vertex configura-tion geometries and 39 distinct configurations when

4

FIG. 2. Perp-space domains corresponding to distinct vertex configurations. (a) The two types of vertex-facedomains, with a fundamental domain of W shown in green. (b) The division of a vertex-face domain into quadrantscorresponding to distinct matching rule markings. (c) Two views of the division of a fundamental domain into cellsby the union of all icosahedral group operations on the vertex-face domain quadrants. Colors are to aid the eye, withmagenta planes corresponding to quadrant divisions and gray to vertex-face domain boundaries.

FIG. 3. The vertex configuration corresponding to thefirst row of Table 1. See text for details.

matching rules are included has been confirmed bydirect computer assisted inspection of regions of Am-mann tilings with tens of thousands of tiles.

Tiles are added to a growing cluster only at siteswhere the choice of what to add is uniquely deter-mined by the requirement of consistency with thevertex catalog. We identify which, if any, of the ver-tices in the catalog represent possible ways of com-pleting a given vertex. Any tiles that are present inall of the possible complete configurations and notalready present in the cluster are labeled forced tiles.At each time step, a forced tile is selected at randomand added to the cluster. The procedure is repeateduntil there are no forced tiles at any vertex on thesurface of the cluster.

In more precise terms, the algorithm may be de-scribed as follows: Let Qw be the set of oriented tiles

FIG. 4. Vertex configurations corresponding to the sec-ond and third rows of Table 1. See text for details.

comprising the vertex w in the catalog. Let T (v) bethe set of tiles that intersect at a vertex v and havealready been placed in a growing cluster. If T (v) isa subset of Qw, then let Tw(v) be the complement ofT (v) in Qw; i.e., Tw(v) is the set of tiles that mustbe added to T (v) to complete the vertex w.

A vertex in a growing cluster is called complete ifit is fully surrounded by tiles. In almost all cases,a complete vertex will have T (v) = Qw for somew. Complete vertices for which T (v) is not in theallowed vertex catalog are defects.

Incomplete vertices may be forced or unforced.Consider all of the sets Qw associated with cata-log vertices that contain T (v) as a subset, and letTf (v) be the intersection of all of those Qw’s. IfTf (v)−T (v) is not empty, then the vertex v is forced,as there is at least one unplaced tile in Tf (v) that

5

exists in all possible completions of v. The tiles inTf (v) are called forced tiles. If Tf (v) is the empty set,then v is an unforced vertex, meaning that there aretwo or more ways to complete the vertex that do notshare any tiles that have not already been placed.

The growth proceeds by the sequential additionof forced tiles. When a tile is added, new forcedvertices may be created, and the growth continuesuntil no forced vertices remain. As long as there areno defects in the cluster, the order in which forcedtiles are added makes no difference. Small differ-ences (discussed in detail below) can arise when thecluster contains defects. In the present work, the or-der of additions is random: at each step a vertex isselected at random from the current set of all forcedvertices, all forced tiles at that vertex are added, andthe list of forced vertices is updated.

Note that the growth procedure does not rely onany global information about the position of ver-tices within the tiling, nor does it rely on informa-tion about the positions or orientations of any tilesbeyond those that share a vertex with the addedtile. In physical terms, the information about dis-tant structures in the growing cluster is tracked onlythrough the requirement that no tiles be added tounforced sites, and this requires only local informa-tion at each surface site.

IV. GROWTH DYNAMICS

A. Worm planes

A key to understanding the growth process gener-ated by the above algorithm is the structure we calla worm plane, which is analogous to a linear wormin the Penrose tilings [3, 26]. A portion of a wormplane is shown in Fig. 5(a). The crucial feature ofthis planar slab of tiles is that the vertices in theinterior of the slab can be moved vertically so asto create a second version of the slab that has ex-actly the same outer surfaces, including the match-ing rule markings, as the original, while the markingson interior faces in two slabs differ, as indicated inFig. 5(b). The operation that moves moves all ofthe interior vertices and changes all markings on theinterior faces accordingly is called a worm flip.

If a portion of the surface of a cluster correspondsto the surface of a worm plane that has not yet beenplaced, it will contain no forced vertices. The wormcan be added in either of its two possible orienta-tions, thus there are two distinct ways to completeany given surface vertex. Once a choice is made forone vertex on the worm plane surface, all of the oth-ers will be forced.

Worm planes are important structural elements

for two reasons. First, the choice of orientation ofa given worm plane must be coordinated within theworm plane itself. If different choices are made forthe orientation of the worm in two half-planes, a lineof defects will necessarily be created where the twohalves of the worm are joined. The growth algo-rithm avoids such defects by filling forced verticesfirst. Once the orientation of a worm plane is deter-mined at a single vertex, the rest of the worm planewill be filled due to forced additions that propagatethe information about the worm orientation to thefull plane.

Second, the orientations of parallel worm planesmust be correlated in subtle ways. In certain config-urations, the necessary orientation of a worm planecan be determined by the orientation of a parallelworm plane that is far away. If arbitrary choiceswere made for the two orientations, the subsequentaddition of forced tiles might eventually lead to con-flicting choices for the orientation of a worm planetransverse to the first two, thereby generating a lineof defects somewhere between the two original wormplanes.

Our growth algorithm avoids this second problemby simply halting when there are no forced verticeson the surface of the growing cluster. This occurswhen the surface consists entirely of worm planes ori-ented such that no forced vertices occur along edgesor at the corners of the faceted cluster. (We havecharacterized the possible dihedral angles and solidangles at the corners that have no forced vertices,but we omit the details here as they are not relevantto the main results.) To avoid this type of arrest inthe growth, we introduce special seeds that nucleateinfinite growth as described below.

A perfect, infinite Ammann tiling contains wormplane regions with 15 different possible normal vec-tors, corresponding to the planes of mirror symmetryof the icosahedron. Typical worm plane regions arebounded by intersecting worm planes with differentnormal vectors. At these intersections, the orienta-tion of one worm plane can force the orientation ofthe other. There may be as many as four intersect-ing infinite worm planes in the tiling.

B. Seeds for growth: Triacontapods

Consider a finite, closed surface comprised ofmarked rhombic faces. If the surface can be foundwithin a perfect Ammann tiling, we refer to it asthe surface of a “legal” cluster of tiles. If it can-not, we call it an “illegal surface.” Given any legalcluster as a seed, growth through the addition offorced tiles must eventually halt. To see why, con-sider the structure of the finite cluster in perp-space.

6

(a)

(b)

(a)

(b)

FIG. 5. a) Worm planes of opposite orientations. Thesetwo planes are composed of different tiles, yet have thesame matching rule markings on their top and bottomsurfaces. Prolate tiles are dark and light purple; oblatetiles are dark and light gray. White rhombi are tilefaces that are not part of any tile in the worm plane.(b) Matching rule markings on the boundary of a wormplane that specify its orientation.

Recall that distinct positions of the window withinE⊥ specify distinct tilings. Thus the vertices of aninfinite tiling are uniquely determined only when thelocation of the window is fixed. We know, however,that any finite portion of an Ammann tiling can befound (infinitely many times) in any Ammann tiling,which means that the finite cluster cannot preciselyfix the location of the window. By definition, forcedgrowth cannot rule out any of the possible windowsthat contain the points in the original cluster. Inother words, forced growth can never result in aperp-space point being placed outside the the hulldefined as the intersection of all windows W thatcontain the points that have already been placed.Growth of an infinite tiling, however, must produceperp-space points that fill an entire window. Thusforced growth from a legal seed cannot yield an in-finite tiling.

In order for forced growth to proceed indefinitely,we must begin with an illegal seed containing a de-fect that determines the precise location of the win-dow. Such a seed can be constructed via analogyto the decapod seeds that generate infinite forcedgrowth in the Penrose tilings.[10, 12–14] For theAmmann tilings, a suitable seed is a triacontapod,a rhombic triacontahedron with exterior matchingrule markings. An example is shown in Fig. 6.

���������

FIG. 6. A legal triacontapod and its unfolded net offaces. There exists an Ammann tiling in which 15 infiniteworm planes intersect at this triacontapod.

FIG. 7. Triacontapod seeds dictate the orientation ofworm planes. A seed is shown with a worm plane rep-resented symbolically by the horizontal gray plane. Thematching rule dot circled in blue dictates the orientationof the worm plane. To satisfy the matching rules, tilesin the interior of the worm plane must be oriented suchthat the dot (red or black) on each face perpendicular tothe plane lies on the same side of it.

C. Legal seeds

The markings on the triacontapod of Fig. 6 areconsistent in the sense that this configuration doesappear in the Ammann tilings, and the triacontahe-dron can be filled in with tiles that obey the match-ing rules everywhere. There is exactly one Am-mann tiling that has 15 infinite worm planes all in-tersecting to form this triacontapod. Four of thesepass through the triacontapod to form perfect wormplanes; the others are disrupted in the interior of thetriacontapod but are otherwise perfect. The orien-tation of each worm plane is dictated by the mark-ings on two opposite faces of the triacontapod, asillustrated in Fig. 7. In more general cases (i.e., tri-acontapod defects) we will assign dots on the seed’ssurface manually without worrying about whetherthe interior of the seed can be consistently tiled.Thus we drop the color distinction between the fourtypes of tiles when showing a triacontapod. Thereis only one legal triacontapod, up to symmetry op-erations on the icosahedron, shown in Fig. 6. Anyother pattern of marks on the triacontapod makesfor an illegal seed.

When we speak of using a triacontapod as a seedfor growth, we assume that the seed includes prolatetiles covering all of the faces of the triacontapod.

7

One such tile is shown in Fig. 8(a). The red dots ona triacontapod determine the orientations of theseprolate tiles and hence the positions of 30 verticeslike the one marked by a black sphere in Fig. 8(a),whose normal projection onto the triacontapod facein question lies within that face. Each of these ver-tices lies in the interior of a worm plane, determiningits orientation. It is instructive to examine the loca-tions of these 30 vertices in perp-space. Figure 8(b)shows their locations for the case of a legal triaconta-pod, obtained from Eq. (4) using indices taken fromEq. (2). The figure shows one possible perp-spacewindow containing those points (not to be confusedwith the real space triacontapod!). For the windowshown, the points lie precisely on exterior facets, andfor the tiling determined by this window, the tria-contapod lies at the intersection of 15 infinite wormplanes. As must be the case for a finite legal seed,however, there exist other windows that contain allof the points. Roughly speaking, the points all lie inone hemisphere of the window shown, and the win-dow can be shifted in the direction of the pole ofthat hemisphere and still contain all 30 points.

D. Illegal seeds

In order to force growth to infinity, a seed mustfully constrain the position of the perp-space win-dow. [14] We can arrange for a triacontapod touniquely determine the window by choosing themarkings such that the black dots of Fig. 8(b) fallon facets that do not all lie in any single hemisphere.This can be accomplished, for example, by movingthe red mark that specifies the orientation of theprolate tile in Fig. 8(a) to the opposite corner of theface it lies on. The resulting flip of the tile causesthe black dot in perp-space to jump to the oppositeface of the window.

An example of an illegal seed is shown in Figure 9,and a plot of the forced vertices in perp-space for thesame seed is shown in Figure 10. The location of theperp-space window is fixed; attempting to shift thewindow in any direction will move at least one vertexoutside of the window. This implies that there is atmost one tiling that is consistent with the matchingrules everywhere outside the seed, and so it is possi-ble, but not guaranteed, that the seed forces growthto infinity.

For the 2D Penrose tilings, Onoda et al. pointedout that there exist tilings that obey the matchingrules everywhere outside an illegal decapod and that,for some illegal decapods, the surface of any clustercontaining the decapod must always have at leastone forced vertex. (See also [12–14].) For such de-capods, the sequential addition of forced tiles never

(b) Perp-space positions of

(a) A triacontapod seed

black spheres in (a)

FIG. 8. (a) A triacontapod with a prolate tile attached.The orientation of the tile is dictated by the covered reddot on the triacontapod surface. The black sphere marksthe vertex on the prolate tile that lies in the interior ofa worm plane. (b) Two views of a perp-space windowand the 30 projected vertices (black spheres from (a))for a legal triacontapod seed. The perp-space window,shown as a transparent yellow triacontahedron (not to beconfused with the real space triacontapod), is displayedfrom two opposite perspectives. The 30 vertices lie inone hemisphere of the window.

halts and never produces a matching rule violation.

For the 3D Ammann tilings, the situation is differ-ent: any tiling that contains an illegal triacontapodmust contain matching rule violations outside theseed. To see this, consider the vertices of the illegalseed shown in Figure 11. Notice that the red dots,circled in black, are three-fold symmetric about thevertex circled in blue. Such a configuration doesnot appear in any vertex in the catalog. Similarly,

8

FIG. 9. An unfolded net representation of an illegal seedthat constrains the perp-space window. (Compare toFig. 6.)

Perp-space positions of vertices associated with an illegal seed

FIG. 10. Perp-space positions of projected vertices(black spheres of Fig. 8(a) for an illegal triacontapodseed that fully constrains the perp-space window. Thedifference between this figure and Fig. 8(b) is that twovertices have been moved to the opposite side of the win-dow.

for the lower image, it can be determined by in-spection that while the vertex circled in blue canbe completed without any matching rule violations,the dots circled in black will force the creation of avertex that cannot be legally completed. In general,growth from any seed with either a three-fold sym-metric vertex, as illustrated in the upper image, ora vertex with a chiral pattern of dots as illustratedin the lower image, must produce additional defects.An exhaustive search through all illegal seeds revealsthat each possesses at least two such vertices.

Though the creation of defects during the growth

Triacontapod defects

FIG. 11. The two varieties of triacontapod vertex con-figurations (circled) that force defects to appear duringgrowth. Every illegal triacontapod contains at least twovertices in this class.

might be expected to prevent forced growth fromproceeding, it turns out that the algorithm can anddoes accommodate these defects and still generatesa space-filling tiling by adding only to forced ver-tices as originally defined. The tiles surrounding theillegal vertices get added due to forcing from othernearby vertices. A matching rule violation occurson a single face shared by two tiles that get incorpo-rated into the bulk as growth proceeds. The preciselocation of the mismatch may depend on the orderin which forced tiles are added, but it must occursomewhere along the row of tiles that share facesparallel to the mismatched face. As new tiles areadded in the vicinity of a defect, the mismatch inworm plane orientations requires the creation of anew defect nearby, giving rise to a meandering lineof defects sites radiating outward in the direction ofgrowth.

Moreover, such defects do not disrupt the over-all quasiperiodic order. It remains true that forcedgrowth can never produce a vertex that lies outsidethe perp-space hull determined by the tiles that havealready been placed. This means that the only de-fects in the tiling occur outside the triacontapod seedlie within the infinite worm planes, whose interiorvertices lie on the boundaries of the perp-space win-dow. These defects must manifest as vertices that

9

���������

FIG. 12. A tiling grown from the seed of Figure 9.This cluster contains approximately 100,000 tiles. Largerclusters have been grown from this seed, and none hasyet encountered a dead surface.

0 5000 10000 15000 20000 250000.00

0.05

0.10

0.15

0.20

0.25

0.30

Ncluster

Qforced

FIG. 13. Fraction of surface vertices in the cluster ofFig. 12 that are forced, plotted as a function of the num-ber of tiles in the cluster during growth.

lie on opposite facets of the window. The bulk ofthe tiling is therefore defect free, and as the incon-sistent worm plane orientations on two halves of aninfinite worm plane meet along a line of defects, thenumber of defects is expected to grow only linearlywith cluster radius and therefore have a vanishinglysmall density in the infinite tiling.

An example of simulated growth from the seed ofFigure 9 is shown in Fig. 12. It appears that thegrowth proceeds to infinity, as will be discussed fur-ther below. Figure 13 shows Qforced, the number offorced surface vertices divided by the total numberof surface vertices, as a function of the total num-ber of tiles in the growing cluster of Fig. 12. Qforced

does not show dips to very low values that would beassociated with growth spurts between nearly com-pletely unforced surfaces.

FIG. 14. Defects confined to worm planes. Here all de-fective faces are shown from a cluster of approximately1000 tiles grown from the illegal seed of Figure 9. Thefour panels show different views of the same defect struc-ture. Defect faces in different planes are given differentcolors.

0 5 10 15 20 25 300

10

20

30

40

50

Volume1/3

Numberofmismatchedfaces

FIG. 15. Number of defects in the cluster of Fig. 12 asa function of cluster volume at the time a given defectappears. The volume is measured in units of the volumeof the oblate tile.

The defects are confined to a subset of the infi-nite worm planes, as shown in Fig. 14. In this case,defects appear in the infinite worm planes becauseopposite faces of the seed specify different orienta-tions for a given plane. As growth proceeds, thegiven plane is thus divided into two halves of oppo-site orientation, and a line of defects forms wherethe two halves meet. Figure 15 shows the numberof defects as a function of cluster radius, confirmingthe expected linear relationship.

10

E. The probability of infinite growth

Numerical simulation suggests that the seed ofFigure 9 does indeed force growth to infinity. Mul-tiple clusters containing several hundred thousandtiles have been grown from this seed, and none hasever halted due to a lack of forced sites. We can-not rule out, however, the possibility that growthcould be stopped if the order in which forced tilesare added conspires to enclose the seed in a legalsurface. A defect line separating regions of a wormplane with opposite orientations could bend intoloop, leaving an enclosed portion (containing part ofthe seed) that is flipped but showing no deviationsfrom a perfect worm plane on the surface of the clus-ter. If the seed were hidden in this way in a legalcluster, the relevant window for describing furthergrowth would not be constrained by the structure ofthe seed and would no longer be uniquely specified,so growth would eventually halt for the reasons de-scribed above. We have observed this phenomenonin simulations, with the simplest example being thefollowing. There is an equatorial band around theseed, contained within a worm plane, where five tileson one side are oriented one way and the five tilescompleting the band on the other side are orientedthe other way. If one simply refuses to choose ver-tices on any of the tiles on one side of the band ascandidates for forced additions, the growth inducedby choices made on the other side (and out of theplane in question) can add tiles that surround theentire original band making all of the exposed tilesin the equatorial band around the growing clusterhave the same orientation.

The probability of choosing the sequence of forcedadditions in a way that erases the memory of theseed is clearly quite small and decreases rapidly asthe cluster size increases. If we assume that the clus-ter grows at a roughly uniform rate in all directions,then the worm plane containing the rays of defectsis a growing disk with two points on its circumfer-ence marking the points where the defect rays hitthe surface. Each time a new layer of tiles is addedto the cluster, the defect moves randomly on theboundary by a distance of the order of on tile edge.Thus the two endpoints of the defect rays executingrandom walks on the surface with fluctuations thatgrow as

√r, where r is the disk radius. The disk cir-

cumference, however, grows at a rate proportionalto r, making it exponentially improbable that thetwo walks will meet unless they do so at a very earlystage. We conclude that infinite growth with onlyinfinitesimal phason strain occurs in this model witha probability of order one, where the precise valueincreases rapidly with the size of the cluster that istaken to be the initial seed.

The example of infinite growth shown here illus-trates a subtle feature of the growth rule. One mightthink that a seed could be used for which there areno worm planes with inconsistent orientations ontwo half planes. The defect in the seed would beencoded in the relative orientations of intersectingworm planes rather than any discrepancies within asingle worm plane. Indeed, the example used herebegins with such a seed, as can be seen by the factthat for every pair of opposite faces of the perp-space window both dots appear on the same face.(When an infinite worm plane is divided into twohalves, the interior vertices from the two differenthalves project onto opposite faces of the window.)As forced growth proceeds, however, tiles are placedthat override the orientations dictated by the origi-nal seed, creating half-plane defects.

V. DISCUSSION

We have shown that the information required togrow a nearly perfect, infinite icosahedral quasicrys-tal without any backtracking to correct mistakes canbe stored in local neighborhoods of the surface sitesat all times during the growth. The growth is nu-cleated by a small seed and proceeds through theaddition of new tiles to randomly selected surfacesites, where the probability of attachment is deter-mined by the configuration of existing tiles sharing asingle vertex. The resulting infinite cluster containsa vanishing density of defects; its diffraction pat-tern would contain the dense set of infinitely sharpBragg diffraction peaks characteristic of quasicrys-tals with icosahedral symmetry, and the relative in-tensity of any diffuse scattering would vanish in theinfinite system size limit. The grown sample can becharacterized as a quasicrystal with only infinitesi-mal phason fluctuations corresponding to inconsis-tent choices of which faces of the perp-space accep-tance window are taken to be closed, but includingno points that lie outside the closed window.

An interesting feature of the growth in the 3Dicosahedral case is the necessary inclusion of a lin-ear density of defects associated with a boundarybetween two halves of an infinite worm plane thathave opposite orientations. When growth is at-tempted from seeds that do not induce such bound-aries, it halts at surfaces where there are no forcedsites. Thus the defects actually promote more per-fect growth than can be guaranteed in systemswhere they are eliminated through annealing at earlystages. Once the cluster has grown to any desiredsize and growth is considered complete, it would bepossible to anneal out these lines of defects. Theenergy barriers involved would be very low because

11

motion of a line of defects can be accomplishedthrough sequences of flips in which it is never nec-essary to create more than two extra matching ruleviolations in the sample, though selecting such a se-quence through a probabilistic process would likelytake a very long time. We emphasize, however, thatthe as-grown sample would still be, for all practicalpurposes, a perfect quasicrystal.

We conjecture that the growth of real, rapidlyquenched materials is an approximation of the idealprocess described here. The ideal process requiresthe probability of growth at unforced sites to bestrictly zero. A nonzero probability would lead tooccasional additions of the wrong type of tile at avertex, which would give rise to a finite density ofmatching rule defects. There are three classes ofsuch defects.

The first type of defect involves simple mistakesthat create illegal sites where local growth stops un-til other nearby forced additions promote a correc-tion. In the second type of defect, two conflictingchoices are made for the orientation of a portion ofa worm plane, giving rise to additional forced ad-ditions that propagate along the worm plane untilintersecting worm planes are reached that dictatethe proper choice. In such cases, it is possible thatan incorrectly oriented portion of a worm plane willbe buried deep below the surface by the time thecorrect choice is forced. Defects of this type, whichare confined to the interior of a single worm plane,do not disrupt the long range quasiperiodic transla-tional order unless they give rise to the third typeof defect, discussed below. The effects associatedwith these first two types of defect may be mini-mized via annealing of a surface layer of finite depth,with the density of the benign intra-worm defects de-creasing with increasing depth of the solidificationfront. A process of this type appears to have beenobserved directly by Nagao et al. in experiments ona decagonal phase [19]. We conjecture that similarprocesses occur in computer simulations of growthmechanisms that involve an advancing solidificationfront [15, 17].

The third type of defect presents a greater chal-lenge to the realization of strict quasiperiodic trans-lational order. These arise when the relative orien-tations of two different parallel worm planes, whichmay be far apart, conflict with one another. Thedefects associated with this type of phason fluctua-tion become visible as matching rule violations onlyafter additional growth has filled in a bulk region be-tween the two planes. At that point, the existenceof the phason fluctuation is revealed, but there isno local indication of where the real problem lies.Avoiding this type of defect requires the introduc-tion of an illegal seed, which allows forced growth

to dictate the proper worm orientations before anincorrectly oriented worm grows too large. Becausedefects of this type, which are not identifiable by lo-cal tests, do generate phason strains that can disruptthe quasicrystalline order, the growth of a perfectquasicrystal requires a strong separation of scalesbetween the rates of addition at forced and unforcedsites. Further study of the dependence of the sizephason strains on the ratio of the two rates and onthe depth of the solidification front should providetestable predictions for systems in which the tem-peratures of the solid and supercooled liquid can becontrolled. The present work shows that the limitingcase does allow for essentially perfect growth.

We note for mathematical completeness that onecould consider an intermediate option as well inwhich the probability of adding at an unforced siteis taken to be infinitesimal (but nonzero) so thatforced growth proceeds until a completely unforcedsurface is reached, at which point a random choiceis made for the addition of a single tile. Such a rulewould never give rise to either of the first two typesof defects described above. An estimate of the typi-cal size a cluster would reach before a choice is madethat creates a defect of the third type requires an un-derstanding of the set of possible unforced surfaces,which is beyond the scope of the present work.

ACKNOWLEDGMENTS

We thank Trinity College of Duke University forits support of this project through a Dean’s SummerResearch Fellowship for C. H.

12

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13

TABLE I. The complete vertex catalog. Each row represents one vertex type, and the 39 rows of the table constitutethe entire catalog up to rotations. Within each row, each column represents one face. In a given box, the twonumbers specify the icosahedral star vectors (see Eq. (1)) forming the edges of the face, with overbars denotingnegative directions; ab indicates that the four vertices of the face are 0, ea, −eb, and ea − eb. The arrow indicatesthe locations of the matching rule dots: an up arrow indicates that a dot is placed near the vertex at the origin, andand a down arrow indicates that a dot is placed near the opposite vertex. The number next to the arrow indicatesthe location of the second dot: the dot is placed near the vertex located at the tip of the corresponding star vector.Complete tiles can be directly inferred from these faces.

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