The metallic ratios as limits of complex valued transformations

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1 Available online at Chaos, Solitons and Fractals 4 (009) 3 The metallic ratios as limits of comlex valued transformations Sergio Falcón *,Ángel Plaza Deartment of Mathematics, University of Las Palmas de Gran Canaria, 3507-Las Palmas de Gran Canaria, Sain Acceted November 007 Abstract We study the resence of the metallic ratios as limits of two comlex valued transformations. These comlex variable functions are introduced and related with the two geometric antecedents for each triangle in a articular triangle artition, the four-triangle longest-edge (4TLE) artition. In this way, the fractality of a geometric diagram for the classes of dissimilar generated triangles is also exlained. Ó 007 Elsevier Ltd. All rights reserved.. Introduction One of the simlest and more studied integer sequence is the Fibonacci sequence [ 9] ff n g n¼0 ¼f0; ; ; ; 3; 5;...g wherein each term is the sum of the two receding terms, beginning with the values F 0 ¼ 0; and F ¼. On the other hand the ratio of two consecutive Fibonacci numbers converges to the golden mean, or golden ratio, / ¼ ffiffi þ 5, which aears in modern research in many fields from Architecture, Nature and Art [0 7] to theoretical hysics [8 4]. For instance, El Naschie rooses a quantum golden field theory (QGFT), based on the golden mean binary. In this theory the classical quantum field theory is included [5]. The aer resented here was originated for the astonishing resence not only of the golden ratio but also of the general metallic ratios [] in a recursive artition of triangles in the context of the finite element method and triangular refinements... Grid generation and triangles Grid generation and, in articular, the construction of quality grids is a major issue in both geometric modeling and engineering analysis [6 9]. Many of these methods emloy forms of local and global triangle subdivision and seek to maintain well shaed triangles. The four-triangle longest-edge (4TLE) artition is constructed by joining the midoint of the longest edge to the oosite vertex and to the midoints of the two remaining edges [30,3]. The two subtriangles with edges coincident with the longest edge of the arent are similar to the arent. The remaining two subtriangles form a similar air that, in general, are not similar to the arent triangle. We refer to such new triangle shaes as dissimilar to those receding. The iterative artition of obtuse triangles systematically imroves the triangles * Corresonding author. Tel ; fax address sfalcon@dma.ulgc.es (S. Falcón) /$ - see front matter Ó 007 Elsevier Ltd. All rights reserved. doi0.06/j.chaos

2 S. Falcón, Á. Plaza / Chaos, Solitons and Fractals 4 (009) 3 in the sense that the sequence of smallest angles monotonically increases, while the sequence of largest angles monotonically decreases in an amount (at least) equal to the smallest angle of each iteration [7,3]. In this aer, we use the concet of antecedent of a (normalized) triangle to deduce a air of comlex variable functions. These functions, in matrix form, allow us to study the fractality of a geometric diagram which arises at studying the sequence of dissimilar triangles generated by iterative alication of the 4TLE artition to any initial triangle [9].. Normalized triangles, antecedents and comlex valued functions Since we are interested in the shae of the triangles, each triangle is scaled to have the longest edge of unit length. In this form, each triangle is reresented for the three vertices ð0; 0Þ; ð; 0Þ and z ¼ðx; yþ. Since the two first vertices are the extreme oints of the longest edge, the third vertex is located inside two bounding exterior circular arcs of unit radius, as shown in Fig.. In the following, for any triangle t, the edges and angles will be resectively denoted in decreasing order r P r P r 3, and c P b P a. Definition. The longest-edge (LE) artition of a triangle t 0 is obtained by joining the midoint of the longest edge of t 0 with the oosite vertex (Fig. a). The four-triangle longest-edge (4TLE) artition is obtained by joining the midoint of the longest edge to the oosite vertex and to the midoints of the two remaining edges (see Fig. b). In the 4TLE scheme, subdivision leads to subtriangles that are similar to some revious arent triangles in the refinement tree so generated. Other subtriangles may result that are not in such similarity classes yet and we refer to these as new dissimilar triangles. We define the class C n as the set of triangles for which the alication of the 4TLE artition roduces exactly n dissimilar triangles. Let us begin by describing a Monte Carlo comutational exeriment used to visually distinguish the classes of triangles by the number of dissimilar triangles generated by the 4TLE artition. We roceed as follows () Select a oint within the maing domain comrised by the horizontal segment and by the two bounding exterior circular arcs. This oint ðx; yþ defines the aex of a target triangle. () For this selected triangle, 4TLE refinement is successively alied as long as a new dissimilar triangle aears. This means that we recursively aly 4TLE and sto when the shaes of new generated triangles are the same as those already generated in revious refinement stes. (3) The number of such refinements to reach termination defines the number of dissimilar triangles associated with the initial triangle and this numerical value is assigned to the initial oint ðx; yþ chosen. (4) This rocess is rogressively alied to a large samle of triangles (oints) uniformly distributed over the domain. (5) Finally, we grah the resective values of dissimilar triangles in a corresonding color ma to obtain the result in Fig. 3. Definition. (4TLE left and right antecedents) A given triangle t nþ, has two (different) triangles t n, denoted here as left and right antecedents, whose 4TLE artition roduces triangle t nþ. As an examle, triangle t nþ in the diagram with vertices ð0; 0Þ; ð; 0Þ and z in Fig. 4a, has left antecedent t n with vertices z; ð0; 0Þ; and z þ infig. 4b, and right antecedent t n with vertices z; ð; 0Þ; and z infig. 4c. z=(a,b) γ 0 α0 β 0 (0,0) (0.5,0) (,0) Fig.. Diagram for reresenting shae triangles.

3 S. Falcón, Á. Plaza / Chaos, Solitons and Fractals 4 (009) 3 3 a r3 β 0 γ 0 r r α 0 b A C t 0 t t t0 B Fig.. (a) LE artition of triangle t 0, (b) 4TLE artition of triangle t Fig. 3. Subregions for dissimilar triangle classes generated by Monte Carlo simulation. t n+ (0,0) t n+ (,0) (a) Triangle t n+ t n+ (0,0) (,0) (0,0) (,0) (b) Left antecedent t n (c) ight antecedent t n Fig. 4. Two antecedents for the 4TLE artition of triangle t n. Theorem 3. The relation between the aex of a given triangle z in the right half of the diagram and the aices of its left and right antecedents may be mathematically exressed by the mas f L ðzþ ¼ zþ ; and f ðzþ ¼ z, comlex z. (See Figs. 5,6 where transformations f L (z) and f (z) are deduced). emark. Notice that for z having 6 eðzþ 6 then eðf LðzÞÞ 6 eðf ðzþþ (and hence the left / right terminology given to these comlex functions). Theorem 4. The class searators determined exerimentally in Fig. 3 may be generated mathematically as a recursive comosition of left and right mas f L ðzþ, and f ðzþ. Function f is a Moebius transformation (also homograhy or fractional linear transformation) [3,33], while function f L is an anti-homograhy. Both transformations may be considered as mas of the comleted comlex lane C ¼ C [ fg onto itself. f is a conformal ma, and hence it reserves angles in magnitude and direction, and straight

4 4 S. Falcón, Á. Plaza / Chaos, Solitons and Fractals 4 (009) 3 lines and circles are transformed into straight lines and circles. On the other hand, f L is not conformal, but angles are reserved in magnitude and reversed in direction, as the comlex conjugation. Also f L takes circles to circles (straight lines count as circles of infinite radius) (see Fig. 5). A general Moebius transformation hðzþ ¼ azþb a b can be defined by the matrix H ¼ ; whose elements are constant real numbers and its determinant is not null to avoid the constant transformation. In a similar way, an anti- czþd c d homograhy resents the form hðzþ ¼ azþb and also has the same associated real matrix. czþd Moebius transformation f is obtained by the ordered alication of an inversion f ðzþ ¼, a rotation z f ðzþ ¼e i z ¼ z and a translation f 3 ðzþ ¼z. So, f ðzþ ¼ðf 3 f f ÞðzÞ ¼f 3 ðf ðf ðzþþþ. Therefore, if we note by the matrix associated to f and by i the resective matrix associated to transformation f i, then ¼ 3 (see Fig. 6) 0 ¼ ¼ Anti-homograhy f L results as the roduct of the inversion g 3 ðzþ ¼, symmetry with resect to real axis g z ðzþ ¼z and translation g ðzþ ¼zþ. So, by noting with L, and L i the matrices, resectively associated to f L and g i for i ¼ ; ; 3, we have L = L 3 L L, and so, L ¼ 0 ¼ Theorem 5. () Both transformations f and f L transform real numbers in real numbers. () Both transformations transform uer half lane (and lower half lane) into itself. (3) If jimpj > then jimf ðpþj < jimpj, for f ¼ f and f ¼ f L, so in this case f ðpþ is closer to the real axis than P. Proof. () It is trivial. () Let P be a oint on the uer half lane P ¼ a þ bi with b 0 and consider function f. Then f ðpþ ¼ a bi ¼ aþbi b, and Imf ð aþ þb ðpþ ¼ with the same sign that ImP, sop and f ð aþ þb ðpþ are in the same half lane. b (3) If jimpj ¼jbj >, then jimf ðpþj ¼ j j 6 ð aþ þb jbj <. Note that Imf b LðPÞ ¼. h ðþaþ þb Fig. 5. Geometric transformations to reconstruct the left antecedent t n of t nþ f L ðzþ for eðzþ P.

5 S. Falcón, Á. Plaza / Chaos, Solitons and Fractals 4 (009) 3 5 Fig. 6. Geometric transformations to reconstruct the right antecedent t n of t nþ f ðzþ for eðzþ P. As a consequence, observe that the iterative alication of functions f and f L in any order converges to a real number. Theorem 6. The image by f of the second quadrant C ¼fz C; eðzþ 6 0; ImðzÞ P 0g is the half circle fz C; z 6 þ eih ; 0 6 h 6 g. Proof. Let P ¼ a þ bi be a oint in C. Then f ðpþ ¼ aþbi, and if d denotes the distance from f ð aþ þb ðpþ to oint ð=; 0Þ, then d a ¼ð ð aþ þb Þ b þð Þ ¼ a þ ¼ a þb 6. h ð aþ þb ð aþ þb 4 4 ð aþ þb 4 Corollary 7. The image of the iterated alication of f to the comleted comlex lane C is the circle fz C; z 6 þ eih ; 0 6 h 6 g. Theorem 8. f ðpþ ¼f L ðpþ if and only if eðpþ ¼. Proof. If P ¼ þ bi then f ðpþ ¼ ¼ f 3 bi LðPÞ. ecirocally, let P ¼ a þ bi and suose that f ðpþ ¼f L ðpþ. Then therefore, a ¼ þ a, soa ¼. h a þ ð aþ þb b ð aþ þb i ¼ aþ þ ðþaþ þb b ðþaþ þb i, and.. Fixed oints We study here the fixed oints of transformations f and f L, and, as a consequence, of any combination of these functions. Note that every Moebius function of the form f ðzþ ¼ azþb has at most two fixed oints since equation f ðzþ ¼z is czþd equivalent to a second degree equation. We call attractive fixed oint to the fixed oint of Moebius transform f ðzþ which is limit of the sequence ff n ðzþg nn for any initial comlex z. In other case, the fixed oint of f will be called reulsive. However, one of the fixed oints of the anti-homograhy f L has the articularity that if P is a oint in a neighborhood of the fixed oint, then the sequence ff n ðpþg nn converges to it but it does ivoting around it. That is, aroaching to it and going far away, alternatively. For this reason, this fixed oint will be called isolated fixed oint, since it is only the limit of the constant sequence. In conclusion, one of the fixed oints of transformation f L is isolated and the other one is attractive as it will be shown later on. A transformation of the comlex lane is arabolic if the determinant of its associated matrix is and the square of its trace is 4. A necessary and sufficient condition for a transformation to be arabolic is that it has a unique fixed oint. In addition, any non-arabolic transformation resents two fixed oints. The transformation is ellitic if its trace verifies trace < 4, and it is hyerbolic if trace > 4.

6 6 S. Falcón, Á. Plaza / Chaos, Solitons and Fractals 4 (009) 3 The Moebius transformation f is arabolic since the determinant of its associated matrix is and its trace is. However, although matrix L also holds these roerties the anti-homograhy f L is not arabolic because it is not a Moebius transformation. In fact, f L is hyerbolic. Theorem 9. z ¼ is the unique fixed oint of f. Proof. Since f ðzþ ¼ it is enough to solve equation z ¼. z z Note that, as a consequence, f is arabolic, and hence the iterative alication of f with any initial oint converges to the fixed oint z ¼. Theorem 0. / ¼ ffiffi þ 5 and / ¼ ffiffi 5 are the two fixed oints of f L. Proof. It is enough to solve the equation ¼ z. / is the well-known golden ratio. h zþ In the next section, we will study the iterated alication of the functions f and f L to any oint of the comlex lane C. We will rove that the anti-homograhy f L is related with the classical Fibonacci sequence ff n g. h 3. Transformations f n and f n L We consider here the alications obtained after n comositions of transformations f and f L, that is f n and f n L. Theorem. f n is defined by f n ðzþ ¼ð nþþzþn nzþnþ. Proof. By using the associated matrices, the roof is straightforwardly obtained since the associated matrix to f is n 0 ¼, then n 0 ¼ ¼ n þ n is the associated matrix to f n nþ n, as it can be easily verified by induction. h Corollary. Sequence ff n ðzþg nn converges to the fixed oint z ¼. Proof. It is trivial since for any initial oint z C lim n! ð nþþzþn nzþnþ ¼. h Note that for any n P ; trð n Þ¼ and the ole of ðf Þ n is z ¼ nþ n. Although f L is not a Moebius transformation, ðf L Þ k does, for k ¼ ; ; 3;..., while for odd exonent ðf L Þ kþ is an anti-homograhy. For this reason, the iterative alication of f L will be written as function of variable w, where w ¼ z if n is even, and w ¼ z if n is odd. Theorem 3. The iterative alication of f L is defined by fl n Proof. Since the associated matrix to f L is L ¼ 0 to fl n, as it can be easily verified by induction. h ðzþ ¼F n wþf n, then L n ¼ 0 F nwþf nþ, where F i is the ith Fibonacci number. n ¼ F n F n F n F nþ is the associated matrix Corollary 4. Sequence ffl nðz 0Þg nn converges to the fixed oint z ¼ if the initial value z / 0 /, while fl n ð /Þ ¼ / is an isolated fixed oint, where / ¼ ffiffi þ 5 is the golden ratio. Proof. Let us suose z /, then by taking limits in the exression of f n L lim n!f n L ðzþ ¼ lim n! F n wþf n F nwþf nþ ¼ lim n! F wþ n F n ¼ wþ/ ¼ wþ/ F n F wþ F nþ /wþ/ n F n ¼, where it has been taking into account that lim /ðwþ/þ / k! F k ¼ / F and also lim k þ F k! F k ¼ lim k þ F k k! F k F k ¼ /. And hence, if w þ / 0, then lim n! fl nðzþ ¼ /, while the other fixed oint is lim n!fl n ð /Þ ¼ /. h Note that the iterated alication of transformation f L to any initial oint gives a convergent sequence to its unique fixed oint z 0 ¼. On the other hand, the iterated alication of transformation f to any initial oint gives a sequence of oints with real arts alternating around the fixed oint z 0 ¼ /. F k

7 S. Falcón, Á. Plaza / Chaos, Solitons and Fractals 4 (009) ate of convergence of sequences ff n ðz 0Þg nn and ff n L ðz 0Þg nn Let fa n g be a convergent numerical sequence such that lim n! a n ¼ L. The rate of convergence of fa n g is usually given by the quotient j an L a n j, i.e. the minor this quotient is the faster the sequence converges. L In our case, we choose as initial oint z 0 ¼ / which is the midoint of the two attractive fixed oints of the two transformations f and f L ; and, resectively. In this situation for the sequences ff n / ðz 0Þg nn and ffl nðz 0Þg nn the rates of convergence are f ðf ð / ÞÞ ¼ and f Lðf L ð / ÞÞ ¼ , which imlies that sequence ff L nðz 0Þg nn converges f ð / Þ faster than ff n ðz 0Þg nn does. f L ð / Þ 3.. Other roerties of f n L traceðl n Þ¼F n þ F nþ and its ole is z ¼ F nþ F n. In addition, by the roerty of revious aragrah, and taking into account Cassini s identity ðtracel n Þ ¼ðF n þ F nþ Þ ¼ F n þ F nþ þ F n F nþ ¼ F n þ F nþ þ F n þ ð Þnþ. So ðtracel n Þ > 4 for n ¼ ; ; 3;... and hence these transformations are of hyerbolic tye having two different fixed oints as we already knew Inverse transformations To obtain the inverse transformations of f and f L it is sufficient to find the inverse matrices of the resective associated matrices, having in mind that the multilying factor corresonding to the inverse of the determinant, is not necessary for obtaining the inverse transformation because the associated matrix reresents a quotient in which this multilying factor acts as a multilying factor both for the numerator and for the denominator, and hence, it can be eliminated from the inverse matrix [33]. Therefore, in the sequel, and L are for the matrices associated to the inverse transformations f and fl which are not exactly the same that the inverse of the associated matrices and L. Note, that the fixed oints of a given transformation and its inverse are the same, although their characters are not necessarily equal. In this way, an attractive fixed oint for a transformation may be a reulsive fixed oint for its inverse transformation, and vice versa. In fact, the attractive fixed oint of transformation f L is an isolated fixed oint of f and fl are resectively reresented by the matrices ¼ 0 and recirocally. So, transformations f L ¼. Then, f 0 ðzþ ¼z and f z L ðzþ ¼ zþ. z Following the same argument as before it can be easily roved that (a) f n ðzþ ¼ ðnþþzþn, and therefore, lim nzþn n!f n ðzþ ¼. (b) fl nðzþ ¼ F nþw F n F nwþf n, where w ¼ z if n is even, and w ¼ z if n is odd. Therefore, lim n! fl n ðzþ ¼ /, ifz /. In the next section, se will study the roduct of transformations f and f L and their roerties. L and 4. Product of transformations f and f L It is clear that not only iterative alications of single transformations f and f L may be considered, but also any combination of (finite) roducts of such transformations. This is the focus of this section. Proosition 5. The comosition of transformations f and f L is not commutative, but it is associative. 0 0 Proof. Since the roduct of matrices and L is not commutative because L ¼ ¼, while L ¼ 0 0 ¼ ; then the comosition of transformations f 3 and f L cannot be commutative. On the other hand, since matrix multilication is associative, the same is true for the comosition of three or more transformations f and f L [33].

8 8 S. Falcón, Á. Plaza / Chaos, Solitons and Fractals 4 (009) 3 About the inverse of the roduct of these transformations, it is easy to check that (a) ðf f L Þ ¼ fl (b) f f L fl f. h f, ðf L f Þ ¼ f f L. 4.. Poles of these transformations Given a Moebius transformation or an anti-homograhy, its fixed oints and the corresonding oles to this transformation and to its inverse function are the oosite vertices of a arallelogram [33,34] called characteristic arallelogram of the transformation. Since the fixed oints and also the oles of transformations f and f L and of their integer owers are real numbers, the fixed oints are the extreme oints of a segment in the real axis. The midoint of this segment is also the midoint of the segment which extreme oints are the ole of the transformation and the ole of its inverse. That is, if z 0 and z 0 0 are the fixed oints of one of these transformations, say f, and z is its ole and Z is the ole of its inverse transformation f, then z 0þz 0 0 ¼ zþz. Moreover, since the fixed oints and the oles of these transformations are real numbers, the oles z ; Z are into the segment with extreme oints z 0 and z 0 0 at the same distance from its center. Hence the sum of the fixed oints is equal to the sum of its ole lus the ole of its inverse transformation z 0 þ z 0 0 ¼ z þ Z. Finally, since the normalized matrix of a transformation is A ¼ Z z 0 z 0 0. For examle, it is easy to find that z the fixed oints of transformation f L f L f are ffiffi 3 3, and its ole is 5. Therefore, the associated matrix with determinant is A ¼ 3 5, which is similar to matrix L L ¼ 3, in the sense that both generate the same 5 transformation. Note here that since the roduct of these transformations is not commutative in general, it makes no sense to try to find a formula for certain number of alications f and f L without any order. Instead, below we study the iterated alications of two comositions f k f L and f k L f. 5. Transformations of the tye ðf k f L Þ n The comosition of two of such functions has as associated matrix the ro-duct of the matrices associated to the two initial transformations. Similarly, any articular combination of transformations f and f L is determined by the roduct of the corresonding matrices in the same order. For instance, transformation ðf f L ðzþþ ¼ f ðf L ðzþþ ¼ f ð Þ¼ zþ zþ L ¼ 0 0 ¼ zþ zþ ¼ could be given more easily by the matrix roduct Therefore, from now on, we will substitute the use of the transformations f and f L by the use of the resective associated matrices and L. Let us find the roduct k L associated to the comosition f k f L which will be used below. It is easy to rove that k k þ k ¼ for all k P ; and so k k þ k 0 L ¼ ¼ k. k þ k k þ k k 5.. k-fibonacci numbers k-fibonacci numbers have been recently introduced and studied in different contexts [35 39]. We relate here these numbers with the comlex variable functions f and f L. Definition 6. For any integer number k P, the k-fibonacci sequence, say ff k;n g nn is defined recurrently by F k;0 ¼ 0; F k; ¼ ; and F k;nþ ¼ kf k;n þ F k;n for n P Particular cases of the revious definition are

9 If k ¼, the classical Fibonacci sequence is obtained F 0 ¼ 0; F ¼ ; and F nþ ¼ F n þ F n for n P ff n g nn ¼ f0; ; ; ; 3; 5; 8;...g. If k ¼, the Pell sequence aears P 0 ¼ 0; P ¼ ; and P nþ ¼ P n þ P n for n P fp n g nn ¼ f0; ; ; 5; ; 9; 70;...g. If k ¼ 3, the following sequence aears F 3;0 ¼ 0; F 3; ¼ ; and F 3;nþ ¼ 3F 3;n þ F 3;n for n P ff 3;n g nn ¼f0; ; 3; 0; 33; 09;...g. The relation between matrix k L and the kth Fibonacci sequence is given by the following roosition. Proosition 7. For any integer n P holds ð k LÞ n F k;nþ F k;n F k;n ¼ ðþ F k;nþ F k;n F k;n þ F k;n Proof (By induction). For n ¼ k L ¼ k ¼ F k; F k; F k; ; k F k; F k;0 F k; þ F k;0 since F k;0 ¼ 0; F k; ¼ ; and F k; ¼ k. Let us suose that the formula is true for n ð k LÞ n ¼ F k;n F k;n F k;n F k;n F k;n F k;n þ F k;n S. Falcón, Á. Plaza / Chaos, Solitons and Fractals 4 (009) 3 9 Then ð k LÞ n ¼ð k LÞ n ð k LÞ ¼ F k;n F k;n F k;n k F k;n F k;n F k;n þ F k;n k ¼ ðk ÞF k;n þ F k;n F k;n F k;nþ F k;n F k;n ¼ kf k;n F k;n þ F k;n F k;nþ F k;n F k;n þ F k;n ecall that last relation means that ðf k f L Þ n ðf k;nþ F k;n Þw þ F k;n ðzþ ¼ ; ðf k;nþ F k;n Þw þ F k;n þ F k;n where w ¼ z if n is odd, and w ¼ z if n is even. 5.. Fixed oints and ole of transformation f k f L For obtaining the fixed oints of this transformation it is enough to calculate lim n! ðf k f L Þ n ðzþ or, equivalently, ðf k;nþ F k;n ÞwþF k;n F lim n! ðf k;nþ F k;n ÞwþF k;n þf k;n. Note that lim k;nþr n! F k;n ¼ r r, where r is the ositive characteristic root of the olynomial ffiffiffiffiffiffiffi associated to the recurrence relation, r ¼ kþ k þ4 [35 37]. Then by dividing numerator and denominator of the revious limit between F k;n, and having in mind that the fixed oints of these transformations are real numbers, it is obtained lim n! ðf k f L Þ n ðzþ ¼ ðr rþwþr ¼ rððr ÞwþÞ ¼ r in the case z r, while if z ¼ r, then ðr Þwþrþ ðrþþððr ÞwþÞ rþ lim n! ðf k f L Þ n ð rþ ¼ r and therefore, z ¼ r is the unique attractive fixed oint. Note that the ole of each transformation ðf k f L Þ n is recisely zðk; nþ ¼ ð þ F k;n k F k;n Þ which tends to the nonattractive fixed oint for n!. Particular cases are If k ¼, the classical Fibonacci sequence is obtained F 0 ¼ 0; F ¼ ; and F nþ ¼ F n þ F n for n P, and therefore, the associated matrix to the transformation ðf k f L Þ n ¼ fl n is Ln ¼ F n F n, which underlines the relation between anti-homograhy f L and the classical Fibonacci sequence. Note that matrix L n is very similar F n F nþ

10 0 S. Falcón, Á. Plaza / Chaos, Solitons and Fractals 4 (009) 3 to the Fibonacci matrix Q defined by Q ¼ F F, and then Q n ¼ F nþ F n [4]. F F 0 F n F n The two fixed oints are z and z, being z ¼ / ¼ / ¼ ¼ /þ / / r ¼ ffiffi 5 which is the limit oint of the sequence ffl nðzþg nn if z / and hence z is attractive; z ¼ ¼ r / if z ¼ / which is the reulsive fixed oint. If k ¼, the Pell sequence aears P 0 ¼ 0; P ¼ ; and P nþ ¼ P n þ P n for n P. The associated matrix is ðlþ n ¼ P n þ P n P n. Its fixed oints are z P n P n þ P ¼ r ¼ þ ffiffi ffiffi r nþ ¼ ffiffi þ þ and z ¼ r ¼ ffiffi, where r ¼ þ ffiffi is the silver ratio []. Note that sequence fðf f L Þ n ðzþg nn converges to z for all z r and it converges to z if z ¼ r. The ole of each iteration is zðnþ ¼ ð þ P n P n Þ and the sequence of oles results f ; 3 ; 7 7 ; ;...! ffiffi g. If k ¼ 3, the following sequence aears F 3;0 ¼ 0; F 3; ¼ ; and F 3;nþ ¼ 3F 3;n þ F 3;n for n P ff n g nn ¼f0; ; 3; 0; 33; 09;...g. Let r 3 denote the bronze ratio r 3 ¼ ffiffiffi 3þ 3 ffiffiffi []. The fixed oints in this case are r 3 3, and therefore, the sequence fðf 6 f LÞ n ffiffiffi þ 3 ðzþg nn converges to. 6 r 3 þ ¼ þ Any recursive sequence obtained by iterative alication of any combination of transformations f and f L will converge to some real number which will be between the limit of sequence ffl nðzþg nn ;, and the limit of sequence / ff nðzþg nn ;. 6. Transformation of the tye ðf L f k Þ n The associated matrix to this comosition can be obtained as follows. In the revious section we have already obtained that ð k LÞ nþ ¼ F k;nþ F k;nþ F k;nþ kf k;nþ F k;nþ þ F k;n Consequently k ðl k Þ n L ¼ F k;nþ F k;nþ F k;nþ kf k;nþ F k;nþ þ F k;n And, hence ðl k Þ n ¼ð k Þ F k;nþ F k;nþ F k;nþ L kf k;nþ F k;nþ þ F k;n And, therefore ðl k Þ n k k þ F k;nþ F k;nþ F k;nþ ¼ ; k k þ kf k;nþ F k;nþ þ F k;n 0 ðl k Þ n ¼ F k;nþ ðk ÞF k;n kf k;n ð3 kþf k;n F k;nþ þðk ÞF k;n For obtaining the fixed oints it is sufficient to solve the equation ½F k;nþ ðk ÞF k;n Šw þ kf k;n ¼ z; ð3 kþf k;n w þ F k;nþ þðk ÞF k;n considering that the fixed oints will not deend on n. Since the fixed oints are real numbers, this yields to the following equation ½F k;nþ ðk ÞF k;n Šx þ kf k;n ¼ð3 kþf k;n x þ½f k;nþ þðk ÞF k;n Šx from which we obtain the fixed oints. It is worthy to be noted that the sequence of the numerators of sequence fzðnþg without sign is referenced in [40] as A00333, and the sequence of denominators is referenced as A0545.

11 S. Falcón, Á. Plaza / Chaos, Solitons and Fractals 4 (009) 3 Table Symbolic sequences, associated matrices, fixed oint and oles Symbolic sequence Matrix Attractive fixed oint n n þ n n nþ L n F n F n F n F nþ ð k LÞ n F k;nþ F k;n F k;n F k;nþ F k;n F k;n þ F k;n ðl k Þ n F k;nþ ðk ÞF k;n ð3 kþf k;n kf k;n F k;nþ þðk ÞF k;n ðlþ n F 3;n F 3;n 3F 3;n F 3;n 4F 3;n þ F 3;n ðllþ n n n n, where a 3a n a n a n ¼ sn s n n a a a being s ¼ þ ffiffiffi 3 sþs ffiffi 3 Isolated fixed oint Pole þ n / / F F nþ n rk rkþ r k k þ F k;n F k;n k þr k k 3 ffiffiffi 5 3 k rk k 3 k 3 ffiffiffi 4 þ F 3;n 5þ 3 F k;nþ ðk þ F Þ k;n F 3;n 3 ffiffi þ an an ffiffiffiffiffiffiffi zðkþ ¼ 3k k þ4, being the attractive fixed oint the ositive value of zðkþ ¼ k þr k ð3 kþ k 3 zðkþ ¼ k r k. k 3 In addition, the ole of this transformation corresonds to the real number zðk; nþ ¼ k 3 k þ F k;nþ ; F k;n which yields to the non-attractive fixed oint when n!. Particular cases are and reulsive fixed oint If k ¼, we have the matrix L n ¼ F n F n, associated to the transformation fl n, where F n is the nth classical Fibonacci number. Its fixed F n F nþ oints are ¼ ffiffi þ 5 (attractive) and / ¼ ffiffi þ 5 (reulsive). The oles are the real numbers zðnþ ¼ F nþ / F n, which of different values of n gives the sequence f ; ; 3 ; 5 ; 8 ;...! /g. 3 5 If k ¼, the following matrix aears ðl Þ n ¼ P n þ P n P n, where P P n P nþ þ P n is for the nth classical Pell number. Its fixed oints are z ¼ ffiffi n (attractive) and z ¼ þ ffiffi (reulsive). The oles are zðnþ ¼ þ P nþ P n and the sequence of oles results f3; 7 ; 7 ; 4 ;...! þ r 5 ¼ þ ffiffi g. 3 Evidently, the fixed oints of these transformations are recisely the fixed oints of their inverse alications, and also the midoint of the fixed oints coincides with the midoint of the oles of the transform and the corresonding inverse transform. 7. Conclusions With the aim to exlain a geometrical diagram we have studied in this aer different combinations of two comlex functions, an homograhy and anti-homograhy. The same rocess can be alied to similar functions that can aear The sequence of numerators (without sign) is the classical Fibonacci sequence A in [40]. 3 The sequence of numerators is the sequence A00333 in [40].

12 S. Falcón, Á. Plaza / Chaos, Solitons and Fractals 4 (009) 3 in other scenarios. This study has been motivated by the arising of two comlex valued mas to reresent the two antecedents in an secific four-triangle artition. Acknowledgement This work has been suorted in art by CICYT Project number MTM C0-0 from Ministerio de Educación y Ciencia of Sain. Annex. Examles Here we show some examles of functions defined by a few combinations of transformations f and f L. Table shows several functions of this tye, the associated matrix, its corresonding fixed oints and the ole. eferences [] Hoggat VE. Fibonacci and Lucas numbers. Palo Alto (CA) Houghton-Mifflin; 969. [] Livio M. The golden ratio the story of Phi, the world s most astonishing number. New York Broadway Books; 00. [3] Vajda S. Fibonacci and Lucas numbers, and the golden section. Theory and alications. Ellis Horwood Limited; 989. [4] Gould HW. A history of the Fibonacci Q-matrix and a higher-dimensional roblem. Fibonacci Quart 98; [5] Kalman D, Mena. The Fibonacci numbers exosed. Math Mag 003; [6] Stakhov A. The golden section in the measurement theory. Comut Math Al 989;7(46) [7] Stakhov A. The generalized rincile of the golden section and its alications in mathematics, science, and engineering. Chaos, Solitons & Fractals 005; [8] Stakhov A. The generalized golden roortions, a new theory of real numbers, and ternary mirror-symmetrical arithmetic. Chaos, Solitons & Fractals 007;33() [9] Kilic E. The Binet formula, sums and reresentations of generalized Fibonacci -numbers. Eur J Combin, 007. doi0.06/ j.ejc [0] Sinadel VW. In Williams Kim, editor. The metallic means and design. Nexus II architecture and mathematics. Edizioni dell Erba; 998. [] Sinadel VW. The family of metallic means. Vis Math 999;(3). Available from htt//members.triod.com/vismath. [] Sinadel VW. The metallic means family and forbidden symmetries. Int Math J 00;(3) [3] Casar DLD, Fontano E. Five-fold symmetry in crystalline quasicrystal lattices. Proc Natl Acad Sci USA 996;93(5)47 8. [4] Greller AM, Matzke EB. Organogenesis, aestivation, and anthesis in the flower of Lilium tigrinum. Bot Gaz 970;3(4)304. [5] Kirchoff BK, utishauser. The hyllotaxy of Costus (Costaceae). Bot Gaz 990;5() [6] Mitchison GJ. Phyllotaxis and the Fibonacci series. Science 977;96(487)70 5. New Series. [7] Stein W. Modeling the evolution of stelar architecture in vascular lants. Int J Plant Sci 993;54()9 63. [8] El Naschie MS. Quantum mechanics and the ossibility of a Cantorian sace time. Chaos, Solitons & Fractals 99; [9] El Naschie MS. On dimensions on Cantor sets related systems. Chaos, Solitons & Fractals 993; [0] El Naschie MS. Statistical geometry of a cantor discretum and semiconductors. Comut Math Al 995;9()03 0. [] El Naschie MS. Deriving the essential features of the standard model from the general theory of relativity. Chaos, Solitons & Fractals 005; [] El Naschie MS. Non-Euclidean sacetime structure and the two-slit exeriment. Chaos, Solitons & Fractals 005;6 6. [3] El Naschie MS. On the cohomology and instantons number in E-infinity Cantorian sacetime. Chaos, Solitons & Fractals 005;63 7. [4] El Naschie MS. Stability analysis of the two-slit exeriment with quantum articles. Chaos, Solitons & Fractals 005;69 4. [5] El Naschie MS. Towards a quantum golden field theory. Int J Nonlinear Sci Numer Simul 007;8(4) [6] Knu PM, Steinberg S. The fundamentals of grid generation. Boca aton (FL) CC Press; 994. [7] Plaza A, Suárez JP, Padrón MA, Falcón S, Amieiro D. Mesh quality imrovement and other roerties in the four-triangles longest-edge artition. Comut Aided Geom Des 004;(4) [8] Plaza A, Suárez JP, Padrón MA. Fractality of refined triangular grids and sace-filling curves. Eng Comut 005;0(4)33 3. [9] Plaza A, Suárez JP, Carey GF. A geometric diagram and hybrid scheme for triangle subdivision. Comut Aided Geom Des 007;4()9 7. [30] ivara MC. Algorithms for refining triangular grids suitable for adative and multigrid techniques. Int J Num Method Eng 984; [3] ivara MC, Iribarren G. The 4-triangles longest-side artition of triangles and linear refinement algorithms. Math Comut 996;65(6)

13 S. Falcón, Á. Plaza / Chaos, Solitons and Fractals 4 (009) 3 3 [3] Marsden JE, Hoffman MJ. Basic comlex analysis. New York WH Freeman; 999. [33] Schwerdtfeger H. Geometry of comlex numbers. New York Dover Publications, Inc.; 979. [34] Jacobsthal E. Ueber die Klasseninvariante ähnlicher Abbildungen. Kon Norske Vid Selskab Forhandlinger 95;Part I(5)9 4. [35] Falcón S, Plaza Á. On the Fibonacci k-numbers. Chaos, Solitons & Fractals 007;3(5)65 4. [36] Falcón S, Plaza Á. The k-fibonacci sequence and the Pascal -triangle. Chaos, Solitons & Fractals 007;33() [37] Falcón S, Plaza Á. The k-fibonacci hyerbolic functions. Chaos, Solitons & Fractals 006; [38] Falcón S, Plaza Á. On the 3-dimensional k-fibonacci sirals. Chaos, Solitons & Fractals 007; [39] Falcón S, Plaza Á. Onk-Fibonacci sequences and olynomials and their derivatives. Chaos, Solitons & Fractals 007; [40] Sloane NJA. The on-line encycloedia of integer sequences,