Dihedral Escherization

Craig S. Kaplan1 David H. Salesin2,3

1School of Computer ScienceUniversity of Waterloo

2Microsoft Corporation 3Computer Science and EngineeringUniversity of Washington

Abstract“Escherization” [9] is a process that finds an Escher-liketiling of the plane from tiles that resemble a user-suppliedgoal shape. We show how the original Escherization al-gorithm can be adapted to thedihedral case, producingtilings with two distinct shapes. We use a form of theadapted algorithm to create drawings in the style of Es-cher’s printSky and Water. Finally, we develop an Es-cherization algorithm for the very different case of Pen-rose’s aperiodic tilings.

Key words: Tilings, tessellations, Escher, aperiodic, Pen-rose tilings

1 Introduction

The Dutch artist M.C. Escher had a singular gift for in-corporating mathematics into art. His work continues todelight and fascinate, a mix of paradox and harmony, ofwhimsy and order. In particular, he produced a large col-lection of ingenious tessellations [14], made from motifsresembling people, animals, and fantasy creatures. Thesearch for interlocking forms was for Escher a lifelongpursuit and source of both frustration and inspiration.

In the SIGGRAPH 2000 conference, Kaplan andSalesin presented “Escherization” [9], a method for auto-matically discovering Escher-like tessellations with tilesthat resemble arbitrary user-supplied shapes. Given agoal shapeS, their system uses continuous optimizationto hunt through a parameterized space of tilings in searchof a tile shapeT that best approximatesS. The quality ofthe approximation is determined via an efficient polygoncomparison metric [1].

The Escherization algorithm they describe is able tofind Escher-like tilings that aremonohedral, i.e., madeup of copies of a single motif. Escher also created manytessellations featuring two or (less frequently) more mo-tifs. Dihedral (two-motif) tessellations are particularlyimportant in his work. Some of his most famous prints(for example,Sky and Water, Verbum, andMetamorpho-sis II) make use of one or more dihedral tilings [2]. Fur-thermore, the use of multiple motifs agrees with Escher’spredisposition to imbue his work with narrative structure.The interplay between different motifs in a single design

provides an opportunity for contrasts, for harmony or dis-cord, for interaction and drama.

In this paper we extend the Escherization algorithm ofKaplan and Salesin to the dihedral case. We may beginby analogy with their work, formulating adihedral Es-cherization problem:

Problem (“D IHEDRAL ESCHERIZATION”): Givenclosed plane figuresS1 andS2 (the “goal shapes”),find new closed figuresT1 andT2 such that:

1. T1 andT2 are as close as possible toS1 andS2, respectively; and

2. T1 andT2 admit a dihedral tiling of the plane.

We present a solution to the dihedral Escherizationproblem as an extension to the algorithm given by Ka-plan and Salesin. As discussed in Section 3, we augmentthe representation of a tile shape with a curve that splits itinto two pieces. We optimize over this new configurationspace using an objective function that compares the twopieces with two goal shapes. In Section 4, we show howa restricted version of the general splitting process canproduce what Dress callsHeaven and Hell patterns[5].In Section 5, we develop a tool that uses Heaven and Hellpatterns to generate drawings in the style of Escher’s printSky and Water. In Section 6, we present an alternativeformulation of dihedral Escherization that works on Pen-rose’s aperiodic tile setsP2 andP3. We conclude in Sec-tion 7 with a discussion of directions for future work.

2 Mathematical background

In this section we briefly define some of the concepts oftiling theory that are relevant in this work. A definitivetreatment of the subject is given by Grunbaum and Shep-hard [7]; Kaplan and Salesin [9] provide an overview.

A tiling of the planeis a countable collection of tilesthat cover the plane without any gaps or overlaps. A tilingisk-hedralwhen every tile is congruent to one ofk differ-ent shapes, calledprototiles. The casesk = 1 andk = 2are referred to asmonohedralanddihedralrespectively.

A tiling vertexis a point where three or more tiles meet.A tile’s boundary can be subdivided into a collection oftiling vertices connected by arcs calledtiling edges.

The symmetriesof a tiling are rigid motions of theplane that map the tiling onto itself. A tiling with transla-tional symmetries in two non-parallel directions is calledperiodic. There are many non-periodic tilings, but of-ten the prototiles of such tilings can also be used to con-struct periodic tilings. A more remarkable situation oc-curs when every tiling that can be constructed from a setof shapes is non-periodic; such sets are calledaperiodic,as are the tilings that can be built from them.

For any two congruent tiles in a tiling, there will bea rigid motion of the plane that maps the first tile ontothe second. If this motion also maps the entire tiling toitself then the two tiles are said to betransitively equiva-lent. Transitive equivalence partitions the tiles of a tilinginto equivalence classes calledtransitivity classes. Atiling with exactly one transitivity class is calledisohe-dral; more generally, ak-isohedraltiling is one withktransitivity classes. Differently-shaped tiles will neces-sarily belong to different classes, and so ak-hedral tilingwill be at leastk-isohedral, though it may possibly havemore transitivity classes.

The combinatorial properties of every isohedral tilingcan be summarized using a compact string called aninci-dence symbol[7]. By writing down all possible incidencesymbols and eliminating those that cannot correspond tolegal tilings, Grunbaum and Shephard showed that theisohedral tilings can be partitioned into precisely 93 com-binatorial types labeled IH1, . . . , IH93. Each type corre-sponds to a different way that a tile can be surrounded byits neighbours.

3 Split isohedral Escherization

Our search for a useful space of dihedral tilings be-gins with Escher himself. A meticulous note-taker, hecarefully documented his exploration of two-motif sys-tems [13, 14]. In every case, he starts with one of hismonohedral systems and draws a path through the pro-totile to break it into two shapes. When that division iscopied to all other tiles, the result is a dihedral tiling.

We can apply a similar process to the isohedral tilings.We augment the description of an isohedral prototile witha “splitting path,” a path that starts and ends on the pro-totile’s boundary. A splitting path naturally subdivides aprototile into two shapesT1 andT2. When every tile in anisohedral tiling is subdivided by the splitting path, the re-sult is a “split isohedral tiling,” a dihedral tiling with pro-totilesT1 andT2. Every such tiling will be 2-isohedral,though there also exist 2-isohedral tilings that cannot beconstructed using a splitting path (see Section 7).

In our implementation, the splitting path is stored as apiecewise linear path in a local coordinate system. Theisohedral prototile’s boundary is parameterized by ar-

clength and the start and end positions of the path arerecorded using two real values between 0 and 1. Once co-ordinates for the start and end positions are determined,the splitting path can be transformed into place.

The splitting process can be applied to prototiles fromany of the 93 isohedral types. As in the case of monohe-dral Escherization, the choice of isohedral type is discreteand cannot be made within the framework of a continu-ous optimization. Following Kaplan and Salesin, we runmultiple per-tiling-type optimizations in parallel, gradu-ally winnowing down the pool of candidates until onlythe most successful type remains.

The monohedral Escherization algorithm searched aconfiguration space where each tuple of floating-pointvalues encoded the shape of an isohedral prototile. Someof the values controlled the positions of the tiling ver-tices; the rest controlled a non-redundant description ofthe edge shapes. To handle the split isohedral case, weenlarge the search space to include parameters for the po-sition and shape of the splitting path. At each step in theoptimization, we construct the split isohedral prototile,extract shapesT1 andT2, and compare them with user-supplied goal shapesS1 andS2 using the metric of Arkinet al. [1]. The two comparisons yield two non-negativereal numbersd1 andd2; we usemax(d1, d2) as an objec-tive function for dihedral Escherization, forcing both tileshapes to resemble their respective goal shapes as closelyas possible. As in the monohedral algorithm, we peri-odically subdivide the splitting path along with the edgeshapes, giving the algorithm a chance to pick up finer de-tails in the goal shapes.

Let S′1 andS′

2 denote reflections of goal shapesS1 andS2. To find the best split isohedral tiling corresponding tothe goal shapes, two instances of the Escherization algo-rithm are required: one withS1 andS2, and one withS1

andS′2 (or S′

1 andS2). Although the shape comparisonmetric is insensitive to translation and rotation, it doesdistinguish between a shape and its reflection. It mighthappen thatS1 andS2 interact more favourably if one isreflected.1

The split isohedral Escherization process is illustratedin Figure 1. Figure 7 shows some results obtained usingthis process. One might guess that because of the need tomatch two goal shapes simultaneously, dihedral Escher-ization would have a lower success rate than monohedralEscherization. We have found that the additional degreesof freedom offered by the splitting path help to compen-sate for the added complexity of the problem, and that thedihedral and monohedral optimizations have comparable

1Note that only the relative parity matters here; the flexibility ofthe isohedral tilings guarantees that the case(S1, S2) is equivalent to(S′

1, S′2), and that(S′

1, S2) is equivalent to(S1, S′2).

success rates. Speeds are comparable as well; the dihe-dral optimizer usually converges in 10–20 minutes. Aninteractive viewer also allows the user to watch the opti-mization in progress and abort it if no promising solutionsseem likely.

4 Heaven and Hell Escherization

Some of Escher’s dihedral tilings, such asHeaven andHell [14], have additional geometric structure. Each tilecan be given one of two colours so that adjacent tilesnever share a colour. Furthermore, each colour is the ex-clusive domain of one of the two prototiles; in Heavenand Hell, every angel is white and every devil is black.This sort of colouring is possible when every tiling ver-tex is surrounded by an alternating sequence ofA andBtiles, or equivalently, when everyA tile shares edges onlywith B tiles (and vice versa).

Aesthetically, such tilings are particularly effective be-cause each transitivity class of tiles plays the role ofground to the other’s figure: theB tiles exactly fill thenegative space created by theA tiles. Moreover, the factthat the colours can be unambiguously associated withtile shapes allows them to become part of the personali-ties of those shapes, as in the white angels and black dev-ils of Heaven and Hell. Escher used this particular spaceof tilings to produce some of his best-known prints.

The class of 2-isohedral tilings with this additionalstructure were enumerated by Dress [5], who dubbedthem “Heaven and Hell patterns.” Based on an analysisusing Delaney symbols [4], he classified the Heaven andHell patterns into 37 distinct types.

Of the types enumerated by Dress, 29 can be expressedas specialized versions of split isohedral tilings. The ad-ditional structure comes from a careful choice of loca-tions for the endpoints of the splitting path. Dress’s clas-sification shows that an endpoint will always be eitherone of the tiling vertices of the underlying isohedral pro-totile, or the midpoint of one of its tiling edges. If theisohedral prototile hasn tiling vertices, we can enumeratethis set of locations asL = {1, 1 1

2 , 2, 2 12 , . . . , n, n + 1

2},where a whole numberk refers to a tiling vertex andk + 1

2 refers to the midpoint of the edge fromk to k + 1.The numbering of the tiling vertices can be taken fromthe order in which they appear in the enumeration byGrunbaum and Shephard [7]. Each of the 29 types basedon splitting can then be given the notation(IHm; a, b),whereIHm denotes one of the isohedral types, and wherea, b are members ofL.

We represent a prototile for the Heaven and Hell tiling(IHm; a, b) by starting with the representation for thesplit isohedral prototile of the same type, and fixing theendpoints of the splitting path according to the locations

a andb. Once the degrees of freedom controlling the end-points are removed from the configuration space, the re-mainder of the dihedral Escherization algorithm can beapplied as is.

In Dress’s paper, some of the types of Heaven and Hellpatterns can be seen as subsumed under other types, inthe sense that a type with symmetric prototiles is merely aspecial case of an asymmetric parent. Dress’s explicit or-dering makes it easy to recognize that the 29 Heaven andHell tiling types representable as split isohedral tilingscan be summarized using twelvefundamentaltypes withasymmetric prototiles. Using the notation given above,the twelve types are as follows:

(IH1; 1, 4), (IH2; 2, 5), (IH3; 2, 5),(IH5; 1, 4), (IH27; 1 1

2 , 4), (IH31; 1, 3),(IH33; 1, 3), (IH41; 1, 3), (IH43; 1, 3),

(IH47; 2 12 , 4 1

2 ), (IH52; 1, 3), (IH55; 2, 4)

Eight remaining types in Dress’s classification are notaccounted for by split isohedral tilings. These tilings havedifferent proportions ofA andB tiles. We do not con-sider Escherization over such tilings here, although wemention them again in Section 7.

Figure 3 gives an example of Heaven and Hell Escher-ization.

5 Sky and Water designs

Escher’s printSky and Wateris a very special applicationof Heaven and Hell patterns. What starts out in the centerof the print as a dihedral tiling of stylized fish and birdsevolves towards the top and bottom of the print into real-istic figures: birds above and fish below. Escher used thisdevice in many prints and sometimes multiple times in asingle print (as inVerbumandMetamorphosis II).

It is critical that the central tiling where the birds andfish meet be a Heaven and Hell tiling. The stylized birdsevolve into the background for the realistic fish (and viceversa), and so the tiling needs to be colourable with onecolour for each tile shape.

Escherization is especially well suited to the creationof Sky and Water designs because the realistic goalshapes are already part of the process that leads to thestylized tile shapes. To turn a Heaven and Hell tiling intoa Sky and Water design, it suffices to gradually blend thetile shape into the goal shape as tiles are placed succes-sively farther from a given “interface line.”

We extended the basic Heaven and Hell Escherizationalgorithm with a suite of interactive tools for construct-ing Sky and Water designs. One tool lets the user specifyan interface line and a set of tiles to draw. Another toollets the user add decorations to tiles with monochromaticvector-based strokes. Each stroke is a sequence of Bezier

Funky Chickens(IH51)

Figure 1: A summary of our process for split isohedral Escherization. On the left, two goal shapes S1 and S2 aretraced from images. Next, the isohedral tile and splitting path are shown at a late stage in the optimization. The qualityof this configuration is judged by breaking the tile into two shapes T1 and T2, which are then compared with S1 andS2. The optimization attempts to minimize the value at the right, the maximum of the two comparisons. A finishedexample based on these goal shapes appears on the right.

Figure 2: An example of a Sky and Water design, basedon the goal shapes of Figure 7d.

The Owl and the Pussycat(IH27; 1 12 , 4)

Figure 3: An example of Heaven and Hell Escherization.

The Pentalateral Commission(P3)

Figure 4: An Escher-like design created by hand, basedon Penrose tile set P3. The tile shapes were discoveredafter a few minutes of interactive exploration, and deco-rated using Adobe Illustrator.

curves with user-specified widths; the curves are fit tothe user’s drawing gestures using the method of Schnei-der [15]. Additionally, each stroke is given a “priority”that determines how far from the interface line the tilemust be before the stroke is drawn. This approach al-lows for a prioritized sequence of strokes ordered by theirimportance in expressing a stylized version of the goalshape.

Finally, a renderer assembles the final drawing, takingthe output of the other two tools as input, together withcolours for theA andB tile shapes. For every tile, therenderer interpolates that tile with its corresponding goalshape by an amount determined from its distance to theinterface line. The interpolation is carried out so that anytiles that touch or cross the line are set to the tile shape,and maximally distant tiles are set to the goal shape. TheA tiles are then drawn over a solid background of theBtile colour, and vice versa. Finally, the strokes are warpedinto place and drawn if they have sufficiently high prior-ities. Figure 2 shows an example of an Escherized Skyand Water design.

6 Escherization using Penrose tiles

The most widely known aperiodic tilings are those dis-covered by Penrose [7, 12]. In particular, he demon-strated two tile sets that yield dihedral aperiodic tilings:P2, made up of kites and darts, andP3, made up of thinand thick rhombs. Like the Mandelbrot set, the Penrosetilings are ambassadors of mathematical beauty to a gen-eral audience.

Unfortunately, Escher did not live to see the develop-ment of Penrose tilings, and so we can only imagine whatsorts of creatures he might have discovered in them. Pen-rose himself, who corresponded regularly with Escher,expresses his regret at the missed opportunity [11]. Healso gives an example of what Escher might have drawn:a modification ofP2 where the kites and darts have beenturned into chickens. Other artists have also created de-signs based on Penrose tiles [6].

In order to enforce aperiodicity in the tile setsP2 andP3, the tiles must be augmented with matching con-ditions that determine the legal ways one tile may beplaced next to another. One important way to expressthese matching conditions is to deform tile edges so thattiles only fit together in prescribed ways. Grunbaum andShephard give such geometric matching conditions forP2 andP3 [7]. In both cases, the matching conditionsare boiled down to two non-congruent paths and the waythey are arranged around the two tiles in the set.

The geometric matching conditions suggest an Escher-ization algorithm for Penrose tiles. The tiling vertices re-main fixed, and the optimization operates on the degrees

Figure 5: An illustration of how a tiling vertex param-eterization can be derived for the Penrose kite and dart.The original edges are modified using Grunbaum andShephard’s edge matching conditions [7]. When adjacentedges coincide, they are removed, displacing the tilingvertices between the edges. The kite and dart each haveone unconstrained tiling vertex. The others are all im-plied by the original matching conditions.

of freedom in the two fundamental edge shapes. Theseedge shapes are assembled into two tile shapes that arethen compared against two goal shapes as usual.

However, this interpretation of the possible shapes ofPenrose tiles is limited, as can be seen from Grunbaumand Shephard’s reproduction of Penrose’s aperiodicchickens [7]. They superimpose the chickens on top ofthe corresponding unmodified tiling. The registration ofthese two tilings reveals that the chickens have tiling ver-tices that are different from those of the original tiling!There are evidently additional degrees of freedom to thePenrose tilings that must be explored and exploited if weare to extend the reach of aperiodic Escherization.

By experimenting with the geometric matching condi-tions, we have discovered an extended set of points thatcan be parameterized like the tiling vertices of an iso-hedral tiling. We call these points the prototile’squa-sivertices. The quasivertices include all the points on aprototile’s boundary that act as tiling vertices somewherein a Penrose tiling, and some additional points that areforced into existence by them.2

Figure 5 shows how the ordinary geometric matchingconditions yield a new set of parameterizable points forthe kite and dart. The fundamental edge shapes are mod-ified so that they partially overlap. The overlapping re-gions can then be excised from the tiles, producing newtiles that no longer share all of the tiling vertices of theoriginal kite and dart. This process necessarily introducesother vertices into the shapes of the two tiles. Note thata tiling produced from these excised tiles will be visu-

2In an isohedral tiling, transitivity guarantees a unique configurationof tiling vertices around every copy of the prototile. No such guaranteeexists forP2 or P3, and we must consider points that are tiling verticeson some instances of a prototile but not others.

Figure 6: Extended matching rules for Penrose tiling setsP2 (above) and P3 (below). The edge labels are not re-lated between the two sets. Matching is enforced by iden-tifying pairs of interlocking edges. The interlocking pairsare (a, d), (b, h), (c, e), and (f, g) for P2, and (a, g),(b, e), (c, d), and (f, h) for P3.

ally indistinguishable from one produced using equiva-lent tiles with degeneracies. However, they behave dif-ferently from the point of view of the shape metric. Ournew parameterization of the excised tiles is compatiblewith the shape metric.

From Figure 5, we conclude that the quasivertices ofthe kite and dart can be parameterized using four real-valued parameters, determined by the positions of thetiling vertices created at the tips of the two excised re-gions. Similarly, four parameters suffice to parameterizethe Penrose rhombs. Once the free parameters are under-stood, we can derive explicit formulae for the positions ofthe quasivertices. See the thesis by Kaplan [8] for details.

Like tiling vertices, quasivertices partition tile bound-aries into edges that behave like tiling edges. We mustderive new matching rules so that congruent edges arecontrolled by a single set of parameters within the opti-mization. By analogy with the incidence symbols usedfor isohedral tilings, edge shapes can be specified by la-beling edges around each tile and indicating adjacencyrules for the labels. Figure 6 shows a set of labels andadjacency rules forP2 andP3.

The edge shapes, combined with the four parameterscontrolling the tiling vertices, yield a configuration spacesuitable for Penrose Escherization. Note that this param-eterization cannot in general represent both a particularpair of tile shapes and the reflections of those shapes, and

that in each of the Penrose setsP2 andP3 the two pro-totiles are fundamentally different shapes. For these rea-sons, given two goal shapesS1 andS2 and their reflec-tionsS′

1 andS′2, we must optimize for the eight combi-

nations(S1, S2), (S′1, S2), (S1, S

′2), (S′

1, S′2), (S2, S1),

(S′2, S1), (S2, S

′1), and(S′

2, S′1).

Rendered designs based on Penrose tilings are givenin Figure 8. In general, it is much more difficult to dis-cover satisfactory Escherizations using Penrose tilings.The range of possible tile shapes is limited and peculiar,always having many sharp angles. More obviously, thereare fewer “tiling types” than in the split isohedral case;we no longer have the luxury of hunting over many dif-ferent types for one that happens to be particularly wellsuited to a given pair of goal shapes. The results are there-fore less successful than in the isohedral and 2-isohedralcases, but interesting nevertheless for their connection tothe interaction (both mathematical and personal) betweenEscher and Penrose. On the other hand, an interactive ed-itor for Penrose tiles still allows profitable forward explo-ration of the space of tilings. Figure 4 shows an exampleof a tiling that was not Escherized but developed fromscratch in a few minutes and decorated in a cartoon style.

7 Discussion and future work

The first dihedral Escherization technique presented hereis a natural extension of the isohedral method of Kaplanand Salesin — we subdivide isohedral prototiles with asplitting path. By delving deeper into the tiling theory lit-erature, we can restrict our technique to Dress’s Heavenand Hell patterns, and then use those patterns to createSky and Water designs. We also take a fresh look at Es-cherization in the context of the aperiodic Penrose tilings.We close this paper by briefly discussing the issues thatarise with our technique, and suggesting ideas for futurework.

The split isohedral method can be used to constructthose 2-isohedral tilings for which the two prototiles oc-cur in equal amounts. All of Escher’s two-motif sys-tems have this property, and so we consider the methodsatisfactory for reproducing his work. However, thereare also many 2-isohedral tilings with different relativeamounts of the two prototiles. To understand these ad-ditional tiling types, we must take a closer look at themathematics behind 2-isohedral tilings.

Delgado-Friedrichset al.carry out a complete enumer-ation of the 2-isohedral tilings [3]. They prove a gen-eral result that every(k + 1)-isohedral tiling can be con-structed from ak-isohedral tiling through a combinationof two operations:SPLIT and GLUE. The SPLIT opera-tion is identical to our use of a splitting path. They showthat when the prototiles of a(k + 1)-isohedral tiling are

(a)Strange ’Tractors(IH28)(b) Godel, Bach (Braided):

an Eternal Escherization(IH2)

(c) Pen/Rose Tiling(IH1) (d) Rembrandt and Mrs. van Rijn(IH1)

Figure 7: Examples of dihedral Escherization using the split isohedral tile method. Manysource images appear courtesy of FreeFoto.com .

(a)Busby Berkeley Chickens(P2) (b)A Walk in the Park(P3)

Figure 8: Examples of dihedral Escherization based on parameterizations of Penrose tile sets P2 and P3.

asymmetric, they can be derived from the prototiles ofa k-isohedral tiling via the application ofSPLIT to oneprototile. TheGLUE operation erases the edge betweentwo adjacent tiles, producing tilings with symmetric pro-totiles. It is through use ofGLUE that we can construct2-isohedral tilings with different relative amounts of thetwo prototiles.

Because our technique is based only on theSPLIT

operation, it does not immediately generalize to all 2-isohedral tilings. A full generalization comes in theform of combinatorial tiling theory[4], in which a De-laney symbolsummarizes the combinatorial structure ofa tiling. Delaney symbols can be used to describek-isohedral tilings for anyk; they also generalize to dimen-sions greater than two and to non-Euclidean geometry.

In principle, we might use Delaney symbols as a basisfor parameterizing the shapes of all 2-isohedral tilings,yielding a new dihedral Escherization algorithm. Indeed,given anyk user-supplied goal shapes, such a systemcould discover ak-isohedral tiling that approximates allof them. Unfortunately, Delgadoet al.show that there areover a thousand 2-isohedral types. In general, the numberof k-isohedral types grows very quickly withk. Becausethe choice of tiling type is discrete, it quickly becomesinfeasible to search all possible types by launching inde-pendent continuous optimizations. It might be possible torun an optimization on a carefully chosen subset of De-laney symbols that are particularly well-suited to Escher-ization. As a benchmark fork-isohedral Escherization,consider Escher’s symmetry drawing 71, a remarkableperiodic tessellation featuring twelve distinct bird motifs.

Taken to the limit,k-hedral Escherization becomes akind of packing problem. Ask grows very large, we be-come more interested in simply fitting allk goal shapestogether plausibly in a kind of puzzle. Such a large blockof shapes no longer has as much aesthetic appeal whenrepeated across the plane. Escher’s printPlane Filling1 is an example where a number of one-of-a-kind figuresare assembled to tile a finite region [2]. As we move frommonohedral and dihedral tilings to cases such as this one,the importance of tiling theory is lessened. An approachlike that of Jigsaw Image Mosaics [10] might be moreappropriate.

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