Geometrical tile design for complex neighborhoods

Transcription

1 COMPUTATIONAL NEUROSCIENCE ORIGINAL RESEARCH ARTICLE pulished: 23 Novemer 2009 doi: /neuro Geometricl tile design for complex neighorhoods Eugen Czeizler* nd Lil Kri Deprtment of Computer Science, University of Western Ontrio, London, ON, Cnd Edited y: Hv T Siegelmnn, University of Msschusetts Amherst, USA Reviewed y: Enrico Formenti, Lortoire d Informtique de Mrseille, Frnce Yuriy Brun, University of Southern Cliforni, USA Mtthew J Ptitz, Low Stte University, USA *Correspondence: Eugen Czeizler, Deprtment of Informtion Technologies, Åo Acdemi University, Turku 20520, Finlnd e-mil: eczeizle@ofi Current ddress: Deprtment of Informtion Technologies, Åo Akdemi University, Turku 20520, Finlnd Recent reserch hs showed tht tile systems re one of the most suitle theoreticl frmeworks for the sptil study nd modeling of self-ssemly processes, such s the formtion of DNA nd protein oligomeric structures A Wng tile is unit squre, with glues on its edges, ttching to other tiles nd forming lrger nd lrger structures Although quite intuitive, the ide of glues plced on the edges of tile is not lwys nturl for simulting the interctions occurring in some rel systems For exmple, when considering protein self-ssemly, the shpe of protein is the min determinnt of its functions nd its interctions with other proteins Our gol is to use geometric tiles, ie, squre tiles with geometricl protrusions on their edges, for simulting tiled pths (zippers) with complex neighorhoods, y rions of geometric tiles with simple, locl neighorhoods This pper is step towrd solving the generl cse of n ritrry neighorhood, y proposing geometric tile designs tht solve the cse of tll von Neumnn neighorhood, the cse of the f-shped neighorhood, nd the cse of 3 5 filled rectngulr neighorhood The techniques cn e comined nd generlized to solve the prolem in the cse of ny neighorhood, centered t the tile of reference, nd included in 3 (2k + 1) rectngle Keywords: tile systems, tiled pths, geometric tiles, complex neighorhoods INTRODUCTION During the lst decde, rekthroughs in DNA mnipultion techniques hve generted wide rnge of dvnces in severl res of science: genetics, iology, medicine, ut lso nnoengineering nd computer science Regrding computer science in prticulr, severl new directions of reserch hve een estlished: DNA computing, DNA code-design, ioinformtics, computtionl modeling, nd others However, s usul in mthemtics nd computer science, the theoreticl foundtions of these new reserch topics lie deep in well estlished theoreticl ckgrounds The principle of self-ssemly is one of the key concepts of nnosciences It is the process y which ojects ggregte independently, without externl force or guidnce, to form complex structures This process cn e found in nture t ll levels: toms interct with one nother nd crete molecules, molecules my ggregte to form mcromolecules, proteins self-ssemle into protein complexes, etc This nturl principle hs een mimicked successfully y rtificil nd semi-rtificil self-ssemly systems Exmples re the mcroscopic plstic tiles of Rothemund (2000) which ssemle on n oil/wter surfce, simulting in this wy one-dimensionl cellulr utomton, or the self-ssemly of some led structures on copper surfce from Plss et l (2001) On the other hnd, scientists hve lso used the intrinsic properties of iomolecules, such s the Wtson Crick complementrity of DNA molecules, in order to construct vrious self-ssemly systems which re cple of wide rnge of computtions or tsks Exmples re the DNA nnostructures performing it-wise cumultive XOR (Mo et l, 2000), inry counters (Brish et l, 2005), moleculr switches etween two conformtions (Liedl et l, 2006), DNA wlkers moving long trck (Shermn nd Seemn, 2004), utonomous moleculr motors (Bth et l, 2005), nnoscle DNA shpes nd ptterns (Rothemund, 2006), fixed-width cellulr utomt (Fujiyshi et l, 2008), nd mny others Tile systems re one of the most importnt mthemticl models of self-ssemly systems (see eg, Adlemn, 2000; Rothemund et l, 2004) A Wng tile is n oriented unit squre with ech edge covered y specific glue Tiles re plced on the two-dimensionl plne, nd two djcent tiles stick together if nd only if they hve the sme glue on their utting edges Wng tiles, first proposed y Wng (1961), hve een extensively studied efore finding their nturl pplictions to self-ssemly More recently, Wng tiles nd some of their vrints, eg, the Tile Assemly Model (Winfree, 1998; Rothemund, 2001), hve een widely used s models for self- ssemly nd used to design nd nlyze successful prcticl experiments (see eg, Rothemund et l, 2004; Brish et l, 2005) Although quite intuitive, the ide of glues plced on the edges of tile is not lwys nturl for simulting the interctions occurring in some rel systems For exmple, when considering protein self-ssemly, the shpe of protein is essentil in determining its function nd its interctions with other proteins Tiles of vrious shpes hd een previously designed nd studied in the literture for specific purposes For exmple, Roinson (1971) designed six polygonl tiles (squres with notched edges nd corners) in the context of solving the prolem of minimiztion of the numer of tiles tht llowed only periodic tilings of the plne In this pper, we design geometricl tiles, ie, squre tiles with protrusions on their edges, for the purpose of simplifying neighorhood reltionships in tiling systems Our min gol is to simulte ny tiled pth ( zipper ) tht uses tiles with n ritrrily complex neighorhood, y rion of new geometricl Frontiers in Computtionl Neuroscience wwwfrontiersinorg Novemer 2009 Volume 3 Article 20 1

2 Czeizler nd Kri Geometricl tile design for complex neighorhoods tiles with much simpler neighorhood reltionship Nmely, the only requirement of tiling with the new tiles is tht no overlp occurs The reson we focus on tiled pths, either zippers or rions, insted of simple totl or prtil tilings, is threefold First, from theoreticl point of view, due to their underlying pth, the trnsition from zippers with complex neighorhoods to rions of geometricl tiles is much more complex thn simple trnsition etween two, possily prtil, tilings (the first using complex neighorhood nd the second using geometric tiles) Indeed, preserving the underlying pth of the zippers when trnsforming them into rions cn e non-trivil The second nd third resons re more ppliction oriented Note tht the zipper nd the rion structures re closely relted to protein structures Although we cn see protein s vlid tiled three-dimensionl structure, these orgnic compounds re mde of liner chins of mino cids, which re folded into their ctul three-dimensionl shpe The finl reson for considering pths ssocited with tilings comes from the prcticl chllenge of creting self-ssemling nno-wires nd nno-circuits This spect hs een prticulrly studied in reference with the Tile Assemly Model (see eg, delorimier et l, 2002; Cook et l, 2004; Brun nd Reishus, 2009) The first step towrds solving the proposed prolem ws done in Czeizler nd Kri (Sumitted) where we introduced motif construction, sed on geometricl tile design, tht solved the prolem in the cse of Moore neighorhood This pper is the next step towrds the solution to the generl cse We nmely consider other nturl extensions of the von Neumnn nd Moore neighorhoods, consisting of elements plced further wy from the reference tile In prticulr, we consider the cse of the neighorhood tht is tll version of the von Neumnn neighorhood (the neighorhood of tile consists of the tile t the Est, the tile t the West, two tiles to the North nd two tiles to the South), the f-shped neighorhood, nd the 3 5 filled rectngulr neighorhood The techniques cn e comined nd generlized to solve the prolem in the cse of ny neighorhood included in Moore-type rectngulr neighorhood, centered t the tile of reference, nd of width 3 nd heighk + 1 For ll three cses, ny zipper cn e simulted y rion mde of new geometricl tiles in which the tiles shpes, nd not their glues, determine the self-ssemly process, nd the neighorhood dependency is gretly simplified The geometricl tile design ensures the trnsmission of informtion t distnce (cross severl tiles), s well s cross severl informtion chnnels At the sme time, the design stisfies the requirement of no overlpping etween ny two geometric tiles The pper is orgnized s follows The next section presents sic definitions nd nottions In Section From Complex Neighorhoods to Geometric Tiles we mke the trnsition from tiles with complex neighorhoods to geometric tiles with simple, ie, locl, neighorhoods The three following susections ddress the cse of the tll von Neumnn neighorhood, the f-shped neighorhood, nd the 3 5 filled rectngulr neighorhood In the lst section we show how to comine nd generlize these techniques to solve the prolem of neighorhood simplifiction y geometric tile design in the cse of ny neighorhood, centered t the tile of reference, nd included in 3 (2k + 1) rectngle PRELIMINARIES A Wng tile is n oriented unit squre, ie, squre which cnnot e rotted or reflected, whose edges re leled y symols from finite lphet X, clled glues Thus, ech tile t is uniquely determined y the four glues of its North, Est, South nd West edges s: t = (t N, t E, t S, t W ) X 4 The positions of the tiles on the plne re indexed y pirs of integers, (i, j) Z 2 A tile system T is finite collection of tiles We sy tht two tiles t nd t stick on the North-South direction if nd only if t N = t S, tht is, if the North end of t nd the South edge of t hve the sme glue Similrly, we cn define the sticking property on the Est-West, South-North nd West-Est directions A totl tiling of the plne is mpping T : Z 2 T which ssigns to every position from Z 2 tile from T We sy tht tiling T is vlid on position (i, j) Z 2 if the tile on position (i, j), denoted s t(i, j), sticks on the North-South, Est-West, South-North nd West-Est directions to the tiles t(i, j + 1), t(i + 1, j), t(i, j 1), nd t(i 1, j) respectively; here, we ssume tht there exists tile on ll positions of the plne A prtil tiling of the plne is mpping T D from domin D Z 2 to T We sy tht T D is vlid if for ny tile within the domin there re no mismtches etween the glues of tile nd the glues of its existing neighors More formlly, for ll (i, j) D, if (i, j + 1) D then t(i, j) N = t(i, j + 1) S ; if (i + 1, j) D then t(i, j) E = t(i + 1, j) W ; if (i, j 1) D then t(i, j) S = t(i, j 1) N ; if (i 1, j) D then t(i, j) W = t(i 1, j) E If no confusion cn rise, we refer to vlid tilings (either totl or prtil ones) simply s tilings A pth P is succession of djcent positions in the plne Formlly, finite (resp infinite) pth is mpping P : I Z 2 where I = {1, 2, 3,,n}, n 2, (resp I = N) such tht for ll 1 i n 1 (resp for ll i 1), if P(i) = (x, y) for some x, y Z, then P(i + 1) {(x, y + 1), (x + 1, y), (x, y 1), (x 1, y)} A T- tiled pth is contiguous succession of tiles Formlly, it is pir (P, r) where P : I Z 2 is pth nd r : rnge(p) T is mpping ssigning tiles to ll positions of the pth A T-tiled pth is clled T-rion if P is injective, ie, the pth is not self-crossing, nd for ny position in the pth (except for the lst one, if ny), the tile nd its successor must gree on their glues on the corresponding utting edges A T-zipper is specil type of T-rion where if two tiles re djcent, even if they re not on consecutive positions on the pth, they still must gree on their glues on their utting edges Given rion (resp zipper) (P, r), we refer to P s the underlying pth of the rion (resp of the zipper) A neighorhood vector (or simply neighorhood) is finite collection of pirs of integers, descriing the reltive position in spce of those tiles which interct with given tile Formlly (Kri, 2005; Adlemn et l, 2009), neighorhood is n n-tuple N = (ν 1, ν 2,,ν n ), ν i Z 2 \{(0, 0)}, 1 i n, where ν i ν j for ll 1 i < j n, nd for ll 1 i n there exists unique j such tht ν i = (x, y) nd ν j = ( x, y), where x, y Z This lst restriction on the content of the neighorhood vector is referred to s the symmetry condition nd it is due to the fct tht whenever tile Frontiers in Computtionl Neuroscience wwwfrontiersinorg Novemer 2009 Volume 3 Article 20 2

3 Czeizler nd Kri Geometricl tile design for complex neighorhoods t intercts with nother tile t, then lso t intercts with t The neighors of position (x, y) Z 2 re the positions (x + x i, y + y i ) where ν i = (x i, y i ) for 1 i n In this pper, we consider the wy in which tile intercts with its neighors to e similr to the cse of clssicl Wng tiles, tht is using glues Although the tiles re represented s unit squres nd plced on the nodes of squre lttice (exctly s in the cse of clssicl tiles), we cn think of tile t s hving n virtul edges, ech leled y glue nd ech ssocited to exctly one neighoring tile Given neighorhood vector N = (ν 1, ν 2,,ν n ) nd tile t plced on position (x, y) Z 2, the neighoring tiles of t re ll the tiles plced on the neighoring positions of (x, y), tht is, the tiles on positions (x + x i, y + y i ), where ν i = (x i, y i ) for ll 1 i n Then, ech tile t is uniquely determined y n n-tuple of glues, t = (t 1,,,t n ), where ech glue t i X, 1 i n, corresponds to the virtul edge ssocited with the neighoring tile plced on the reltive position ν i In order to simplify future considertions, for tile t, we use t ( x i, y i) to refer to the glue t i, where ν i = (x i, y i ) Then, for two neighoring tiles t nd t such tht t is plced on the position ν i reltive to the position of t, the glues t nd ( x i, y i) t ( xi, yi) correspond to the common virtul edge etween the two tiles Note tht the existence of the element ( x i, y i ) in the neighorhood N is gurnteed y the symmetry condition Given neighorhood vector N with n elements, tile system using this neighorhood is finite set T N X n of tiles If no confusion cn rise regrding the neighorhood, we cn omit N from the nottion Then, (prtil) tiling is (prtil) function T : Z 2 T If the function is defined for (x, y) Z 2, we denote y t(x, y) the tile on this position Given neighorhood vector N = (ν 1, ν 2,,ν n ) nd some D Z 2, we sy tht (totl or prtil) tiling T : D T is vlid on position (x, y) D if for ll ν i = (x i, y i ) such tht (x + x i, y + y i ) D, we hve t( xy, )( x y tx xiy y i, i) = ( +, + i) ( xi, yi), ie, the two neighoring tiles hve the sme glue on their common virtul edge We sy tht T is vlid if it is vlid on ll positions (x, y) D Next, we give couple of exmples illustrting the previous notions Exmple 1: let N e the well known von Neumnn neighorhood vector, N = ((0, 1), (1, 0), (0, 1), ( 1, 0)), pointing to the positions to the North, Est, South nd West of given tile Tile systems using this neighorhood re exctly the clssicl Wng tile systems defined in Section Preliminries The neighors of given tile t(x, y) re those tiles plced to the North, Est, South nd West directions nd the virtul edges of t(x, y) re simply its edges Exmple 2: let N e the tll von Neumnn neighorhood vector, ie, N = ((0, 1), (0, 2), (1, 0), (0, 1), (0, 2), ( 1, 0)) In this cse, the neighors of tile re the tiles plced one nd two steps to the North nd to the South, nd the tiles plced immeditely to the Est nd to the West Here, esides the four norml edges of the tile, ie, the four geometricl edges of the unit squre, we hve two dditionl virtul edges, one corresponding to the tile two steps to the North, nd the other corresponding to the tile two steps to the South In Figure 1 we present vlid prtil tiling using this neighorhood Note tht djcent tiles do not necessry hve the sme glues on their virtul edges; for instnce, in Figure 1, the North virtul edge of the tile t 1 hs to gree only with the South virtul edge of Exmple 3: nother well estlished instnce is the Moore neighorhood vector, N = ((0, 1), (1, 1), (1, 0), (1, 1)(0, 1), ( 1, 1), ( 1, 0), ( 1, 1)) In this cse, the neighors of given tile re ll those tiles on the eight surrounding positions Here, it is more intuitive to consider the corners of the tiles s the virtul edges, see Figure 2 for vlid prtil tiling using this neighorhood The notions of pth, tiled pth, nd zipper re generlized in the nturl wy for the cse of complex neighorhood vectors The notion of pth (resp tiled pth) remins unchnged independent of the contents of the neighorhood vector, ie, mpping P : I Z 2 (resp pir (P, r)) where the element P(i + 1) cn e plced only to the North, Est, South or West of P(i), exctly s in the cse of clssicl Wng tile systems In the cse of zippers we require tht t 1 FIGURE 1 A prtil tiling using the tll von Neumnn neighorhood The gry sections on the orders of the tiles represent the virtul edges FIGURE 2 A prtil tiling using the Moore neighorhood; oth the edges nd the corners of ech tile re leled y glues Frontiers in Computtionl Neuroscience wwwfrontiersinorg Novemer 2009 Volume 3 Article 20 3

4 Czeizler nd Kri Geometricl tile design for complex neighorhoods the underlying pth is not self-crossing nd, moreover, ny two tiles plced one on the neighorhood of the other (not necessrily succeeding or even djcent to ech other) must hve the sme glues of their common (mye virtul) edges FROM COMPLEX NEIGHBORHOODS TO GEOMETRIC TILES The principle of mtching glues, lthough quite intuitive nd very esy to e modeled mthemticlly, is not lwys nturl for simulting the interctions within some self-ssemly systems Sometimes, the geometry of the ojects involved in these systems cn ply the centrl role in their self-ssemly Thus, mjor requirement for two or severl ojects to successfully ggregte is tht when doing so, the ojects will simply not overlp For instnce, in the cse of protein self-ssemly, the shpe of the protein, ie, its folding, determines oth its function nd the wy it intercts with other proteins The most nturl question rising t this moment is whether one cn modify the structure of the tiles from the previous section such tht insted of using glues s wy to control self-ssemly, the shpe of the tiles would govern the self-ggregtion process The cse of finite zipper structures is prticulrly interesting, s this would correspond to sequence of mino cids forming protein, or sequence of proteins forming complex The cse of the clssicl von Neumnn neighorhood hs lredy een ddressed y, eg, Roinson (1971), Adlemn et l (2002, 2009), Kri (2003), where glues were replced with pirs of mtching umps nd dents However, in this cse, ech tile hs utting edges with ll its neighors, thus mking the construction very intuitive In the cse of more complex neighorhood vectors, tiles my hve virtul edges with some non-utting neighors Thus, we must find wy to trnsmit informtion from one tile to its distnt neighors In Czeizler nd Kri (Sumitted) we present wy of doing so for the cse of tile systems using the Moore neighorhood vector from Exmple 3 However, lso in this cse, the neighors of tile re still surrounding it in some sense, mking the construction esier When simulting tiling using glues nd complex neighorhood dependency y tiling sed only on the non- overlpping principle, one of the first methods tht one could try is the following First, one simultes the first tiling y nother one where tiles use the von Neumnn neighorhood, ie, using Wng tiles Then, sed on the lredy known constructions, one simultes the Wng tiling y nother one where tiles use only their shpes in order to self-ssemle One of the methods used in the literture (see eg, Kri, 1989; Adlemn et l, 2002, 2009), for simulting lrger neighorhood y smller one, is to scle up the construction Tht is, one cretes some mcro-tiles, which cpture the informtion of oth the initil tile nd its neighors For instnce, in Adlemn et l (2002, 2009), for the cse of totl tilings of the plne, the uthors simulte the Moore neighorhood dependency y the von Neumnn neighorhood dependency The uthors replce the initil tiles (let us cll them mini-tiles) with set of mcro-tiles, consisting of locks of 3 3 mini-tiles with no mismtching inside the locks Then, two such mcro-tiles cn e plced one North of the other if nd only if the ottom 2 3 mini-tiles from the first mcro-tile re exctly the sme s the top 2 3 mini-tiles from the second mcro-tile Similrly, one defines the South-North, Est-West nd West-Est sticking property Then, y performing the following trnsformtion, there exists one-to-one correspondence etween totl tilings of the mini-tile system using the Moore neighorhood, nd totl tilings of the mcro-tile system using the von Neumnn neighorhood (Adlemn et l, 2002, 2009) For given totl tiling using mini-tiles we replce ech tile t(i, j) on position (i, j) with the mcro-tile corresponding to the 3 3 lock contining t(i, j) in the middle nd ll eight surrounding tiles For the converse, the correspondence is otined y ssociting to ech mcro-tile the mini-tile plced in the middle of tht prticulr 3 3 lock This technique cn e generlized successfully for ny neighorhood vector, y tking k k locks s mcro-tiles, for sufficiently lrge k However, it is essentil to notice tht this scling up technique works only if we consider totl tilings of the plne If, on the other hnd, we consider prtil tilings, then this method genertes errors, s we cn design some vlid prtil tilings using mcro-tiles which do not correspond to ny vlid tilings using mini-tiles, see, eg, Figure 3 Another method which one could consider for simulting tiling using some complex neighorhood y tiling using Wng tiles, A B C t 2 t 1 t 3 t 1 t 3 t 1 t 1 t 1 FIGURE 3 (A) A set of mini-tiles, (B) vlid tiling using mcro-tiles, emphsizing the center mini-tiles, (C) the ssocited non-vlid tiling using mini-tiles Frontiers in Computtionl Neuroscience wwwfrontiersinorg Novemer 2009 Volume 3 Article 20 4

5 Czeizler nd Kri Geometricl tile design for complex neighorhoods could e the following We modify the wy we ssocite glues to the sides of the squre unit tiles, such tht insted of hving only one glue on ech edge, we could hve severl Then, we use some of these extr glues in order to trnsfer informtion from one tile to its distnt neighors y using the tiles plced in etween Similrly to the previous cse, this method cn e used successfully for the cse of totl tilings However, lso here, in the cse of prtil tilings, this method cn generte errors, since we do not hve tile in ech position of the plne In this pper we propose more direct pproch for this prolem, tht is, we replce ech tile from the initil tile system with severl tiles with complex structures In the following sections we present severl tile designs constructed for some prticulr complex neighorhood vectors, ccomplishing the desired requirement: the shpe of the tiles is the sole fctor determining the self-ssemly process We cll these tiles geometric tiles (or simply tiles if no confusion cn rise) to underline the importnce of their shpes Since we wnt to eliminte glues, the notion of vlid tiling (totl or prtil) must e updted Although t first look the geometric tiles tht we propose here do not hve regulr contour ny more, y mking n strction of the umps nd dents plced on their edges, ech tile hs ctully squre shpe Thus, similr to the previous cses, we require tht ll tiles re plced on the nodes of squre lttice Note tht this requirement cn e esily imposed geometriclly (see eg, Roinson, 1971; Kri, 2008) y dding pirs of mtching umps nd dents on the ctul edges of the tiles, such tht the only wy tht these tiles cn e plced ner ech other without overlpping is y plcing them on the nodes of squre lttice, see Figure 4 Since the geometricl tiles re plced on the nodes of squre lttice, the notions of geometricl tiling nd geometricl tiled pth hve the sme mening s in the cse of Wng tiles We sy tht geometricl tiling (resp geometricl tiled pth) is non-overlpping if no two tiles re overlpping Note tht we my refer to non-overlpping tiled pths lso s rions This is ecuse, similrly to the cse of Wng tiles, the underlying pth FIGURE 4 Tiles with mtching umps nd dents, forcing their plcement on the nodes of squre lttice of this construction imposes qusi-liner neighorhood to ll tiles; nmely, ech tile (except the first nd the lst, if ny) hs predecessor nd successor Even more, y using stndrd methods employed in previous ppers (see Adlemn et l, 2002, 2009; Czeizler nd Kri, Sumitted), one cn esily trnsform such non-overlpping tiled pth into rion of Wng tiles In the following sections we show tht in the frmework of geometric tiles, the notions of non-overlpping tiling nd nonoverlpping tiled pth re equivlent to tht of vlid tiling nd vlid zipper structure, respectively THE TALL VON NEUMANN NEIGHBORHOOD VECTOR In this section we consider the tll von Neumnn neighorhood vector from Exmple 2, N = ((0, 1), (0, 2), (1, 0), (0, 1), (0, 2), ( 1, 0)), which contins six elements, s descried more suggestively y the pttern: (,) 02 (,) 01 ( 10, ) ( 10, ) (, 0 1) (, 0 2) Let T X 6 e tile system using this neighorhood vector, where X is the set of possile glues of the edges We design geometricl tile system G inducing direct correspondence etween vlid (prtil) T-tilings nd T-zippers nd non-overlpping G-tilings nd G-tiled pths, respectively Given tile t T, we denote y (t N, t NN, t E, t S, t SS, t W ) the ordered set of its glues, insted of (t (0, 1), t (0, 2), t (1, 0), t (0, 1), t (0, 2), t ( 1, 0) ) Let n e the totl numer of glues, ie, X = n For ech tile t = (t N, t NN, 1 2 n t E, t S, t SS, t W ) T, we construct n geometric tiles gt, gt,, gt G, clled the geometricl vrints of t The reson for which we need to introduce multiple geometricl vrints for ech tile t will e explined in detil lter, when we introduce some specific constructions, clled the spike, the ig dent, nd the sheth The se shpe of ll geometric tiles is squre However, on the edges of these squres we plce specific umps nd dents in order to simulte the glues of the originl tile We use the Est nd West edges of the geometric tiles to simulte the glues corresponding to the Est nd West neighors At the sme time, the North nd South edges re used to simulte the glues corresponding to the four neighors plced one nd two steps to the North nd to the South We discuss first the Est nd West edges of the geometricl vrints of given tile t First, to ech glue from X we ssocite unique position long the Est nd the West edges Then, in ll geometricl vrints of t we plce ump on the Est edges on the unique position ssocited to the glue t E nd dent on the West edges on the unique position ssocited to the glue t W, see eg, Figure 5 Thus, if the glues of two horizontlly djcent tiles re mtching, then the ump nd the dent of the corresponding edges of the two ssocited geometric tiles re plced exctly on the sme position, ie, they fit ech other For exmple, if the glue on the Est edge of tile t is the sme s the glue on the West edge Frontiers in Computtionl Neuroscience wwwfrontiersinorg Novemer 2009 Volume 3 Article 20 5

6 Czeizler nd Kri Geometricl tile design for complex neighorhoods (norml) ump nd dent the spike the sheth the ig dent FIGURE 5 A geometric tile for the cse of the tll von Neumnn neighorhood of tile t, ie, t E = t W, then the ump on the Est side of ny of the t geometricl vrints fits exctly with the dent on the West side of ny of the t geometricl vrints However, if for two tiles t nd t we hve t E t W, then the ump from the Est edge of ny of the vrints of t overlps with the West edge of ny of the vrints of t Let us consider now the North nd South edges of the geometricl vrints of t The structure of these edges is more complex, s here we use umps nd dents oth to communicte with neighoring tiles nd to llow the propgtion of informtion Thus, oth edges re split in three regions, nd to every glue we ssocite unique position in ech of these regions The first regions of the North nd the South edges of the geometricl vrints of t re used in order to communicte with the tiles plced immeditely to the North nd to the South, respectively Similr to the cse of Est nd West edges, we plce ump on the North edge nd dent on the South edge of ech of the vrints The ump on the North edge is plced on the unique position from the first region ssocited to the glue t N, while the dent on the South edge is plced on the unique position from the first region ssocited to the glue t S The second nd the third regions of the North nd respectively the South edges of the geometricl vrint of t re used to simulte the glues on the virtul edges of t Thus, on the second region of the North edge of ech of the geometricl vrints of t we plce n elongted ump, clled spike This spike is used to simulte the glue t NN, nd thus it is plced (on ll geometricl vrints) on the unique position from the second region of the North edge ssocited to it The spike is long enough to cross the first geometric tile to the North nd to rech the South edge of the geometric tile plced two steps to the North Moreover, the spike shifts its position, see Figure 5, such tht when it reches the second tile to the North, the spike is plced on the unique position from the third region ssocited to the glue t NN If this second geometric tile is vrint of tile t such tht t SS = t NN, then there exists mtching dent in the third region; otherwise n overlp will occur As previously suggested, on the third region of the South edge of ech of the geometricl vrints of t we plce wider dent, clled the ig dent This ig dent is used to simulte the glue plced on the South virtul edge of t, tht is t SS, nd it is plced (on ll geometricl vrints of t) on the unique position ssocited to the glue t SS from the third region of the South edge, see Figure n Until now, ll the geometricl vrints of t, tht is gt, gt,, gt, hve identicl shpes However, the difference etween them is given y the sheth which is plced on the second region of the South edge nd the third region of the North edge of the geometricl vrints The sheth is pir of mtching ump nd dent, see Figure 5, which perfectly surrounds the spike of the tile elow Due to this sheth, geometric tiles cn propgte informtion from the tile elow, to the tile ove Since there re n different possile positions for the spike of the geometric tile plced elow, ech of the vrints of t will cover for exctly one of these possiilities, see Figure 6 (thus, the required numer of different geometricl vrints for t) Note tht the ig dent is constructed wide enough such tht it ctully mtches oth the spike (of the tile plced two steps elow) nd the possile sheth (of the tile plced one steps elow) surrounding the spike In Figure 7 we disply non-overlpping tiling using geometric tiles which is ssocited to the vlid tiling from Figure 1 Note tht the tiling from Figure 7 is only one of the possile non-overlpping tilings which cn e ssocited to the tiling from Figure 1 For instnce, the top right geometric tile cn e replced y ny other geometricl vrint of the sme tile, without cusing ny overlpping The reson for which we hve to use the spike, the ig dent, nd the sheth insted of using norml (smll) umps nd dents is the following We wnt ech geometric tile to ct s n intermedite etween the tile elow nd the tile ove However, we cnnot e sure whether given prtil tiling T D (or similrly zipper construction) etween ny two tiles which re plced two steps North of ech other, there is lso third tile plced in etween Tht is, even if (x, y), (x, y + 2) D, it is not necessry tht (x, y + 1) D Thus, we must mke sure tht in oth cses, ie, whether there exists or not middle tile t(x, y + 1), the informtion regrding the glue t(x, y) NN is trnsferred from t(x, y) to t(x, y + 2) In our cse, this is ccomplished y the mtching spikes, ig dents, nd sheths; see for exmple the tiles g 1 nd g 2 from Figure 7 for the cse when there is tile in etween, nd the tiles g' 1 nd g' 2 for the other cse Oservtion 1: oth the spike nd the sheth hve step-like form due to the following reson We design the geometric tiles such tht ll of them hve the sme squre se shpe, up to the positions of their umps nd dents long their edges (including here lso the spike, the ig dent, nd the sheth) Thus, if we would hve chosen the spike (nd hence lso the sheth) to hve stright form then, the spike, the ig dent, nd the sheth, would ll e plced in the sme region of the South nd the North edges of geometric tile Thus, for instnce, in the cse when for some tile t(x, y) we hve t(x, y) SS = t(x, y) NN then, in ll the geometricl vrints of t(x, y), the spike nd the ig dent will ctully overlp with ech other Even more, in tht prticulr geometric vrint of this tile, t(x, y), in which lso the sheth corresponds to the glue t(x, y) SS, the spike nd the ig dent re going to overlp lso with the sheth, mking the construction much more complicted Also, Frontiers in Computtionl Neuroscience wwwfrontiersinorg Novemer 2009 Volume 3 Article 20 6

7 Czeizler nd Kri Geometricl tile design for complex neighorhoods A B C the sheth corresponding to the glue the sheth corresponding to the glue FIGURE 6 The geometricl vrints of tile, for the cse when X = {, }: (A) the tile, (B) the first geometricl vrint, (C) the second geometricl vrint g 1 g 1 g 2 g 2 FIGURE 7 A non-overlpping tiling of geometric tiles, for the cse of the tll von Neumnn neighorhood in mny other cses, the spike nd the sheth of geometricl tile plced on position (x, y) will overlp with the spike, the ig dent nd the sheth of the geometricl tile plced on position (x, y + 1) Although such design is not impossile, y llowing the spike nd the sheth to hve step-like shpe we overcome ll of these overlpping prolems Theorem 1: let N = ((0, 1), (0, 2), (1, 0), (0, 1), (0, 2), ( 1, 0)) e neighorhood vector, nd let T X 6 e tile system using this neighorhood Then, we cn construct geometricl tile system G such tht for ny surfce D Z 2, there exists one-to-one correspondence etween vlid T D -tilings nd non- overlpping G D -tilings, up to replcing some of the tiles with ny of its geometricl vrints Frontiers in Computtionl Neuroscience wwwfrontiersinorg Novemer 2009 Volume 3 Article 20 7

8 Czeizler nd Kri Geometricl tile design for complex neighorhoods Proof: Let T D : D T e vlid tiling, nd let G e geometricl tile system constructed s descried ove Then, for ech (x, y) D we replce the tile t(x, y) with one of its geometricl vrints s follows: If (x, y 1), (x, y + 1) D, then we replce t(x, y) with ny of its geometricl vrints If either (x, y 1) D or (x, y + 1) D, then we replce t(x, y) with the geometricl vrint whose sheth mtches either the spike of the geometric tile elow or the ig dent of the geometric tile ove, respectively Note tht independently of which geometricl vrints of the tile elow nd the tile ove we use, the spike nd the ig dent re the sme If oth (x, y 1), (x, y + 1) D, then we replce t(x, y) with the geometricl vrint whose sheth mtches the spike of the geometric tile elow Since the tiling T D is vlid, we must hve t(x, y 1) NN = t(x, y + 1) SS Thus, the sheth of the chosen geometricl vrint lso mtches the ig dent of the geometric tile ove In ll of the ove cses we do not otin ny overlpping etween the spikes, ig dents, nd sheths of the geometric tiles Moreover, since we ssumed the tiling T D to e vlid, we lso conclude tht there is no overlpping etween the norml umps nd dents of ll djcent geometric tiles Thus, the tiling G D : D G otined y replcing the tiles from T with the corresponding geometric tiles from G s descried ove, is non-overlpping Moreover, except for the first cse mentioned ove, ie, for some (x, y) D, oth (x, y 1), (x, y + 1) D, where we cn replce the tile t(x, y) with ny of its geometricl vrints, on ll the other cses ech tile cn e replced only y one of its geometricl vrints For the converse, let us ssume tht G D : D G is nonoverlpping tiling From the definition of the geometric tile system G, ech geometric tile from G D is vrint of (unique) tile in T We construct the tiling T D : D T y replcing ech tile from G D with the corresponding tile from T Since there re no overlps in G D etween umps nd dents nd lso etween the spikes nd the ig dents, we conclude tht ny tile from T D must gree on its glues with ll of its neighors from the domin D Thus, the tiling T D is vlid, nd it is uniquely otined strting from the tiling G D The conceptul difference etween zipper nd prtil tiling is given only y the presence of pth Moreover, since in oth generlized Wng tile systems nd geometricl tile systems the tiles re plced on the nodes of squre lttice (ctully the sme squre lttice), the notion of pth remins unchnged Thus, the following result is direct consequence of the previous theorem Corollry 1: let N = ((0, 1), (0, 2), (1, 0), (0, 1), (0, 2), ( 1, 0)) e neighorhood vector, nd let T X 6 e tile system using this neighorhood Then, we cn construct geometricl tile system G such tht for ny non self-crossing pth P there exists one-to-one correspondence etween T-zippers (P, r) nd non-overlpping G- tiled pths (P, s), up to replcing some of the tiles with ny of its geometricl vrints THE f -NEIGHBORHOOD VECTOR In this section we consider the neighorhood vector N = ((0, 1), (0, 2), (1, 2), (1, 0), (0, 1), (0, 2), ( 1, 2), ( 1, 0)), which contins eight elements nd is descried y the following pttern: (,) 02 (,) 12 (,) 01 ( 10, ) ( 10, ) (, 0 1) ( 1, 2) ( 0, 2) This neighorhood descries the shpe of the letter f nd it is otined y dding to the tll von Neumnn neighorhood vector from the previous section the two symmetric elements (1, 2) nd ( 1, 2) The geometric tiles which we use in this cse re otined y dding to the previous geometric tiles new pir of mtching spikes, ig dents nd sheths, which re used to send, receive nd respectively trnsfer the informtion from given tile to the one corresponding to the reltive position (1, 2) The new spike, clled spike II, is plced long the Est edge of the geometric tiles, nd it points upwrds, see Figure 8 This structure is used to simulte the glue corresponding to the virtul edge which links tile to its neighor plced on the reltive position (1, 2) Contrry to the cse of the first spike, the strting point of the spike II is the sme, independently of which glue it stnds for However, the length of its first horizontl segment uniquely determines which glue it simultes Before spike II reches the South edge of the tile plced on the reltive position (1, 2) it crosses oth the tile plced on the reltive position (1, 0) nd the tiled plced on the reltive position (1, 1) Thus, we construct two new sheths, clled sheth II nd sheth III, which enle the crossing of spike II through these two tiles, without overlpping The new ig dent, clled ig dent II, is plced on the South edge of the geometric tiles, nd it corresponds to the glue of the virtul edge which connects tile to its neighor plced on the reltive position ( 1, 2) Thus, it hs to mtch the spike II of the geometric tile plced on this position In fct, s we explin next, it hs to mtch oth the spike II nd the corresponding sheths surrounding it: the sheth II of the geometric tile plced on the reltive position (0, 2), nd the sheth III of the geometric tile plced on the reltive position (0, 1) The sheth II strts from the West edge of the geometric tiles nd it surrounds the spike II of the geometric tile to the left, ie, the tile plced on the reltive position ( 1, 0) Thus, the sheth II crosses the geometric tile plced ove it [ie, the tile plced on the reltive position (1, 0)], resemling in some sense spike The sheth III strts from the South edge of the geometric tiles nd it surrounds oth the spike II of the geometric tile plced on the reltive position ( 1, 1) nd the sheth II of the geometric tile plced on the reltive position (0, 1) (which itself lso surrounds the spike II) For etter understnding of these constructions nd how they emed in ech other, in Figure 9 we present prtil nonoverlpping tiling using these geometric tiles As explined in Oservtion 1, ll these constructions, ie, the spike II, the sheth II, nd the sheth III, must hve step-like form in order to void overlpping with similr structures from the surrounding geometric tiles Thus, some new splitting of the edges re necessry s follows The Est nd West edges re split into two regions, while the South Frontiers in Computtionl Neuroscience wwwfrontiersinorg Novemer 2009 Volume 3 Article 20 8

9 Czeizler nd Kri Geometricl tile design for complex neighorhoods the spike the spike II the sheth II (norml) ump nd dent the first horizontl segment the sheth the sheth III the ig dent II the ig dent FIGURE 8 A geometric tile for the cse of the f-neighorhood FIGURE 9 A non-overlpping tiling of geometric tiles, for the cse of the f-neighorhood Frontiers in Computtionl Neuroscience wwwfrontiersinorg Novemer 2009 Volume 3 Article 20 9

10 Czeizler nd Kri Geometricl tile design for complex neighorhoods nd North edges re split into five (two new regions re creted on the left side of the edges) The spike II strts from the upper region of the Est edge As previously mentioned, the strting position of this construction (long the edge) is independent of the glue it stnds for However, the length of the first horizontl segment of spike II is unique for ech of the n different glues Thus, the first verticl segment of the spike II reches the second region of the South edge of the tile plced on the reltive position (1, 1) on the unique position ssocited to the glue simulted y spike II Before reching the South edge of the tile plced on the reltive position (1, 2), the spike II shifts gin, such tht now it intersects the edge on the third region (gin on the unique position corresponding to the sme specific glue) The sheth II must surround the spike II of the tile on the left Thus, it strts from the upper region of the West edge, it shifts horizontlly, it crosses the second region of the South edge of the tile ove, nd, fter shifting gin, it reches the third region of the South edge of the tile plced on the reltive position (0, 2) Note tht the positions from oth the second region of the South edge of the tile ove nd the third region of the South edge of the tile two steps ove in which the sheth II crosses these tiles, must correspond to the sme glue The sheth III strts from the second region of the South edge, it shifts once, nd it reches the third region of the South edge of the tile ove Also in this cse, the position from the second region of the South edge on which it strts must correspond to the sme glue s the position reched on the third region of the South edge of the tile ove In order to mke the trnsition from tiles with glues to geometric tiles, we proceed s follows Given tile system T X 8, for ech tile t T we construct n 3 geometricl vrints, where n = X is the numer of distinct glues All these n 3 geometricl vrints of t hve the sme umps, dents, spikes nd ig dents This is ecuse ll these structures simulte the eight glues of the tile t, nd they re ll plced on the exct position corresponding to the prticulr glues they simulte However, on distinct geometric vrints, the sheths must e plced on different positions long the specific regions of the edges This is due to the fct tht the sheths hve the role of llowing the crossing of spikes from neighoring geometric tile to nother, without overlpping Thus, since these spikes cn e plced on n different plces long the edges (or ctully long one prticulr region of the edges), for ech tile t there must exist geometric vrint corresponding to ech of these n cses Since there re three sheths, for ech tile we must hve n 3 geometric vrints Theorem 2: let N = ((0, 1), (0, 2), (1, 2), (1, 0), (0, 1), (0, 2), ( 1, 2), ( 1, 0)) e neighorhood vector, nd let T X 8 e tile system using this neighorhood Then, we cn construct geometricl tile system G such tht for ny surfce D Z 2, there exists one-to-one correspondence etween vlid T D -tilings nd non-overlpping G D -tilings, up to replcing some of the tiles with severl of its geometricl vrints Proof: Let T D : D T e vlid tiling, nd let G e geometricl tile system constructed s descried ove Then, for ech (x, y) D we replce ech tile t(x, y) with one of its geometricl vrints Since the sheths of the geometric tile g(x, y) re coming in contct with the geometric tiles on positions (x 1, y), (x 1, y 1), (x, y 1), (x, y + 1), (x, y + 2) (if these positions re in the domin D), we must select crefully which geometricl vrint of t(x, y) we should use Depending on whether or not one or severl of the five positions mentioned ove re in the domin of the tiling T D, we mke the following choices (we descrie here just three situtions, the others cses eing very similr): If (x 1, y), (x 1, y 1), (x, y 1), (x, y + 1), (x, y + 2) D, then we cn replce t(x, y) with ny of its geometricl vrints This is ecuse ll the sheths of the geometric tile g(x, y) re not coming in contct with ny other geometric tile If (x, y + 2) D or (x 1, y) D (or oth) then we replce t(x, y) with its geometricl vrint whose sheth II corresponds to the glue t(x, y + 2) ( 1, 2) or t(x 1, y) (1, 2), respectively Note tht if oth (x, y + 2), (x 1, y) D then, since the tiling T D is vlid, it must e tht t(x, y + 2) ( 1, 2) = t(x 1, y) (1, 2) If (x, y + 1) D then we replce t(x, y) with its geometricl vrint which stisfies the following conditions First, its sheth should correspond to the glue t(x, y + 1) (0, 2) Then, its sheth II should e perfectly surrounded y the sheth III of the geometric tile ove (here, there my e severl vrints hving this property) If lso (x, y + 2) D, then the sheth II should correspond to the glue t(x, y + 2) ( 1, 2) Finlly, its sheth III should correspond to the glue t(x, y + 1) ( 1, 2) The other cses re treted in similr mnner By choosing the correct geometricl vrints for the tiles, since the tiling T D is vlid, we do not otin ny overlpping in G D Thus, the tiling G D : D G otined y replcing the tiles from T with the corresponding geometric tiles from G s descried ove, is non-overlpping Moreover, the tiling G D is unique, up to replcing some of its geometricl tiles with other geometricl vrint of the sme tile For the converse, if G D : D G is non-overlpping tiling then, just s in the previous section, we construct the tiling T D : D T y replcing ech tile from G with the corresponding tile from T (ech tile in G is geometricl vrint of unique tile from T ) Since there is no overlpping in G D etween umps nd dents nd lso etween the spikes nd the ig dents, we conclude tht ny tile from T D must gree on its glues with ll of its neighors from the domin D Thus, the tiling T D is vlid, nd it is uniquely otined strting from the tiling G D As in the previous section, the following result is n immedite consequence Corollry 2: let N = ((0, 1), (0, 2), (1, 2), (1, 0), (0, 1), (0, 2), ( 1, 2), ( 1, 0)) e neighorhood vector, nd let T X 8 e tile system using this neighorhood Then, we cn construct geometricl tile system G such tht for ny non self-crossing pth P there exists one-to-one correspondence etween T-zippers (P, r) nd non-overlpping G-tiled pths (P, s), up to replcing some of the tiles with severl of its geometricl vrints Frontiers in Computtionl Neuroscience wwwfrontiersinorg Novemer 2009 Volume 3 Article 20 10

11 Czeizler nd Kri Geometricl tile design for complex neighorhoods THE 3 5 FILLED RECTANGULAR NEIGHBORHOOD VECTOR As finl cse, we consider the neighorhood vector N contining 14 positions, which is descried y the following pttern: ( 12, ) ( 02, ) ( 12, ) ( 11, ) ( 01, ) ( 11, ) ( 10, ) ( 10, ) ( 1, 1) ( 0, 1) ( 1, 1) ( 1, 2) ( 0, 2) ( 1, 2) In this cse, the neighors of tile t(x, y) re ll the tiles contined in the 3 5 rectngle centered on (x, y), except for (x, y) The geometric tiles used in this cse re otined y comining severl geometricl tile designs We comine the geometricl tile design given in the previous section with its symmetric design, nd lso with new tile design which is descried elow In Czeizler nd Kri (Sumitted) we present motif construction of certin shpe, which is used for proving tht tiling using the Moore neighorhood, see Exmple 3, cn e simulted y nother tiling using qusi liner neighorhood, ie, prtil von Neumnn neighorhood This shpe is otined y ttching squre to regulr hexgon nd y plcing severl ump nd dent constructions on its edges, see Figure 10A This motif construction cn e used for designing geometricl tile system which cn simulte the tiles with glues hving the Moore neighorhood As in the cse of the geometric tiles presented in the previous sections, etween the ump nd dent constructions of these geometric tiles we cn distinguish spike, ig dent nd sheth Note tht this shpe, ie, squre ttched to regulr hexgon, cn indeed tile the plne, s it is derived from one of the 11 Archimeden Tilings done y regulr polygons (see Grünum nd Shephrd, 1987) In Figure 10B we present one of the nonoverlpping tilings y geometric tiles which is ssocited to the tiling from Figure 2 By putting together ll of these constructions, we otin geometricl tile design which hs three pirs of norml (smll) umps nd dents, four spikes, four ig dents, nd six sheths In Figure 11 we present oth the design of these geometric tiles s well s non-overlpping tiling otined using this geometricl tile system (the smll umps nd dents re omitted from the pictures) Note lso tht since in this cse we hve six sheths, for ech tile we hve to construct n 6 geometricl vrints, where n = X is the numer of glues Theorem 3: let N e the 3 5 filled rectngulr neighorhood vector, nd let T X 14 e tile system using this neighorhood Then, we cn construct geometricl tile system G such tht for ny surfce D Z 2, there exists one-to-one correspondence etween vlid T D -tilings nd non-overlpping G D -tilings, up to replcing some of the tiles with severl of its geometricl vrints Corollry 3: let N e the 3 5 filled rectngulr neighorhood vector, nd let T X 14 e tile system using this neighorhood Then, we cn construct geometricl tile system G such tht for ny non self-crossing pth P there exists one-to-one correspondence etween T-zippers (P, r) nd non-overlpping G-tiled pths (P, s), up to replcing some of the tiles with severl of its geometricl vrints CONCLUSIONS In this pper we showed how to simulte prtilly-tiled pths (zippers) tht use squre tiles with complex neighorhood reltions, y rions of geometricl tiles tht use very simple, locl neighorhood reltion sed solely on non-overlpping We nmely solved this prolem for the cse of the tll von Neumnn neighorhood ( slightly tller version of the von Neumnn cross-shped neighorhood), the f-shped neighorhood, nd the 3 5 filled rectngulr neighorhood The techniques we used cn e extended to some other prticulr cses of neighorhoods For instnce, the method from Section FIGURE 10 Geometric tiles for the cse of the Moore neighorhood: (A) geometric tile; (B) one of the non-overlpping tilings y geometric tiles which cn e ssocited to the tiling from Figure 2 Frontiers in Computtionl Neuroscience wwwfrontiersinorg Novemer 2009 Volume 3 Article 20 11

12 Czeizler nd Kri Geometricl tile design for complex neighorhoods A B spikes sheths ig dents FIGURE 11 Geometric tiles for the cse of the 3 5 filled rectngulr neighorhood (norml umps nd dents re omitted): (A) geometric tile; (B) non-overlpping tiling y geometric tiles The Tll von Neumnn Neighorhood Vector cn e esily extended for the cse when the von Neumnn neighorhood s verticl rm is ritrrily tll, ie, the neighorhood contins one or severl pirs of the form (0, k), nd their symmetric ones (0, k), where k 2 These positions correspond to those tiles plced k- steps to the North nd to the South For ech such new position (0, k) nd its symmetric (0, k), we hve to dd new spike, ig dent, nd k 1 sheths on the North nd South edges The spike hs to e sufficiently long, ie, little longer thn (k 1)-times the size of the Est edge, while the lengths of the sheths would hve to mtch the spike, ie, decresing from (k 1)- to one-times the length of the Est edge Also, ll these structures would hve to e in step like form, in order not to overlp with the similr structures from the ove nd elow tiles By generlizing the methods descried in Section The f- Neighorhood Vector, we cn further extend the ritrrily tll von Neumnn neighorhood of the previous prgrph y dding to it pirs of the form (1, k) or ( 1, k), nd their symmetric ones ( 1, k) nd (1, k), respectively, where k 2 Here, for ech new position (1, k) nd the symmetric one ( 1, k), we dd one spike strting from the Est edge, one ig dent on the South edge, one sheth on the West nd North edges, nd k 1 sheths on the South nd North edges By putting together ll the ove considertions, we conclude the following Let N k nd N k e the neighorhoods N k = {(i,j) 1 i 1, k j k, nd (i,j) (0,0)} nd Nk = {( i, j) k i k, 1 j 1, nd ( i, j) ( 0, 0)} Informlly, this neighorhood is rectngulr Moore-type neighorhood, centered t the tile of reference, nd with width 3 nd heighk + 1 (or height 3 nd width 2k + 1) Then, for ll tile systems T N, were N is symmetric neighorhood which cn e included in one of the ove neighorhoods, ie, such tht there exists k 0 with N N k or N N k, we cn design geometricl tile system G such tht ny (prtil) T-tiling or T-zipper cn e simulted y non-overlpping G-tiling or G-tiled pth, respectively Note tht such neighorhood included in 3 (2k + 1) rectngle need not e contiguous, ie, my even hve holes in it On the other hnd, we oserve tht n immedite generliztion of the previous techniques cnnot e directly pplied to the cse of completely ritrry neighorhoods [not even for neighorhoods included in 5 (2k + 1) rectngle], since the spikes nd sheths we would hve to dd in these cses would self-overlp Frontiers in Computtionl Neuroscience wwwfrontiersinorg Novemer 2009 Volume 3 Article 20 12