The injectivity of the global function of a cellular automaton in the hyperbolic plane is undecidable

The injectivit y of the global function of a cellular automaton in the h yp erb olic plane is undecidable Maurice Margenster n 1 Lab oratoire d’Informatique Th ´ eorique et Appliqu´ ee, EA 309 7, Universit ´ e de Metz, I.U.T. de Me tz, D´ epartemen t d’Informatique, ˆ Ile du Saulcy , 57045 Metz Cedex, F rance, margens@un iv-metz.fr Abstract. In this paper, we look at the follow ing q uestion. W e consider cellular automata in the hyperb olic plane, see [5,18,9,13] and we consider the global function d efined on all p ossible confi guratio ns. I s the i njectiv- it y of t his function undecidable? The problem w as answ ered p ositively in the case of t h e Euclidean plane by Jarkko Kari, in 1994, see [3]. In the present paper, we sho w that th e answer is also p ositiv e for the hyperb olic plane: the problem is undecidable. Keyw ords : ce llular automata, hyper bolic plane, tessellations 1 In tro duction The glo ba l function of a cellular automaton A is defined in the set of all co nfig- urations. This is a very different p oin t o f view than implementing a n algorithm to solve a given problem. In this latter case, the initial config uration is usually finite. In the case o f the Euclidean plane, the definition o f the set of config urations is very easy: it is Q Z Z 2 , w her e Q is the set of states o f the automaton. In the hyperb olic plane, see [13,11], following what we did in [11], w e hav e the following situation: we consider tha t the grid is the pentagrid or the terna ry heptagrid, see [13]. W e fix a tile, whic h will b e called the centra l cell and, around it, we dispa tc h α secto rs, α ∈ { 5 , 7 } : α = 5 in the case o f the p en tag rid, α = 7 in the c ase of the ternary heptagrid. W e assume that the sectors and the central cell cover the plane a nd the s ectors do not ov er lap, neither the central cell, no r other sector s : call them the basic sectors . Denote by F α the set co nstituted by the cen tral cell and α Fib onacci trees , each one spanning a basic s ector. Then, a config uration of a cellular automa ton A in the hyperb olic plane can b e represented as an element o f Q F α , where Q is the set of states of A . If f A denotes the l ocal tr ansition function o f A , its global transitio n function G A is defined by: G A ( c )( x ) = f ( c ( x )). The injectivity pr oblem fo r a cellular automaton co nsists in a sking whether there is a n algorithm which, applied to a description of f A would indicate whether G A is injective or not. In this pap er, we pr ove that there is no s uc h algorithm and so, the corr e- sp onding problem is undecidable. The present pape r relies o n a pr e vious work by the author , see [17]. In this pa p er, we give a construction which is describ ed in [1 5 ,12], which yields a plane-filling path, ea c h time we can construct a v a lid tiling with a n exception. How ever, in this exceptiona l case, a more car eful anal- ysis of the structure of the path shows that, c hanging a bit the wa y in which triangles and tr apezes are trav ersed by the path, it is also p ossible to carr y out the a rgumen t which is neede d to prove the undecidability of the injectivity . Accordingly , we shall n ot rep eat the co nstruction of the interw o ven tria ngles on which the cons truction o f the mauv e triangles rely . In sec tio n 2, w e more carefully descr ibe the construction o f th e path bas ed on the mauve triangles. In section 3 , w e s ho w how to der iv e the pro of of the main theor em: Theorem 1 Ther e is no algorithm to de cide whether the glob al tr ansition func- tion of a c el lular automaton on the ternary heptagrid is inje ctive or not. Note that it is enoug h to find a particular tiling whos e cellular automata hav e the prop erty that the injectivity of their global function is undecida ble to prov e that the same prop ert y for c ellular automata in the hyperb olic pla ne in g eneral is also undecidable. Ho w ever, it seems imp ossible to transfer the co nstruction of the pa th which w e co nsider in this pap er to the pe n tag r id. 2 A closer lo ok at the mauv e triangles Remind that the ma uv e tria ngles ar e first cons tr ucted on the interw ov en trian- gles. The latter triang les are obtained by the following construction, illustrated by figure 1. W e refer the r eader to [14,15,10] for a detailed account o n the constructio n of the in ter w oven triangles and for their pr o perties. W e also refer him/her to the same pap ers for an acco un t on the implementation of these triangles in the ternary heptag rid of the hype rbolic plane. A t this p o in t, we would like to make the following r emark. In [1 4,15,10], we implemen t the interw ov en triang les in the ternary heptagrid, us ing another tiling as a background. This tiling, called the man ti lla , is a refinemen t of the ter na ry heptagrid by grouping its tile in a particular way . It is p ossible to implement the in terwo v en triang les in a simpler cont ext of the terna ry heptag rid. H ow ever, the spacing imposed by t he mantilla is a go od p oin t which allows to more easily solve a few details of the implementation o f the path. The c o nstruction of this tiling needs a lot o f signals, which en tails a huge nu mber of tiles, around 18,0 0 0 of them, not ta k ing into ac c oun t the sp ecific tiles devoted to the simulation of a T uring machine. The co nstruction of this pap er requires muc h mo re tiles, but we shall not try to count them. Figure 1 Construction of the interwoven triangles in the Euclide an plane: the gr e en signal. The ma uv e triang les are co nstructed from the in terwo v en ones. More pre- cisely , we fo cus our a ttention on the red triang les only: they constitute the basis of o ur construction. The mauve triang les are simply obtained fro m the r ed tri- angles as follows. The vertex of a mauve tr iangle is that of a red tria ngle R . Its legs follow those o f R . They g o on on the same ray after the corner of R , un til they meet the basis of the red phantoms o f the same g eneration as R which are generated by the basis o f R . At this meeting, the legs meet the basis of the mauve triangle which co incide with the basis of the just mentioned red phan- toms. In [15,12], we thoro ughly descr ib e the construction o f the mauve triang les and we r e fer the re ader to these pap ers. Here, we just men tion a few pro perties of these triangles and then, w e shall use them in order to make it prec is e the co nstruction of the mauve tr iangles a nd the p oints attached to them. 2.1 Prop erties of the mauv e triangles F rom the a bov e cons tr uction of the mauve triangles, we may define the g ener- ations of the mauve tria ngles from those of the red triangles: a mauve triangle of the genera tion n is c o nstructed on a r e d triangle of the genera tion 2 n +1. It will be later useful to r ecognize the mauve triangles of genera tion 0. T o this aim, we define a new co lour, calle d mauv e-0 whic h is given to these tria ngles only . According ly , the ma uv e triangle s of generatio n 0 will b e most often called mauv e - 0 triangles . F rom the doubling o f the height with r espect to the red triang les, the mauve triangles lo ose the nice pr o perty that the red triangles ar e either embedded or disjoint. This is no more the case for the mauve tria ngles. Howev er , we can precisely desc r ibe the ov erlapping of ma uv e triangles a nd how their intersections happ en. F rom [15,12], we know that the in ter section o ccurs by a leg of a mauve triangle cutting a basis of ano ther mauve tr iangle. F r om the co nstruction, any mauve tr iangle T of the g e ne r ation n + 1 contains thee mauve triangles o f the generation n with which they hav e no int ersection. They also meet tw o mauve triangles of the prev ious genera tio n. One of them is met at their bas is : the legs of this triang le of the ge neration n cuts the ba s is of T . The o ther mauve tr iangle M of the ge ne r ation n meets T near its vertex. This time, the legs of T c ut the basis of M . W e give a n um be r in [0..3] to the mauve triangles of the genera tion n who se vertex is contained in T , as four iso clines are involv ed by these vertices. Such a nu mber is called the rank o f the triangles . The ra nk is per iodically r epeated on Figure 2 An i l lustr ation of the mauve triangles. the mauve triangles o f the generation n , to the top and to the b o ttom. A tr i- angle of r ank r is called an r -triangl e . If a mauve triangle o f the generatio n n contains T , it is called the hat of T : it is a 3 -triangle. The hat is unique when it ex ists. Note that if we can r epeat the c onstruction of the hat recur siv ely until reaching a ma uv e- 0 triang le, we obtain that the vertex of this mauve-0 triangle is at a distance at mos t h 4 of the vertex o f T . W e call the mauve-0 triangle the remotest ancestor of T , a notion already rema rk ed for the in terwo ven trian- gles. Accordingly , if the vertex of T is on the ba sis of a mauve-triangle o f the same g eneration, then its re mo test a nc e stor exis ts. The tria ngles of the gener ation n which cut the bas is of T a re also 3-tr iangles. W e define the low p oints of a leg o f a triangle, LP for short, as the po in t which is at a distance h 4 from the basis of the triangle, wher e h is the length of the leg. The LP ’s play an imp ortant role: the line which joins the L P ’s of T cuts the 2-tr iangles also at their LP ’s. The in ter section of the ba sis o f T with its 3-triangle s o ccur at their LP ’s. In [15,12], the cons ideration of the r -triangles ha s led to the extension of the no tion o f latitude used in the interw o ven triangles to the c ase of the mauve triangles. How ever, the definition which was there g iv en is not satisfactory . In order to intro duce a better one, we define the prim a ry latitude of a mauve triangle as the set of iso clines which cross its legs, the basis b eing included but the top being e xcluded. This allows us to obtain a par tit ion of the hyper bolic plane by the primary la titudes attached to a given genera tio n. This defines a partition for each g eneration. But the primary latitudes may ov e rlap from one generation to the next one. Now, we can precisely state the prop erties mentioned a b ov e ab out the inter- sections betw een mauve triangle s a nd we ca n als o prov e them a s follows. Lemma 1 L et T b e a mauve triangle of t he gener ation n + 1 . Then, t he primary latitude of T interse cts five primary latitudes of the gener ation n , denote t he m by L − 1 , L 0 , L 1 , L 2 and L 3 . Ther e ar e four t ria ngles of t he gener ation n , T 0 , T 1 , T 2 and T 3 with the fol lowing pr op ert ie s: ( i ) T i b elongs to the primary latitude L i for i in { 0 , .. 3 } ; ( ii ) the vertex of T i +1 is on the b asis of T i for i in { 0 , .. 2 } ; ( iii ) t he LP ’s of T and T 2 ar e on the same iso cline; ( iv ) the le gs of T 3 cut the b asis of T at their LP ’s. When it exists, T − 1 c ontains the vertex of T and it b elongs to the latitu de L − 1 . In this c ase, the vertex of T is on the iso cline which joins the LP ’s of T − 1 . Let R b e the r ed tria ngle whose vertex and legs suppo rt those of T . Its generation ca n b e wr itt en a s 2 m +3 . It contains tw o red latitudes of the g ener- ation 2 m +1 , each one containing a triangle R α , with α ∈ { 1 , 2 } , and we may assume tha t R 2 is b elow R 1 and that the vertex of R 2 belo ngs to the bas is of a red phantom whose vertex b elongs to the basis of R 1 . F rom [14], w e know that R c o n tains several red triangles of the g enera- tion 2 m + 1 which b elong to L 1 . W e pick one o f them: it is R 1 . It ge ne r ates a mauve triang le T 0 which is contained in T , as R 1 is also contained in T . Note that T 1 is gene r ated by a red triangle of the gene r ation 2 m + 1 which is on the second latitude o f this kind which ar e contained in the latitude o f R . Now, the second half of the legs o ccurs in the latitude of r ed phantoms of the gener ation 2 m + 3. W e know that inside r ed phantoms, the structure of the inner trila terals is the same a s inside a red triang le of the same generatio n. This defines R 2 and R 3 which ar e r ed triangles of the gener ation 2 m + 1. They are inside T and we hav e the same relation b et ween R 1 and R 2 as b et w een R 2 and R 3 . And so, we find again t w o re d tria ng les which give rise to T 2 and T 3 . The succes sion from T 0 to T 1 and then fro m T 1 to T 2 and, finally , from T 2 to T 3 is the s a me. By the situation of R 4 , we obtain that the legs o f T 3 cut the basis of T at their LP ’s. W e also see that the ba sis of T 2 is on the iso cline o f the vertex of T 3 . As the distance from the bas is of R 3 to the mid-dis ta nce line of the phant om Q of the generation 2 m + 3 which contains R 2 and R 3 , is the sa me as the distance from the mid- dis tance line of Q to the vertex of R 4 , and that this distance is half the height of a trilatera l of the gener a tion 2 m +1, we have that the mid-dista nce line of Q is b oth on the is ocline of the LP ’s of T and also on the iso cline of the LP ’s of T 2 . This proves the p oin ts ( iii ) a nd ( iv ). The pr oof of the other p oin ts is als o contained in the just given arg umen ts. Now, as s ume that there is a red triangle R 0 of the generation 2 m +1 which is in the same connection with R 1 as R 1 is with res p ect to R 2 . This means that there is a blue tria ngle B of the gener ation 2 m +2 which gene r ates the vertex of R on its mid-dista nce line and the v ertex of B is gener ated by R 0 . No w, R 0 generates a mauve triangle T − 1 which contains the vertex of T as ther e is a red phantom P in the tower of phantoms aro und the vertex of R , whose vertex is on the bas is of R 0 and whose basis gener ates the vertex of R 1 . Accordingly , T − 1 contains the vertex of T as T − 1 also contains P . Now, note tha t the vertex o f T is on the mid-distance line of the phantom P . As the height of the tria ng le T − 1 is twice that of R 0 , we hav e that T is on the iso cline which s upp orts the mid-distance line of P and, according to the just per formed estimation, this iso cline joins the LP ’s of T − 1 . This proves the last assertion o f the lemma. F rom the pro of, we can deduce a w ay to determine the mid-po in t and the LP ’s of a mauv e triangle by means of a finitely genera ted signals. It will b e enough to show that w e hav e just to app end a few sig nals to those which we hav e a lready defined. 2.2 Construction of the LP ’s of a m auv e triangl e Let T be a mauve triangle of the gene r ation n and let h b e its height. Its mid- po in ts, which lay at a distance h 2 are e a sy to determine: it is the co rner o f the red tria ngle R on which T is constr ucted. The determination of the LP ’s needs so me work. In fact, the LP ’s of T are o n the same iso cline a s the mid-distance line of a phantom whose vertex is generated by the bas is o f R . Let P b e the leftmost phantom gene r ated by the basis of R . As P is a pha n tom, its g reen signal may cut the leg of P . How ever, it may not as well, see [14], in which case it is contin ued by a n ora nge signal. Now, the leg o f T meets a lo t of orange sig nals coming fro m all the phantoms which stand close to its b orders. Without further indication, the leg ca nnot disting uish which one co mes fro m P . The constr uction pr oceeds as follows: First, we lo ok at the determination of the cor ner s of T itse lf which is no t that s traigh tforward. A t the cor ners o f R , the mauve signal defining the leg of T go es o n a long the extremal branch of the Fibona cci tree defining R . At the same time, each corner of R sends a signal tow ar ds the other one on the is o cline of the basis of R . Ca ll this signal the bro wn signal. The brown signa l has the la teralit y of the co rner. When the brown signal meets the fir s t vertex of the phantom P , it is a r ed phantom of the genera tion 2 n + 1. The s ignal go es down alo ng the leg of the phantom which has its laterality . It go es along this leg until it meets the corner of P . There, on the iso cline ι of the basis of P , the brown s ignal leaves the leg to run on ι , to the side of its later alit y , until it mee ts the mauve signa l of the leg of T . Then, a mauve signal is sent to the o ther s ide , in or der to meet the ma uv e signal sent by the other corner of T . Now, the problem for the signal is to meet the correct leg, a s it may encoun ter a lot of them along ι , the iso cline of the basis of P , which b elong to smaller generations . The signal may cros s them becaus e it meets them at their LP ’s where this p ossibilit y is for eseen. When the brown signal firs t meets a leg whose laterality is opp osite to its one, it knows that it is not the appropriate leg. Now, the s ig nal do es not cros s the leg: otherwise it will meet the opp osite leg which is also not the right o ne: it might also meet several legs of its la teralit y befo re meeting a leg with an oppo site laterality . As the brown signal cannot count arbitra ry num b ers, it circumv ent s the triangle by climbing along its leg up to the vertex and then go ing down to the appro priate iso cline. In or der to recognize the right iso cline, when the br o wn signal s tarts its c ir cum vent ing pa th, it s e nds ano ther brown signa l, say a light one, with no laterality , which go es on running o n ι , tow a rds the appro priate c orner. As ι is an iso cline o f the L P of the triangle which the brown signal circumv en ts, the brown sig nal cannot meet another light br o wn sig nal meeting the leg: as for red tr ilaterals, the iso cline of a basis is spec ific to any mauve tr iangle. And so, when going down a long the leg, the brown signal meets its light brown o ne, it knows that it has found ι on which it go es on its wa y , still to the side of its latera lit y . And now, the first mauve leg of its latera lit y met by the brown signa l is the right one. Now, we can use the br o wn signal in or der to lo cate the LP ’s of T . Cons ider the time when the previous brown signal is going down along the appropr iate leg of P . When the brown signal meets the mid-p oin t o f P , it knows that it is the iso cline o f the L P ’s of T . And so, the brown signal sends a purple signa l of the same laterality as the brown sig nal tow a rds the side o f its laterality on the iso cline ζ o f the mid-distance line of P . This signal also cir cum ven ts the mauve tr iangles w hich it meets. Now, the signa l is able to reco gnize ζ during the circumv ention of phantoms tha nks to the following. W e k now that the pur ple signal meets s maller ma uv e triangles at their LP ’s. By induction, we assume that a simila r sig na l a rriv es to the LP ’s from inside the mauve triangle M , crea ted at the time of the co nstruction of M . Note that in any case , such a sig nal is stopp ed by the leg of M . Now, the a rriving signa l from the mid-p oin t of the leg of P is deviated to the first part o f the leg of M . When the signa l go es down on the other le g , it identifies its LP by the arr iv al o f a s imilar sig nal of the appr o priate laterality which is stopp ed by the le g. T his allows the signal to aga in find ζ and to go on its r oute o n this iso cline. Due to the later alit y of the purple signal and to the fact that its laterality is unchanged a nd tha t it must ma tch the mauve leg it meets from inside, such a signal cannot b e pr esen t if it is no t sent by a brown s ignal for detection purp ose. Let us lo ok clo s er at this assertio n. Consider a mauve triangle M 0 and, inside it, a mauve tr iangle M 1 in a place where the iso cline of the LP ’s of M 1 do not meet those of M 0 . Ass ume that the g eneration o f M 0 is the s uccessor of the generatio n o f M 1 . Assume to o that an inner signal σ o f M 1 go es out unt il the leg of M 0 . If this ha pp ens, this may confuse a signal which is circumv en ting M 0 . Now, as σ go es o ut from the LP of M 1 , σ also circumv en t M 1 and, together, all the other maximal inner ma uv e triangles of a generatio n no t gre a ter than that of M 1 whose LP ’s are on the same iso cline as those of M 1 . Consequently , σ rea c hes b oth legs o f M 0 . Now, σ has a s ing le la teralit y a nd when it meets a mauve leg from inside, its latera lit y m ust b e the same as that of the leg. But here, this no t p ossible, a nd so σ cannot go o ut of M 1 . And so, this ens ures the detection of the L P ’s . Note that the detection of the LP ’s of inner mauve tr iangles with the sa me iso cline ζ do es not a lter the pro cess due to the fact that the leg sto ps the signal. There are tiles with a circumv enting signa l, with a laterality indep endent on that o f the leg, and tiles without it but, in bo th cases, the finishing s ig nal inside the triang le is present with the appro priate laterality . Thes e tw o kinds of tiles, and no others, force the right choice, a gain as latera lit y ca nnot b e changed. Now, the mauve signal which defines the basis of a mauve triangle is a ls o emitted by the vertices of the mauve triangle of the s ame g eneration but whose primary laterality is just b elow the co nsidered one. Now, in mauve triang les, a basis must b e stopp ed by its corner s. T o handle this situation, we remar k that the constr ain t of later a lit y of the brown sig nal allows us to for bid the e xistence of s uch a signal in b etw een tw o contiguous red phantoms of the gener ation n with their v ertices on the basis of R . Now, if a ma uv e triang le is missing, the left-hand side brown signal go es to the left until it meets the corner of a ma uve triangle which exists within the considered latitude. Now, such a meeting fr om o utside o f the corner is ruled out: it is enough to forbid s uc h a config ur ation for the tile defining the corner . And so, if ther e is no r ed tr iangle, the br own sig nal will b e des tro y ed, as the phantom without brown signal do es ex ist within the same latitude. F or the mauve basis, it is an analo gous s itua tion: it may b e emitted by the vertex of a ma uv e triangle and it may b e no t. If it is e mitted a nd if there is no leg to s top it, the basis meets a corner of a mauve tr iangle from outside. As the co rner s to ps the mauve basis which is inside the tr ia ngle, it is enoug h to no t allow a n y mer ging at the level of this tile. And so , if the mauve legs do not g o down, the basis cannot exist: it is ruled out by exter nal corners. Now, if the v ertex ex is ts, the legs exist and also the corners , due to the br o wn sig nal a nd the co rners force the pr esence of a unilateral mauve ba sis in b et w een them. W e can now state: Lemma 2 The mauve t ria ngles t o gether with the determination of their LP ’s and mid-p oints c an b e c onstructe d fr om a fin ite set of pr ototiles. Later, inside a tr ia ngle o r a zone in b et ween tw o tr ia ngles o f the same primar y latitude, the path which we shall later desc r ibe will completely cross the primar y latitudes of these domains. This is why it is impor tan t to clearly delimit each zone in each k ind of ar ea. F or this purpo se, we introduce the notions of β -cline which we study in the ne x t subsection, together with its co nstruction. 2.3 The β -clines and their construction W e star t fro m the remark that the basis of a mauve triangle T of the g en- eration n +1 cuts the legs of the 3-tria ngles which have their vertex inside T . Repe a ting this r emark to the 3-tria ngle of the gener ation n , we can c o nstruct a sequence { T i } i ∈ [0 ..n +] such that: ( i ) T n +1 = T ; ( ii ) T i is a 3 - triangle o f the gener ation i for i in [0 ..n ]; ( iii ) the bas is o f T i +1 cuts the leg s o f T i , of cour s e at their LP . An y mauve tria ngle T of a p ositive generation g enerates such a sequence which we call the shado w of T and that n +1 is its generatio n. O f course, if { T i } i ∈ [0 ..n +1] is the shadow o f T n +1 , the sequence { T j } j ∈ [0 ..i +1] is the shadow of T i +1 for i in [0 ..n ]. W e say that the shadow { T j } j ∈ [0 ..i +1] is a trace of the shadow { T i } i ∈ [0 ..n +1] . W e say that a sha dow { T i } i ∈ [0 ..n +1] is a tow er , if it is finite and if it is not the trac e of a shadow o f a bigger g eneration. W e s hall see that there may be a sequence of mauve tria ngles { T i } i ∈ I N in which { T i } i ∈ [0 ..n +1] is a trace of { T i } i ∈ [0 ..n +2] for a n y n . In this case, we say that { T i } i ∈ I N is an i nfin i te to w er . When { T i } i ∈ [0 ..n +1] is a finite to w er, we say that the iso cline of the basis of T 0 is the β -cline of T n +1 and that its ty p e is the rank of T n +1 . F rom the β -clines , we define tw o new p oin ts o n the le gs of a triangle o f a po sitiv e gener ation: the β - and γ -poi n ts . By definition, the β -p oin t of a ma uv e triang le T o f the gener ation n +1 is the int ersection o f its leg with the β -cline of the 2-tr iangles whose vertex in inside T . It is not difficult to see that the β -p oin t lies on the leg in b e tween the LP and the corner . It is at a distance les s than h 12 from the line joining the LP ’s of T , with h b eing the height of T , and as closer to this v alue as n tends to infinity . Constructing the β -cline T o co nstruct the β -cline, we define sig na ls which start from the LP ’s of a mauve triangle of the considered genera tion a nd latitude. Call them the β -signals . The β -sig nals ar e la ter al, with the la teralit y which is opp osite to that of the leg on which they sta rt. They travel a long legs of mauve triangles and along iso clines of a ba sis. The β -s ignals go down alo ng legs of a laterality o pposite to their own one, fr o m an LP to a corne r . When they r un on an iso cline, they g o in the direc tio n of their later alit y . When they meet a corner, they run on the ba sis, in the direction of the other corner . They can freely trav el on this iso cline, until they meet the leg o f a tria ngle of a laterality which is o pposite to their own one and a t their LP . If the leg is of another laterality o r if the meeting p oint is no t in the closed interv al with the LP of the leg and its cor ner as end p oin ts, the β -signals crosses the leg. It is plain that bo th β -s ig nals star ting fr om the opp osite LP ’s o f the same mauve tr iangle will meet, and they ca nnot do that alo ng a leg or at a corner . When they meet, they use a join tile, see [14], in which the r igh t-hand, left-ha nd side β - signal is on the left-, rig ht-hand side part o f the tile. It is plain that bo th β - signals de fine a kind of conv ex hull of this part of the mauve triang le . Note that for tw o consecutive mauve triang les of the same gener ation within the same iso cline, the β -signal whic h starts fro m the low-po in t o f one of them cannot travel on this is ocline to the facing low-p oin t of the other tria ngle. Indeed, on the r igh t-hand side low-point, we hav e a left-hand side β -signa l and on the left-hand side low-p oin t with have a rig h t-hand side β - signal. And so, this would require a join tile with a left-ha nd s ide β - signal on the left-hand side part of the tile: this is ruled out. W e may imp ose an a dditional constraint on the join tile for β -sig nals of opp osite lateralities with the right-hand side s ig nal on the left-hand side of the join tile: the join tile gene r ates a horizo n tal unila teral yellow signal. This s ignal runs on an is ocline only: it ma rks the β -cline. It is impor tan t to note here that the who le iso cline constitutes the β -cline. By cons tr uction, the yello w signal trav els along an iso cline of a basis o f a mauve 0 triangle. Conseq ue ntly , it trav els on an iso cline 5. Accordingly , it meets no basis of a mauve triangle of a p ositive genera tion and no L P , a s LP ’s are alwa ys o n a n iso cline 1 5. And so , the yellow signal will meet legs of triang les. In our study of the shadow of a mauve triang le, we have alrea dy noticed that the s ame β -cline can be s ha red by several tria ngles of different genera tio ns. F or the purp ose of the path, in the case of a β -cline of type 2, we co nsider that it defines a sp ecial signal on the iso cline which is just b elow the β -cline. This means that there is a pre-path sig nal on the iso cline 4 which is just b elow a β -cline of type 2. This signal plays an impo rtan t role as can b e s een further. Note that the pre-pa th signa ls o f a given generation are different fro m those o f the next genera tion: the s ignals corr e s ponding to the g eneration n +1 o ccur on the iso cline 4 o f a β -cline of type 3 in terms of the generation n . W e have a stronger result: Lemma 3 The iso clines of a pr e-p ath signal of the gener ation n ar e differ ent fr om those of the gener ation m for any n , m with n 6 = m . Pro of. Consider the smaller gene r ation, say m . The pr e-path sig na ls of the next generation corr espond to β -clines of type 0 of the generation m . Now, for the generation m +2 we hav e the same cor respondence with the β -clines of type 3 of the genera tion m +1. But, the β -clines of type 3 of a gener ation also b elong to the β -clines of type 3 to o of the prev ious generation. This is due to the fact that the num ber o f β -clines which a re crossed is a multiple of 4. By induction, this pr o v es our claim. 2.4 The β -p oin ts and their construction F or the next section , we need to ma k e clear the connection b et w een a mauve triangle T of the generation n +1 and its inner mauve triang les of the gener a- tion n . T o lo cate the tr iangles o f the just pr evious gene r ation, there is a wa y given by the lo cal num ber ing of the triangles . W e alr eady noticed that the intersection betw een mauve triang les o ccur b et w een a leg and a basis and that with resp ect to the leg, the intersection happ ens at its low p oin t. The consequence is that mauve tr iangles o f the gener ation n which are inside T are cut by the basis of T if a nd only if they ar e 3-tria ngles. Now, the conv erse is tr ue: Lemma 4 L et T b e a mauve t ria ngle of the gener ation n +1 . Its b asis cuts mauve triangles of t he gener ations i for any i in [0 ..n ] . When i = n , the m auve triangle is of typ e 3 . W hen i < n , t he m au ve triangle is of typ e 2 . Pro of. W e alr eady know the prop erty for i = n . If n = 0 we are done. If not, consider i = n − 1 a nd let T 1 be a mauve triangle of the g eneration n and of t yp e 3 which meets the basis of T . It is not difficult to see, by definition of the LP ’s of T 1 , that the line of the L P ’s of T 1 cuts inner ma uv e triangles of the generation n − 1 if and only if they ar e of type 2 . Now, this ca n b e r e peated for these 2-triang les. Now, as ma uve tria ngles which are within the same latitude hav e the same t yp e, we conclude that any mauve triangle of the generation i , with i < n , is of type 2. Now, consider the β -cline of type 2 which corr esponds to the mauve triang les of the genera tion n which are inside T . This b eta -cline gives rise to a pre-path which plays an imp ortant role in the construction of a path which fills T . In particular, it is impo rtan t to recog nize the intersection o f this β -cline with the legs o f T . W e call them the β -p oints o f T . Lemma 4 gives us a way to construct the β -p oin ts of a ma uv e triangle . Note tha t there is no β -p oin t o n a mauve-0 triangle and that the β -point of a mauve triangle T 1 of genera tion 1 is easy to deter mine. Indeed, the line joining the LP o f T 1 cuts inner mauve-0 triangles of type 2. Now, the basis of theses tr ia ngles a re on the same iso cline which is the β -cline pa ssing thro ugh the β -p oin t of T 1 . And so the constructio n is simple: a sil v er sig nal is sent from the LP of T 1 un til it reaches the first mauve- 0 tria ngle of t yp e 2 , T 2 ; the silv er signal goes do wn along the leg of T 2 ; when it re a c hes the corner o f T 2 , it also reaches the β -cline of T 2 ; it follows this β -cline outside T 2 ; this intersection o f the s ilv er s ig nal with the leg o f T 1 defines the β -p oin t of T 1 . The constr uction o f the β -p oin t in the general ca se relies on lemma 4. Now, this time, we cannot star t from the LP of T : it meets only tria ngles of t yp e 2 o f all gener ations fro m 0 to n and so, it is difficult to recog nize those of generation n by a fixed in adv ance amount of means. Now, if we sta rt the silver signal from the co rner, it can b e per formed ac- cording to the following algor ithm: Algorithm 1 The c onstruc tion of the β -p oi nt of a triangle T of the gener ation n +1 . the s ilv er signa l starts from the co rner into t wo dir e ctions; the first directio n follows the basis until it meets the leg of a tria ngle of t yp e 3; it goes along this le g up to the vertex a nd there, it follows the iso cline of the vertex aw ay fr om the leg of T , until it meets a corner which is a cor ner of a tr iangle of type 2, T 2 ; from the co rner o f T 2 , the silver signal follows the β -sig na l co ming from the LP of T 2 which is ab ov e the considered c o rner; then the silver signal even tua lly meets the β -cline defined by the β -signal o f T 2 ; the silver signal g oes back to the leg of T , following the just met β -cline; the se c o nd dire ction follows the leg of T reachin g the c orner a nd go es up a long this leg tow ards the LP of T ; bo th dir ections of the silver signal meet at the intersection of the leg of T with the exp ected β -cline of t ype 2 coming fr om a n int ernal mauve triangle of the generation n : it is the exp ected β - p oint and the intersection stops b oth s ilv er signals. Note that this constructio n als o holds when n = 0. Let us prove that algor ithm 1 is co rrect. F rom lemma 4, when the fir st dir ection of the silver signa l meets a 3-tria ngle, it is a triangle T 3 of the generation n . In fact, the signal meets the leftmost 3- triangle of the g eneration n whos e vertex is inside T . When the signal ar riv es on the iso cline of the vertex of T 3 , it is the iso cline of the bases of the 2 -triangles of the generatio n n which are inside T . Go ing aw ay fro m the initial leg of T , the sig nal ar riv es to the 2 - triangle of the generation n which is the c lo sest to the leg, T 2 . The signal meets T 2 at its cor ner which fac es the leg o f T . At this corner, a ccording to what we have seen, the s ilv er signa l meets the β - signal. As it then follows the β - signal down to the β -cline of T 2 , and then the β -cline itself in the dir e c tion o f the leg of T from where it c o mes, it event ually meets the leg of T . Now, the silver signa l meets legs o f bo th lateralities which canno t always bee n distinguished e v en with the help of the cir cum ven ting technique. And s o, the fir st dir ection of the silver signal g oes on a long the β -cline, co nstan tly . Now, during the same time, the seco nd direction of the sig nal g oes up along the s ame leg of T from the corner to the direction of the LP . Before reaching the L P , it meets the β -cline which is accompa nied by the other silver signal. This allows to fix the β -p oint and bo th direc tio ns of the silver signa l to meet and to stop ea c h other. Note that these silver signals of different mauve triangles ca n meet: this happ ens when the first directio n of the s ilv er signal c oming from T re a c hes the corner of the 2-triangle o f the g eneration n . In this case , a s the sig nal concer ns the current genera tion and its previo us o ne only , there cannot b e more than the o ccurrence o f tw o signals b elonging to different g e ne r ations. In this case, the basis co n tains tw o silver signals: that which it generates and that whic h it receives from the next genera tion. It can b e rea lized in the tiles by giving different horizontal channels to these signals : a n upp er channel for the silver signal o f the same genera tion, as it will go up; a low er channel for the silver signa l coming from the next gener ation as it will g o down. In this wa y , the s ignals o f different triangles do not intersect. Co ns equen tly , as the intersection of b oth directions of a silver s ig nal of the same genera tio n stop e ac h other: Corollary 1 In any mauve triangle of a p ositive gener ation, ther e is a single β -p oint on e ach le g. Abo ut the construction of the β -p oin ts, it c a n be noticed that algorithm 1 can b e pro cessed in the r ev er se or der. This means that it can b e constructed from a mauve tria ngle T of type 2 a nd of the generation n for lo oking at the β -p oin t of the mauve triangle M of the genera tion n +1 which co ntains T if any . The algor ithm may detect if M exists or not and, when it ex its , how to find the β -p oin t. The existence of M is equiv alent to the pre s ence o f a ba sis at the LP ’s of the mauve triang le o f type 3 of the g eneration n reached b y the pre-signa l emitted fro m T lo oking a ft er s uch a bas is. If the basis is re a c hed, the pre-sig nal returns to T in order to trigger the signa ls of algorithm 1 in a reverse orde r which allows to detect the β -p oin t. Acco rdingly , this provides an iterative and bo ttom-up version of alg orithm 1 for the constr uction o f the β -p oin ts. A la st feature ab out the β -p oint is that it a llo ws to differentiate the part of the β -cline of type 2 on which it lies which is contained in the triangle from the part which is outside. Later, we shall see that this differentiation is very impo rtan t. It can easily b e realized, for insta nce as follows, a ccording to the different iation b et ween op en and covered basis in the in terwo v en triangle s . Ea c h β -p oint emits a horizontal signal on its iso cline, outside the triang le to which it b elongs. The signa l is lateral and has the laterality of the leg. In b et ween tw o co ns ecutiv e mauve triangles on the same primary latitude and of the sa me generation, the signals emitted by the opp osite β - points meet thanks to a join-tile which is similar to those use d with the interw ov en tr iangles. On the pa rt where the horizontal signal is present, we shall s a y that the β -cline is co vered . In the par t where it is not pres en t, we shall say that the β - c line is op en . Clea rly , the β - cline is op en inside the mauve triangles of its ge neration a nd it is co vered in-b et ween tw o consecutive such triangles within the sa me la titude. 2.5 The latitude F rom lemma 1, we know the in tersections b et ween mauve triangles of the gen- eration n +1 and those of the generation n . W e hav e to lo ok at a more general situation. F rom the cons tr uction of the interw ov en triangles, we know that the bases and vertices of mauve triangle s characterize the corr esponding tr iangles. This is not the case for the iso cline o f their L P ’s : such an iso cline is the mid-distanc e line of phantoms. W e know that usually , the mid-dista nce line of a phantom may be such a line for s ev er al phantoms of different g e ne r ations. It is the reason why the sa me a mbiguity is attached to the iso clines of the L P ’s as we can s ee fro m lemma 1. Recursively applying the lemma to inner triangles in a fixed ma uve triangle, we o btain that L P ’s o f a tr iangle of the genera tio n n may be crossed by the ba sis o f a triangle of the g e neration n + k , for any p ositiv e k . In ge ne r al, it is not p ossible to predict if such a situation will o ccur. Now, if it o ccurs, w e know tha t inside the mauve triangle of the genera tio n n , the 2-tr iangles will also be cut by this basis, also at their L P ’s. F rom lemma 1, we know that this situation do es not o ccur for the 0- and 1-triangle s which ar e contained in a mauve triangle. These triangles ma y be int ersected b y smaller triangles only , whic h cut their basis or their legs near their vertices. Going back to 2-tria ngles, we can see that if a 2-triang le T is of a gener ation n with n > 0, we can find smalle r triangle s which are a lso 2 -triangles inside T , their legs b eing cut b y the basis of T , at their L P ’s to o. And this can b e rep eated un til we rea c h the generatio n 0 . W e can say the same for 3 -triangles. If such a tria ngle is no t of the genera- tion 0, its bas is cuts triang les of the previous gener ation, and this pro perty can be rep eated recursively . Remember the notion o f shadow of a triangle and the construction of the β -cline. F rom this, w e define the b order line of a pr ima ry latitude of the generation n as a bro ken line as fo llo ws : First, define the b ottom o f a mauve triangle as the broken line which consists of the leg s of the triangle from the LP to the cor ner and the basis. Then, we define the b order l i ne as the iso cline of the LP ’s of the triang le s of the gener ation n of this primar y latitude in whic h ea c h maximal segment which falls inside a mauve triangle M of a gener ation at mo st n − 1 is replaced by the b ottom of M , the same pro cess o f substitution b eing r ecursively applied to the ba sis of the triangle and of the subs tituted triangle s . The term maximal indicates that we take the big gest tria ngle of a g eneration at most n − 1 which is cut by the iso cline. F rom now on, the l atitude of a triang le o f the ge neration n is the set of tiles which is c o n tained b et w een the b order line o f its primar y latitude and the b order line of the same g eneration which is a tta ched to the primary latitude whic h is just ab o v e. W e include a ll the tiles of the low er b order and we include none of the upp er b order. Note that in a b order line, when w e apply the recurs ive pro cess of s ubstitut ion of bo ttoms of triangles starting from a triangle of the gener a tion n , the bases which are the further from the iso cline of the LP ’s are bases of the gener ation 0. They are all on the same iso cline which we ca ll the β - cli ne . Now that the notion o f latitude is clea rly defined, let us lo ok at wha t happ ens betw een tw o consecutive triangles T 1 and T 2 of the sa me generation which b elong to the same latitude. A pr iori, we hav e three situations: ( i ) for b oth T 1 and T 2 , the vertex do es not b elong to a basis of a mauve triangle; ( ii ) the vertex of T 1 do es no t b elong to the basis o f a mauve tr iangle but the vertex o f T 2 do es; ( iii ) ea c h vertex b elongs to a ba sis of a mauve tria ng le. In fact, we hav e: Lemma 5 Consider two mauve triangles T 1 and T 2 of the gener ation n and b elonging to t he same latitu de. Assume that T 1 and T 2 ar e c onse cutive. Then, if the vertex of T i b elong to the b asis of a t ria ngle B i for i ∈ { 1 , 2 } , then B 1 = B 2 . Pro of. Assume that T 1 and T 2 hav e b oth their vertices on a bas is of tria ng les B 1 and B 2 of the same g eneration as s tated in the lemma. Assume that B 1 and B 2 are dis tinct. F ro m what we hav e noted in section 2.1 , a s T i has its vertex on the basis o f a mauve triangle B i , i ∈ { 1 , 2 } , then B i contains the remotest ancestor A i of T i . W e also know that the vertex of A i is at a distance at most the fourth of the heig h t of T i . F rom this, the vertices of A 1 and A 2 are b elow the mid-distance line o f B 1 and B 2 . And so , if we a ssume that B 1 6 = B 2 , even if the vertices of B 1 and B 2 are c lo se to e a c h other, the vertices of A 1 and A 2 are s o far from ea ch other that on the is ocline which joins them there is a nother vertex of a ma uve triangle A 3 in b et ween them. Now, as all mauve triangles genera ted from A 3 exist, there is a ma uv e tr ia ngle T 3 of the same gener ation as T 1 which stands in b et w een T 1 and T 2 . A co n tra diction. And so, B 1 = B 2 . 2.6 The γ -poi n ts and the high p oin ts W e conclude this sec tio n with the notion of γ -p oint and of high p oin t , H P for sho rt, w hich b oth play an imp ortant ro le in the next section. Int uitively , the LP corresp onds to the entry of the path into a tr iangle and the H P co rrespo nds to its exit. The γ - point plays a similar r ole to that of the β -p oin t. The γ -p oint and its cons truc tion The γ -point is defined by the intersection of the leg of mauv e triangle with the β -cline defined by its ha t, if any . The difficult y comes fro m the fa ct that the hat may no t exist while the γ -point can alwa ys defined for a mauve triangle o f a p ositive g eneration. As for the β -p oin t, the γ -point is not defined for a mauve-0 tr ia ngle. F or a mauve triangle T 1 of generation 1, co nsider the a bov e definition when the hat exists. W e remark tha t the β - cline is defined by the basis of the hat as it is a mauve-0 tr ia ngle. Now, the basis of the hat c o n tains vertices o f the 0-tria ngles of generation 0 contained in T 1 . Now, as T 1 exists, its inner 0-triangle s also exist. And so , it is p ossible to define the γ -p oints of T 1 by using its 0-tria ngles only . First, we ca ll firs t p oints , F P for short, the p oint of a leg of a mauve triangle T which is at a distance h 4 from the vertex of T . The co nstruction of the γ -p oint s starts by the de ter mination of the F P ’s which is ea s y: it is not difficult to see tha t the F P ’s of T are the mid-p oin ts of the red tria ngle R whose vertex is the s ame a s that of T . Then, we pro ceed a s follows: t wo γ - signals star t fr o m the F P of T 1 : one to its vertex, along the leg, the o ther ins ide the triangle; the inside sig na l g oes on along the iso cline un til it meets the closest 0-triangle M 0 to this leg of T 1 ; there, it go es up alo ng the leg until it r eac hes the vertex o f M 0 ; the γ -sig nal go es back to the leg of T 1 , following the is ocline of the vertex of M 0 ; the intersection of the γ -signal go ing back to the leg with the γ -signal going up along the leg defined the γ -p oint of T 1 . The justification of this algo rithm r elies on the fact that the iso cline ι which joins the F P ’s of T 1 passes through the L P ’s of the 0-tria ngles which are co n- tained in T 1 . F ro m this fact which is ea sy to establish, and from the intersection prop erties of the mauve triangles, we conclude that a ll o ther mauve triangles which ar e enco un tere d by the iso cline ι are 2-triangle s, a nd that ι passes by the LP ’s of these triangles to o. The general case is not m uch more difficult to establish by the following recursive algorithm. Algorithm 2 The c onstruc tion of the γ -p oint of a triangle T of the gener ation n + 1 . t wo γ -signals star t fro m the F P of T , one a long the leg tow ards the vertex and the s econd inside the tr ia ngle alo ng the iso cline which joins the F P ’s; the ins ide sig nal go es o n until it meets the first 0 -triangle M 0 inside T ; there, meeting M 0 at an LP , it go es up along the leg of M 0 un til it r eac hes the γ -p oint G 0 of M 0 ; there, it go es back to the leg o f T , on the iso cline which pas s es throug h G 0 , cir cum vent ing the inner triangles which it encounters; the intersection of the γ - s ignal going back from G 0 to the leg of T with the γ -signa l climbing along this leg defines the γ -point of T ; the γ -p oint stops b oth γ - signals. The justification of the co nstruction given by a lg orithm 2 is provided by the following lemma. Lemma 6 L et T b e a mauve triangle of the gener ation n +1 . The iso cline which p asses thr ough its F P ’s en c ounters mauve triangles inside T of typ es 0 and 2 only. The me eting o c cu rs at the LP ’s of the inner t ria ngles. The enc ounter e d 0 -triangles b elong t o the gener ation n . The enc ounter e d 2 -triangles b elong to a gener ation i with i < n . Pro of. The conclusion o n the generatio ns o f the inner 0- a nd 2 -triangles enco un- tered b y the iso cline ι which passes thro ug h the F P ’s o f T is a conseq uence o f the fact that the meeting with the legs of the triang les o ccur at their LP ’s, as we hav e seen in lemma 1. Now, the fact that ι mee ts the inner triangles at their LP ’s is easy . The 3-tria ngles of the g eneration n inside T are cut by the basis of T at their LP ’s . Now, the distance in is oclines fro m the basis of T to its F P ’s is 3 h 4 where h is the height o f T . Now, the height of a triangle of the genera- tion n is h 4 . Acco rdingly , for what is the disp osition of the iso clines, ι is in the same p osition with res pect to a 0-tria ngle of the g eneration n as the basis with resp ect to a 3-triang le of the generatio n n . Consequently , inside the 0- triangle of the gener ation n , the is o cline meets inner triangles at their LP ’s and they ar e all of the type 2 as alr eady seen. By synchronization, it is the sa me for the inner triangles, enco un tered by this iso cline. W e hav e an additiona l interesting pr operty: Lemma 7 The iso cline of the γ -p oints of a mauve triangle me ets other mauve triangles at their γ - p oints to o. Pro of. This comes from the structure of the tow er which defines a β -cline . And, by cons tr uction, the iso clines thr ough the γ -p oint s is a β -cline. W e can for m ulate the same remark a bout algor ithm 2 as the o ne which was formulated for alg orithm 1. The construction can als o p erformed in the r e verse order. Aga in we hav e a pr e-signal which detects the exis tence of a containing mauve triangle of the next genera tio n. It is the same signal as pr eviously , lo oking after a bas is at the LP o f a mauve tria ngle of type 3 reached from the consider ed mauve triang le of t ype 0 . If the basis is found, the pre-signa l go es back to its emitting po in t in or der to trigg e r the sig nals of algo rithm 2 in the reverse order. Again, this provides us with an itera tiv e and b ottom-up version of alg orithm 2. The H P F rom the no tion of γ -points, it is easy to define the H P ’s . Indeed, the H P ’s o f a mauve tr iangle T is defined by the following construc- tion. A signa l starts from each F P of T and g oes up alo ng the leg, tow ards the vertex o f T . If ther e is a γ -p oint , then if the β -cline which pass es thro ugh the γ -p oint is a β -cline of t yp e 2 , the H P is the γ -point and the sig nal s tops there. Otherwise, the signa l go es o n climbing along the vertex until it meets the firs t basis which cuts the legs of T if any . If such a basis is encountered, the meeting with the leg s of T define the H P ’s . If not, the H P is the tile o f the leg which is on the iso cline which is just b elow the vertex. This is also the definition o f the H P fo r a mauve-0 tria ngle. 3 An almost plane-filling path Now, we turn to the co ns truction of the pa th. The g e ne r al str ategy which we follow was presented in [15,12], but we s hall ma k e it muc h more precise. The path go es fr om an LP to a H P and then to an LP and so o n. It can b e seen a s a bi-infinite word of the form ∞ (( LP )( H P )) ∞ on the alphabet { LP , H P } . Roughly spea k ing, we fill up a latitude until w e meet legs which cross b oth the upper and the lower b order of the latitude. Then, we go up or down, dep ending on the dire c tion o f the path and into which type o f basic region we fall: the type of a bigg er tria ngle o r of a zone in b etw een tw o bigger triangles. In most cas e s, this strateg y is enough to fill up the whole pla ne. Later, we shall discuss ab out the exceptional cases. 3.1 The regio ns and the path Our fir st task is to define the re g ions which we sha ll inv estigate and then, how the pa th is built o n the basis of what will be c a lled the basic regions . W e have tw o basic re g ions. The first one is the set of tiles defined by a mauve triangle: its b orders and its inside. Remember that the bas is of a mauve triang le contains more than the ma jorit y o f tiles r esulting from the just given definition. It is consider ed as a basic r egion as o nc e the pa th enters a ma uv e tr ia ngle T , it fills up T almost completely befor e leaving T . In fa c t, ther e is a r estriction and the path fills a bigger ar ea. In fact, the path also fills up the space which is contained b etw een the basis of T and the pa r t o f the b order o f the latitude of T which is delimited b y the cor ners of T , the tiles on this b order b eing included. The restrictio n comes fro m the definition o f the latitude: we hav e to withdraw at least the tiles belo nging to the b order of the just upp er latitude of the same generation. An additional res tr iction co mes in the ca s e when the H P is on a n op en β -cline of type 2, a s we shall describ e this later. The other type of a ba sic r egion is defined b y the area in b et ween tw o con- secutive ma uve tria ngles of the same genera tion within the same la titu de. W e alr eady know tha t the just indicated reg ions can b e split into four ho ri- zontal s lices defined by the types of the triangles of the just pr e vious ge ne r ation which are co n tained in thes e regions . Now, if we go from o ne side to ano ther in each slice, a nd if the directions alter na te fro m one slic e to the next one, this even num b er raises a proble m: a pr iori, star ting fr om one side, we go back to the same side. T o solve this pr oblem, we split o ne slice into tw o o nes thanks to the β -cline of type 2: inside a mauve triangle, there is a unique op en β -cline o f t yp e 2 which runs fr o m one le g of the triangle to the other. It is the iso cline of the β -points. This β -cline splits the region o f type 3 into to sub-slices . W e shall use the s econd one to g o ba c k to the o riginal side. As there remain three slic es, we go from the origina l o ne to the o pp osite o ne, as req uir ed. This is the genera l principle for defining the path. Note that this princ iple holds both for triangles an the in betw e en regio n. W e shall no w turn to the precise description. W e shall examine how we fill up the basic regions for generation 0 and w e shall then pr oceed by induction fr o m n to n +1 . In fac t, as we shall s ee, the induction step can b e ba sed o n what is to do for the bas ic reg ions of gener ation 1. F or generatio n 0 The situation o f genera tion 0 is the s implest one. F or a triangle, the tra j ectory o f the path is the following. The path enters the triang le through o ne of its LP ’s, say A . Then, it r uns alo ng the leg of the triangle, down wards, un til it r e ac he s the cor ner. O n this way , the path is in the inside part of the tile which supp o rts the leg. A t the c orner, the path follows the basis, until it reaches the other corner . There, it g oes up alo ng the leg to the next iso cline and there, it go es along the iso cline to the leg o f A . Just b efore reaching the leg , the pa th go es up to the next iso cline and there, it r uns along it un til it reaches the leg, opp osite to A . This ba ck and for th motio n, climbing up by one iso cline each time a leg is reached go es on until the path rea c hes the top of the triangle. There , the pa th exits from the triangle thr o ugh the iso cline − 1 below the vertex or the iso cline − 2, dep ending on the type of the triang le : if the tr iangle is of type 3, the path exists throug h the iso cline − 2 , if not, it exits through the iso cline − 1. The exit B is placed o n this iso cline, on the leg of the triangle which is opp osite to the leg on which A lies. Figure 3 illus tr ates this part of the pa th for a triang le when the topmost iso cline is not o ccupied by ano ther segment o f the pa th. Figure 3 The p ath i nside a triangle of gener ation 0 . F or a part b et ween tw o consecutive ma uv e- 0 triangles within the same la ti- tude, we hav e the three situations which result fro m lemma 5. The ea s iest situation is when tw o cons ecutiv e mauve-0 tr iangles hav e their vertices on the basis of the same ma uv e- 0 triangle . In this ca se, we have a similar zig-zag line as in a triangle. Figure 4 illustrates this situatio n which lo oks like very muc h to what happ ens in a mauve-0 tr iangle. Figure 4 The p ath in b etwe en two c onse cutive triangles of gener ation 0 wi thin the same i so cline. a ( ) c ( ) b ( ) d ( ) Figure 5 A schematic r epr esentation of the p ath: On the left-hand side, inside a mauve-0 triangle. On the right-hand side, in b etwe en two c onse cutive mauve-0 triangles within the same latitude when the vertic es b elong to the same b asis. In or der to desc r ibe what ha ppens in the other situations, we define a sc hematic representation of the zig-zag path of figure 3 a nd 4 in fig ur e 5. Now, as these situation will b e inv o lv ed starting fro m genera tio ns with a p o sitiv e num ber, we po stpone the r epresen tation of the o ther case of basic regions o f gener ation 0 to the s ituation co ncerning genera tion 1. It is important to indicate that the repr esen ta tions o f figur e 3 and 4 are also schematic. In fact, the actual tra jecto r y of the pa th is a bit mor e complex. W e canno t decide that on a leg of a mauve triangle of any genera tion the path strictly go es on the tiles cross e d by the mauve signal and only them. If we do this, we cannot hav e a pa th which go es through a n y tile according to the indicated scenario. Howev er , it is p ossible to s lig h tly change the tra jectory of the path in order to make things p ossible. If the path has to g o a long the leg, there is no problem to organize the ha irpin-turns of the zig-za gs when the pa th is on the left-hand side of the leg. Indeed, the leg is along a line of black no des. There is a ’parallel’ line of white no des which are the left-hand side neighbour s o f the black no des. When the path is on the left-hand side, the organization of the hairpin-turns r equires some care, otherwise, the condition to go thro ugh all tiles once would not b e resp ected. The pa th on the leg follows the fo llo wing pattern. Let α k denote the black no des of the leg, k indicating a n abso lute num b er fo r the is ocline on which the no de is. Let β k be the r igh t- hand s ide neighbour of α k and let γ k be the r igh t-ha nd side neighbo ur of β k . Note that β k is the white son o f α k − 1 and that γ k is the black son o f β k − 1 . F ro m the co nnections be t ween no des o f a Fib onacci in the heptagrid, we know that ther e is no common edge betw een β k − 1 and β k . A solution is the following pa th: go from α k to α k − 1 , then to β k − 1 and then to α k − 2 and fro m there , rep eat the just descr ibed pa tt ern. The pattern of hairpin-turns for an inside path is defined as follows, using the same notations. The path arr ives a t β k from γ k +1 . It g oes to the r igh t from β k on the iso cline k . When it arrives to the leg on the iso cline k − 1, it stops at γ k − 1 from where it g oes to β k − 2 , from wher e it go es back to the right on the iso cline k − 2. See the illus tr ation provided by figure 6. 0 1 2 3 4 5 α β γ Figure 6 The adaption of the p ath close to a le g of a mauve triangle. F or generatio n 1 First, we lo ok a t what happ ens in b et ween t wo consecutive mauve tria ngles of gener ation 1 within the s ame latitude. Denote them by T 1 and T 2 , with T 1 on the left-ha nd side of T 2 . W e shall assume that the path enters a mauve triangle of generation 1 throug h a low p oin t and that it exits the sa me tria ngle through its top, on the leg which is o pposite to the entry p oin t. In fig ure 7, we consider the case when a mauve tr iangle T 1 of gener ation 1 is hatted by a mauve triangle o f gener ation 0. The next mauve triangle of gener a- tion 1 to the r igh t, say T 2 is not hatted a s it can b e e asily concluded from the distance b et ween the corners o f tw o consec utiv e mauve-0 tr iangles. There are necessar ily 0 -triangles o f gener ation 0 in b et w een T 1 and T 2 . Fig - ure 7 illustr ates a schematic situation of the i - triangles of generation 0 whic h we may find in in b etw een T 1 and T 2 . Note that we have 0 -, 1- and 2 - triangles. The 3-triangles are no t r e presen ted as they do not b elong to the la titude of generation 1 defined by T 1 and T 2 . The fig ure illustrates the way o f the path, a ssuming that it exits from T 1 through its r igh t-hand side H P in order to en ter T 2 through its le ft -hand side LP . W e hav e to take into acco un t the b eha viour of the path in the pr imary latitude of H 1 . It again appe ars in the figur e b y lo oking at the config ur ation of the 0- and 1- triangles of generation 0 which are in be t ween T 1 and T 2 . Figure 7 The p ath i n b etwe en two triangles of gener ation 1 . First, the path follows the b order o f H 1 and then climbs along its right-hand leg until it reaches the is ocline which is just b elo w the LP of H 1 . It go es on a long this iso cline until it reaches the left-hand side leg o f T 2 , just b elow the v ertex of T 2 . Next, it follows a zig -zag wa y until it go es back to the p oin t M defined by the cor ner of H 1 . This p oin t M lies on the iso cline ι which is just b elow the basis of H 1 and is on the way up wards taken by the pa th. During the zig- zag, the pa th meets the vertices of 0-triang les of genera tion 0 . As the path inside a triangle never passe s throug h its vertex, the path may cross them, as if it would do if a basis would contain these vertices. Coming back after leaving the closest vertex of such a 0-tria ngle to H 1 and trav eling on the iso cline ι +1, the path arrives to the tile which is befo re the tile of the pa th ab o ve M on ι +1. There, the path go es down to ι a nd, on the tile which is adjacent to M , it go es on the is ocline ι in the direction of T 2 . Now, the path do es not meet T 2 but a 0-tria ngle T 0 of generation 0, w hich it reaches just below the vertex. Accordingly , the path zig- zags down wards, osc illa ting b et ween T 1 and T 0 . By this oscillating motion, the path r eac hes the LP of T 0 : it e nters the tria ngle which it fills according to the motion defined by figure 3. When the path exits from this triangle, it follows the wa y de fined by fig ure 4 until it reaches the nex t 0-tria ngle on its wa y to T 2 . Accordingly , this s equence is re p eated until the last 0-triang le of gener ation 0 befo re T 2 . Now, when the pa th exits fr om the triang le, it is barr ed by the for mer passage of the path on β a nd so the path go es on ι until it reaches T 2 . But, as the path exited from T 1 and as it is clo se to β , it knows that it canno t enter T 2 . And so, it go es downw ar ds and zig- zagging. Now, during this zig-zag , it will meet the LP of the last 0-triang le of gener a tion 0: this LP is clo sed a s the path filled up this tr iangle. W e s hall la ter s ee the mechanism which forces one LP to b e o pen and the other to b e clo sed. And so, going down, still zig-zagg ing, the path will meet the L P of the clos est 1-tr ia ngle of g eneration 0 to T 2 . Here, the LP is free, so that the path enters the tr iangle. Now, w e turn to the route of the path inside a mauve triangle of g e neration 1. Figure 8 The p ath i nside a triangle of gener ation 1 . On the left-hand side: a 0 - or a 1 -triangle. On the right-hand side, a 2 - or a 3 - triangle which is cut by a mauve triangle of a bigger gener ation. In b oth pictures of figure 8, we can see a n o pen β -cline of type 2 which cuts the s trip delimited by the line o f the LP ’s and the basis of the triangle into tw o parts. This is a general featur e . This cut allows to make the path going ba c k nea r the LP throug h which it entered the triangle in order to cr oss the latitude of the 2-triangle s in the direction fro m LP to H P , where LP r efers to the s ide of the triangle thro ug h which the path entered and H P refers to the other s ide as the path will exit through the H P of this other side . The cro ssing of this la titud e inside the triangle ob eys the same principles as in b et ween tw o triangles. When arriving almost to the closed LP , the path go es up to the is ocline which is b elo w the F P . F rom this p oin t, it c rosses the latitude of the 1- triangles, this time in the direction from H P to LP . When it ar riv es to the other side, the path go es up along the leg until it ar r iv e s by o ne iso cline b elow the H P of this leg . F rom there, it cross e s the latitude of the 0-tr ia ngles, in the dire ction fr o m L P to H P . When the cross ing c o mpletes, the path arrives a t the F P from where it go es to the r igh t H P by go ing up a lo ng the leg of the triangle. In b etw een tw o consecutive mauve triang les of generatio n 1 within the sa me latitude, the β -cline 2 which we noticed inside a tria ngle T of g eneration 1 plays a similar role but for a nother latitude: for the o ne which is below the latitude of T . Now, for a basic re gions, there are a lo t of β -clines of type 2 which cro s s the legs of the tria ng les defining these regions. The β - and γ -points tell us which one a re imp o rtan t for the regio n: only tho se which pass through this p oints. The other intersections are not imp ortant. In a ba sic reg ion, we have four sub-la titud es, corr esponding to the four types of mauve triangles o f the previous g eneration. In order to g o in the rig ht directio n, we need to split one such sub-latitude into tw o horizontal ones. The role of the β - a nd γ -p oin ts is to b e the milestones o n the path which indica te where it is po ssible to make this splitting. And so, when the pa th meets a β -cline o f type 2 along a leg, if the p oin t of intersection is neither a β -p oin t nor a γ -one, it knows that it may cros s this β -c line to go o n the zig -zags. In the other cas e, dep ending on which type of po in t is met, the path knows tha t the β -cline m ust b e followed in o rder to c r oss the leg of a triangle. F rom the g eneration n to the gener ation n +1 Figures 7 a nd 8 allow us to prove the induction s tep which allow to establish the path in a basic r egion of the g eneration n +1 once the path is established in any ba sic r egion o f the g eneration n . T o see this po in t, it is enough to start from one o f the figures 7 or 8 . Now, we consider that the big tria ngles b elong to the g eneration n +1 and that the small ones b elong to the genera tion n . Mor eo v er, we ass ume that for the genera tio n n , the path go es from an L P to the o pp osite H P inside a triangle and fro m a H P to the oppo site LP in the region in-b et w een tw o consecutive triangles of the generation n within the same latitude. It is not difficult to see that the same scheme o f the indicated figures a llo w to define the pa th for the ba sic regio ns of the genera tio n n + 1 . Howev er , a tuning is needed here, as the β -clines ar e no more in contact of the ba s es for the ma uv e triangles of the g eneration n +1. T o see this p oin t, co ns ider that we also draw the mauve triangles of the generatio n n − 1, now ass uming that n ≥ 1 . Then, it is not difficult to see that the r egions of the genera tion n split in to regions of the gener ation n − 1 in the same way as those of the generation n +1 split into regions o f the generation n . W e hav e the following gener al pr operty: Lemma 8 L et τ b e a tile of the t ili ng. Then for any non- ne gative n , ther e is a mauve latitude Λ of the gener ation n such that τ ∈ Λ . And then: either τ fal ls within a mauve triangle of gener ation n in this latitude or τ fal ls outside two c onse cu tive mauve triangles of gener ation n and of the latitude Λ and in b et we en them. This pro perty follows immediately from the fact that the latitude o f a mauve triangle exac tly covers that of the cor responding red triangle and the following latitude of red phantoms. 3.2 Additional tuning In o rder to e ns ure the guidance of the path, we pr ovide an additiona l to ol. As indicated in the pr evious section, if o ne L P a llo w s the path to ent er a triangle, the other forbids such a p o ssibilit y . W e hav e the same prop erty for a H P . In fact, it is not difficult to devise sig nals based o n the notion o f la teralit y which allow to ensure this working. It may b e one or the o ther LP , mandatory one of them and never b oth of them. This is p erformed by a signal which runs along the leg s a nd whic h meet at the vertex. E ac h LP sends a signa l to the other which runs a long the leg to the vertex where they meet. If the LP admits the pa th, it sends a signal of its laterality and if not, it sends a sig nal of the other la teralit y . And so, it is enoug h to forbid the meeting of signals of opp osite lateralities. In this way , only unilateral sig nals ar e allowed and they indica te the general motion of the path. Note that once the later alit y is fixed, this allows to place signboards a t appro priate place s. First, the knowledge of which LP is admitting allows to know which H P allows the path to exit fr o m the triangle. This is inside a tria ngle. Now, the same mechanism can b e used to dir ect the path in b et w een tw o consecutive triangles . This time, the information, still g oing from a n L P to ano ther go es thro ugh the co rners a nd takes the route o f the red basis of a pha ntom which runs on the considered iso cline. O n this iso cline there can be corners of the appr opriate generation only . Now, inside a basic region and within a sub- la titude, the direction of the path is the sa me. In fact it is the same all along the latitude, a s ca n b e easily noticed from the fact that there is a shift in the triangles with resp ect with the in b et ween regio ns. Accordingly , the same dir ection o ccurs globally . The c hange of direc tio n happ ens when the path meets the legs of a triangle . This o ccurs for the s tandard hairpins of the zig-zag s . Now, the signal which go es from one LP to the other also allows to place signboar ds at the decisiv e p ositions: the mid-p oin t and the F P ’s, when the path climbs along the leg to go fro m a sub-la titude to the next one. No w, the signal which go es in b et w een tw o conse c utiv e triangles has also to detect the p ossibilit y o f a leg coming from a big ger genera tion: this even t may change the direction of the further motion of the path. F o r this purp ose, the signal circumv ents the mauve triangles it meets on its way . The iso cline of a cor ner contin ues a bas is : accor dingly it meets s ma ller mauve triang les at their L P ’s, the mauve triangles of their g eneration at cor ners ag ain and bigger triangles at v arious place s , except the LP ’s. Accor dingly , when such a meeting o ccurs, the signal knows that it stops here. A last tuning deals with the parity of the num b er of zig -zags in a basic r egion. It is not difficult to notice that the path should a rriv e at pa rticular is o clines in the r igh t direction. As a lready seen, the v arious signboar ds which we hav e constructed allow to do this without problem. As an example, the corners of mauve tr ia ngles play an imp ortant r ole but ther e is not need to signa liz e them: they ar e r ecognizable by their very co nformation which is unique. Now, in orde r that the zig-zag line lea ds a po int on a leg to the opp osite leg, we ne e d an o dd nu mber of zig-za g s. The height of a tria ng le, in terms of iso clines, the basis being included but the vertex b eing ex c luded, is a n even num b er. But, it is not difficult to or g anize one piece of a zig- zag in a given direction on tw o is oclines. It is enough to go up to a no de o f the highest iso cline from its leftmost son, then to g o down to the nex t son fro m the son a nd then to go on until the leftmost son o f the next no de. The need of such a run ca n b e s ig nalized, as the parity o f the num b er of is oclines can easily be computed. I t is enoug h to put signboar ds of the req uir ed p oints three iso clines so oner in or der the path know whether it go es on along a standa rd mo tion or it has to simultaneously cross tw o iso clines on the same motion. 4 Ab out the injectivit y of t he global function of a cellular automaton in the h yp erb olic plane 4.1 Almost filli ng up the plane W e can der iv e tw o cor ollaries fro m lemma 8. The first one is very impo rtan t: Corollary 2 The p ath c ontains n o cycle. Pro of. I there was a cycle, it w ould b e co n tained in so me basic reg ion. Now, the pa th enters the regio n through an L P in ca s e of a triangle, thro ug h a n H P in ca se of an in b et ween region and it exits through the o pposite H P o r L P resp ectiv ely . Accor dingly , there is no cycle in this r egion due to the r ecursive structure of the bas ic regio ns a nd it is plain that there is no cycle in a basic region o f ge ne r ation 0. Another one is also very imp o rtan t: Corollary 3 F or any tile τ , the p ath on one side of τ fil ls up infinitely many mauve t r iangles of incr e asing sizes. Pro of. This a ls o comes from the filling up of the ba sic reg ions and their r ecursiv e structure. The path ca nnot r emain in the same latitude fo r ever. And so, it go es to another latitude, upp e r or low er, dep ending o n the structur e of the underlying mo del implemented by the interw o ven tria ngles. This ensures the conclusion of the co rollary . Another co nsequence is given by the following statement: Corollary 4 If ther e ar e only finite b asic r e gions, t he p ath go es t hr ough any tile of t he plane. Pro of. In this cas e, we hav e a sequence of incr easing and embedded basic regio ns as ea sily follows from the pro of o f coro llaries 2 and 3. 4.2 The exceptional situation Corollar y 4 indica tes that if there are only finite triangles , then we hav e a plane- filling path. Is it p ossible to have infinite tr iangles? The answer is yes: this means that there a re also infinite red triangle s. W e know that this happ ens with the butterfly mo del. No interwo ven triangle c r osses a g iven iso cline 15. As a corolla ry , there is an infinite mauve basis which cro sses infinitely ma ny 2-tr iangles of a n y sizes. Now, this ba sis gives ris e to infinitely many mauve tria ng les, by the very principle of s ync hr onization. And so, this situation is possible. Now, it is the unique one: an infinite triangle has a n infinite ba sis and this ass umption lea ds to what we have just describ ed. In this case, there cannot be a sing le path pas sing thro ug h a ll tiles of the plane once o nly . Indeed, once the path enters an infinite triangle, it c a nnot leav e it. The same for a regio n in b et ween tw o infinite triangles with the vertex o n the infinite basis . And s o, in this ca se, there are infinitely many comp onen ts for the path. How ever, corolla r y 3 is still v a lid for them. 4.3 Pro of of the main theorem W e can now prove: Theorem 2 The inje ctivity of the glob al function of a c el lular automaton on the tern ary heptagrid fr om its lo c al tr ansition function is u nde cidable. The pro of follows the ar gumen t o f [3], with a slight mo dification. In pa rticular, we hav e to bring a new ingredient to the path as describ ed in section 3: we define a dire ction for the path. This can b e introduced by three hues in the colo ur used for the signal of the path. One co lo ur calls the next one and the last o ne calls the first one. The p e riodic rep etition of this pattern toge ther with the order of the co lours define the directio n. This notion of directio n allows to define the successo r of a tile on the path. This can b e formalized by a function δ from Z Z to the tiling such that δ ( n +1) is the successo r of δ ( n ) on the path. Remem be r that in [3], the automato n A T attached to a set o f tiles T has its states in D × { 0 , 1 } × T , where D is the set o f tiles which defines the tiling with the plane filling prop ert y and T is an ar bitrary finite set o f tiles. W e can s till tile the plane as we assume that the tiles o f T are ternar y heptagons but the abutting conditions may b e not obs e rv ed: if it is obser v ed with a ll the neighbo ur s of the cell c , the corres ponding config uration is said to b e correct at c , otherwise it is said i ncorr ect . When the considere d configuratio n is co rrect a t every tile for D or at every tile for T , it is called a realization of the cor responding tiling. Let δ denote the function defining the orientation of the pa th induced by a realiz a tion of D . Here, w e intro duce a difference with [3]. Instead of considering any finite set of tiles, we consider the family { T M } of finite sets o f prototiles defined in [14] wher e M runs ov er the s et of deterministic T ur ing machines with a single head a nd a single bi-infinite ta pe starting their computation fro m the e mpt y tap e. As in [3], the tr ansition function do es no t change neither the D - nor the T -comp onen t o f the state o f a cell c : it only changes its { 0 , 1 } -component x ( c ) which is later on called the bit of c . As in [3 ], we define A T M ( x ( c )) = x ( c ) if the configuratio n in D or in T is inco rrect a t the considered tile. If bo th are cor rect, we define A T M ( x ( c )) = xor( x ( c ) , x ( δ ( c ))). I t is plain that if M does not ha lt, T M tiles the hyperb olic plane and there is a configuration of D and o ne of T M which are realizations of the resp ectiv e tilings. Then, the transitio n function co mputes the xor of the bit of a cell and its s uccessor o n the path. Hence, defining all cells with 0 and then all cells with 1 define tw o config ur ations which A T M transform to the same image: the configur ation where all cells have the bit 0. Accor dingly , A T M is not injective. Conv er sely , if A T M is not injective, we hav e tw o different co nfigurations c 0 and c 1 for which the image is the same. Hence, there is a cell t at which the configuratio ns differ. Hence, the xor was a pplied, which means tha t D and T are bo th cor rect at this cell in these configur ations a nd it is not difficult to see tha t the v alue for each configur ation at the success o r of t on the path must also b e different. And so, following the path in one direction, we hav e a corr ect tiling for b oth D and T M . Now, from corolla ry 3, a s the path fills up infinitely many triangles of increa s ing sizes, this mea ns that the tiling rea lized fo r T M is cor rect in thes e triang les. In particular , the T uring machine M nev e r ha lts. And so, we prov ed that A T M is not injective if a nd o nly if M do es not halt. According ly , the injectivit y of A T M is undecidable. 5 Conclusion The question o f the surjectivity of the globa l function o f cellular automa ta in the h ype rbolic plane is still op en. In the E uclidean case, the undecidability of the sur jectivit y problem is derived fro m the undecida bilit y of the injectivity a s the surjectivity o f the global function of a c e llula r auto maton is equiv a len t to its injectivity on the set of finite configura tions, see [19,20]. Now, in the case of cellula r automata in the h yper bolic plane, this is not at all the case. 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