Transcription

1 This article was originally ublished in a journal ublished by Elsevier, and the attached coy is rovided by Elsevier for the author s benefit and for the benefit of the author s institution, for non-commercial research and educational use including without limitation use in instruction at your institution, sending it to secific colleagues that you know, and roviding a coy to your institution s administrator All other uses, reroduction and distribution, including without limitation commercial rerints, selling or licensing coies or access, or osting on oen internet sites, your ersonal or institution s website or reository, are rohibited For excetions, ermission may be sought for such use through Elsevier s ermissions site at: htt://wwwelseviercom/locate/ermissionusematerial

2 Chaos, Solitons and Fractals (007) 84 5 wwwelseviercom/locate/chaos Abstract Symmetric fractal trees in three dimensions RM Frongillo a, E Lock b, DA Brown c, * a Cornell University, Deartment of Mathematics, Ithaca, NY 14850, USA b Hamilton College, Deartment of Mathematics, Clinton, NY 1, USA c Ithaca College, Deartment of Mathematics, Ithaca, NY 14850, USA Acceted 10 Aril 00 Communicated by Prof G Iovane In this aer, we classify and describe a method for constructing fractal trees in three dimensions We exlore certain asects of these trees, such as sace-filling and self-contact Ó 00 Published by Elsevier Ltd 1 Introduction A fractal tree can be loosely defined as a trunk and a number of branches that each look like the tree itself, thus creating a self-similar object Often, these aear strikingly similar to real trees, and hence are used frequently as tree models Fractal tree models have also been used in other areas, such as antenna construction [1] and mantle melting [] Many other alications of the structure of fractal trees are found in [5 ] Mathematically, these trees have been studied rimarily by Mandelbrot and Frame [,4], who roved and conjectured roerties of symmetric binary fractal trees in the lane Mandelbrot and Frame looked into lane-filling trees, the shae of the canoy (or ti set), and conditions for self-contacting trees Less work has been done concerning fractal trees in three dimensions Recent work in three dimensions has been initiated by Tancheva, Baker, and Maceli as art of the Research Exerience for Undergraduates rogram at Ithaca College These students develoed a method for constructing and visualizing -branch trees They also made conjectures about the shae of the canoy when a tree has ti-to-ti self-contact We continue to generalize the work of Mandelbrot and Frame to three dimensions We show that the canoy of a tree with ti-to-ti self-contact forms a continuous surface, and give conditions for generating these trees In addition, we discuss skew trees, a new class of fractal trees, and construct sace-filling trees that fill boxes in arbitrarily many dimensions Finally, we mention that the search for scaling that results in ti-to-ti contact involves rather interesting constants, deending, of course, on the branching angles In -dimensional trees with two branchings at angle of /, we see that the required scaling ratio turns out to be 1//, where / ¼ ffiffi 1þ 5 is the Golden Ratio [] We see this same scaling ratio * Corresonding author addresses: rmf5@cornelledu (RM Frongillo), elock@hamiltonedu (E Lock), dabrown@ithacaedu (DA Brown) /$ - see front matter Ó 00 Published by Elsevier Ltd doi:10101/jchaos00040

3 RM Frongillo et al / Chaos, Solitons and Fractals (007) required in -dimensional trees when the branching angle is arccosð 1= ffiffi Þ; see Section 5 The aearance of the Golden Mean in the study of fractals and Hausdorff dimension can be found in several references [10 1] Fractal trees First, let us define a fractal tree more rigorously We can do this with three arameters: Definition 1 (Fractal tree) We denote a -dimensional fractal tree by T ¼ðr; u; bþ, where r is the scaling ratio at each level, u is the angle of rotation in the y z lane, and b is the number of branches With these arameters, we can construct the tree using affine transformations (functions of the form f ð~xþ ¼A~x þ ~ bþ Definition (Tree transformations) For a fractal tree T ¼ðr; u; bþ, the corresonding affine transformations are cosðh j Þ sinðh j Þ T j ð~xþ ¼~t þ r4 sinðh j Þ cosðh j Þ cosðuþ sinðuþ 5~x; sinðuþ cosðuþ where h j =(j 1)/b and 1 j b The trunk ~t of the tree is assumed throughout to be the standard vector 0 7 ~e ¼ 4 0 5: 1 These transformations ma the ti of a branch to the tis of its child branches: they scale the branch ti by r, rotate it about the x-axis by an angle u, rotate it about the z-axis by an angle h j, and finally translate it 1 unit in the z direction (see Fig 1) Skew trees, another class of fractal trees, are constructed in the same manner as the regular fractal tree, but with an additional rotation about the trunk at every level of the tree Definition (Skew tree) We will denote by T skew ¼ðr; u; bþ S a skew tree, which has the same transformations as in Definition excet with h j ¼ ðj 1 Þ=b instead of (j 1)/b With the transformations defined above, we define a ath in the tree which we use to refer to secific arts of the tree Paths are given by sequences of transformations Definition 4 (Path) Given a fractal tree T ¼ðr; u; bþ, aath in T is a string of integers s 1 s s k, where 1 s i b The size of the ath, denoted, is the number of integers, k The ti of, ti(), is defined as T s1 T s T sk ð~e Þ: It should be noted that aths in a tree are constructed somewhat counter-intuitively, since the first integer in the ath corresonds to the outermost transformation, and the last integer to the innermost transformation Fig 1 shows the construction of the ð ; ; Þ tree Observe how each ste contains three exact coies of the revious ste coming out 4 of the new trunk This is the reason for the inverted aths; the transformations ush oints from the trunk toward the tis, so the ti of each ath must corresond be the first transformation Fig 1 The construction of the T ¼ð ; ; Þ fractal tree, and a ath in T 4

4 8 RM Frongillo et al / Chaos, Solitons and Fractals (007) 84 5 The main object of study of this aer is the ti set of a fractal tree, which is defined below, and illustrated in Fig H denotes the sace of comact subsets of R with the Hausdorff metric Definition 5 (Ti set) Let T ¼ðr; u; bþ be a fractal tree The ti set or canoy of T, denoted ti ðtþ, is the attractor of the iterated function system {T 1,,T b } Thus, ti ðtþ is the fixed oint of the function F : H! H, where F(S) =T 1 (S) [[T b (S): F ðtiðtþþ ¼ [b i¼1 T i ðtiðtþþ ¼ tiðtþ: Our rimary concerns regarding the ti set are the conditions under which a tree touches itself at the tis This is the notion of (ti-to-ti) self-contact Definition (Self-contact) If two branches of a tree T intersect, T is said to have self-contact If branches of T intersect only at the tis (ti()=ti(q) for distinct aths ; q T), T is said to have ti-to-ti self-contact Note that by symmetry and self-similarity, if two branches intersect, then the tree self-contacts infinitely many times Similarly, if a tree has ti-to-ti self-contact, the ti set will be connected and form a continuous curve, as we show in the next section Connectedness of a self-contacting canoy Fig The ti set of the ð 1; ; Þ tree, which resembles the Sierinski triangle Here we show that the canoy of a tree with ti-to-ti self-contact is connected, and therefore forms a continuous curve Definition 1 (-Connected) A set S is -connected if for any two oints x,y S there is a finite sequence of oints s 1,s,,s n S such that s 1 = x, s n = y, and ks i s i+1 k < for all 1 i < n Such a finite sequence is called an -chain from x to y Lemma (-Connectedness of contractions) Let f:x! Y be a contractive maing with Lischitz constant r < 1 Then if S X is -connected, f(s) is r-connected Proof Let x,y be any two oints in S ByDefinition 1, we can find an -chain s 1,,s n from x to y such that ks i s i+1 k < for all 1 i < n Since f is a contraction, we then have that kf(s i ) f(s i+1 )k rks i s i+1 k < r, so f(s 1 ),,f(s n )isanr-chain from f(x) tof(y) Since we can ick x,y such that f(x),f(y) f(s) are arbitrary, f(s) must be r-connected h Theorem (Connectedness of the attractor of an IFS) Let A be the attractor of the iterated function system {T 1,,T n }, where each T i is a contractive homeomorhism with Lischitz constant r < 1 Let G be the grah with vertices V={T i } and edges E = {(T i,t j ) T i (A) \ T j (A) 5 ;} Then A is connected if and only if G is connected Proof Assume A is connected Then as each T i is continuous, T i (A) is also connected If we also assume that G is not connected, then there exists a nonemty set S ( V such that there are no edges (s,v) E with s S and v V S Thus, T i (A) \ T j (A)=; for all T i S and T j V S We now define sets C and D as follows:

5 C ¼ [ T i ðaþ; D ¼ [ T is RM Frongillo et al / Chaos, Solitons and Fractals (007) T jv S T j ðaþ: Note that C and D are disjoint, but C [ D = T 1 (A) [[T n (A) =A Since each T i is a homeomorhism and A is closed, we have that each T i (A) is closed, and thus C and D are closed as finite unions of closed sets Thus A must be disconnected, which is a contradiction Assume that A is disconnected but G is connected We can then find disjoint closed sets C,D such that C [ D = AAs C and D are disjoint there must be some nonzero distance between them, d If we let d be the diameter of A, we trivially have that A is d-connected By Lemma, since each T i has a Lischitz constant r < 1, we have that T i (A) isrdconnected We now show that F(A) isrd-connected, where F(A) =T 1 (A) [[T n (A) Let x T i (A) and y T j (A) be any two oints in F(A), and let T a1 ; ; T ak be a ath in G from T a1 ¼ T i to T ak ¼ T j Since T ai \ T aiþ1 is nonemty for all i < k (by definition of G), we can define oints s 1,,s k 1, where each s i T ai \ T aiþ1 We can now build a giant rd-chain from x to y, since there is an rd-chain between x and s 1, between each s i and s i+1, and between s k 1 and y Therefore, F(A) =A is rd-connected We can continue this rocess indefinitely, so that F m (A) =A is r m d-connected Since r < 1 there must be some m for which r m d < d, but this is a contradiction since C and D are searated by d h Theorem 4 (Attractors and curves) Let A be the attractor of an IFS {T 1,,T n } such that each T i is a contractive homeomorhism with Lischitz constant r < 1 Then A is ath-connected and a continuous image of [0, 1] Proof By Theorem, A is connected We now show that A is locally connected Let x A be arbitrary Since A = T i (A), there must be some i for which x T i (A) We can also find j for which x T i T j (A), since T i (A) = T i T j (A) Continuing this rocess, we obtain an infinite sequence {s i } such that x F(k) for all k, where F ðkþ ¼T s1 T s T sn ðaþ Now, since A H is comact, and thus closed and bounded as a subset of R n, there must be some finite diameter d of A Since each T i have a Lischitz constant r < 1, we now have that F(k) has diameter r k d for all k, and is connected since A is connected Thus, x is in an arbitrarily small connected subset of A, and since x was arbitrary, A is locally connected We now have by the Hahn Mazurkiewicz theorem that A is a continuous image of the interval [0,1] and thus is a curve h The corollary below follows immediately from Theorem 4, since the canoy of a fractal tree is the attractor of the tree s transformations Corollary 5 (Canoies and curves) If a fractal tree T has ti-to-ti self-contact, its canoy ti (T) will be connected, locally connected, and a curve 4 Aroximating conditions for self-contact Given branching angle u, we outline the method used to comute the value of r required for ti-to-ti self-contact in a symmetric fractal tree In other words, our method determines the greatest scaling ratio ossible without a given tree intersecting itself Note that by self-similarity, it is enough to determine the conditions for contact between the first and second original branches (the branches generated by T 1 and T, resectively) By symmetry, the two branches will intersect if and only if the tree created from the first branch crosses the lane bisecting these two branches (see Fig ) To find the correct ratio for a given angle u, we first determine the ath along the branches that moves toward this lane the fastest (as exlained below) Although this ath must follow an infinite number of branches, it is easy to comute u to a secified length Our rogram, written in Matlab, takes u, the number of branches, and the ath size, then searches for the sequence of branch transformations (starting at branch 1) that moves the greatest distance in the direction of the bisecting lane After determining this ath u to a certain length, we can search for the scaling ratio that brings the ti of the ath closest to the lane If our ath length is reasonably long, this should rovide a very recise aroximation for the ratio required for ti-to-ti self-contact in the infinite case This method is useful in comaring the ratios required for ti-to-ti self-contact among trees with different numbers of branches and different u values Generally, we have found that as the number of branches in the tree increases, the ratio decreases As a function of u, the ratio tyically follows a continuous curve This curve changes dramatically at certain values of u, however, which we will discuss in the next section

6 88 RM Frongillo et al / Chaos, Solitons and Fractals (007) 84 5 Fig A to down view of a -branch tree The dotted line reresents the lane bisecting branches 1 and Fig 4 A -branch tree with u = /4 and r = 5507, conditions that should aroximate ti-to-ti self-contact The first 0 transformations of the ath starting at the first branch are drawn in blue, and its symmetric comlement is drawn in red As illustrated, they come very close to connecting at the tis (For interretation of the references to color in this figure legend, the reader is referred to the web version of this article) This method can also be used to find and comare self-contact ratios among skew trees Tyically, these ratios will vary significantly from that of theion-skew counterarts (see Figs 4 and 5) 5 Mathematical results involving conditions for self-contact When looking at the aths giving us ti-to-ti self-contact, three things become aarent First, the same ath often works for a wide range of u values Second, the aths tyically change at critical oints in the r u lot And third, aths tyically end with an infinite sequence of the same transformation These three characteristics are helful in uncovering mathematical exressions for the conditions of self-contact In the four branch case, the ath 11 seems to work for u 17 rad to u 00 During this interval, the r u lot forms a smooth curve Then, when u 00, the ath changes to 14 and the r u lot abrutly changes direction We can find exactly where this change occurs, then, by solving for when the ath 14 brings us closer to the lane bisecting the first two branches than 11 This haens to occur recisely at u ¼ arccosð1 ffiffi Þ This method can be used to find exactly where the r u lot changes in manycases (see Fig and Fig 7) In the four branch case, u ¼ arccosð1 ffiffi Þ haens to give the maximum ratio for ti-to-ti self-contact Trees with a different number of branches also tyically have their maximum ratio at a oint where the ath changes significantly, allowing us to solve for u at this oint

7 RM Frongillo et al / Chaos, Solitons and Fractals (007) Fig 5 An r u lot for,, 4 and 5 branch trees The two branch lot at the to is identical to that found by Mandelbrot and Frame in the lanar case [] Fig An r u lot for skew,, 4 and 5 branch trees In Table 1, we give recise mathematical exressions for the maximum scaling ratios in the, 4 and branch cases Here, we have been able to find closed formulas for the oint at the ti of the infinite ath (that is, the ath of infinite size) When the dot roduct of this oint and the normal to the lane bisecting branches 1 and is equal to zero, this oint will lie on the lane Hence, we can solve for the correct scaling ratio Using this method, we have found r u equations yielding ti-to-ti self-contact for 4 branch, branch, skew and skew 5 branch trees Our equations, along with the aths giving us self-contact, are included in Table for the skew -branch trees; for shorthand we define

8 0 RM Frongillo et al / Chaos, Solitons and Fractals (007) 84 5 a ¼ 1 Sace-filling trees Fig 7 Path changes in the 4-branch tree Note that u ¼ arccosð1 ffiffiffi Þ gives the maximum value of r Table 1 Maximum branching ratios Branches Exression for u Exression for r Aroximation of r arccosð 1 ffiffi Þ, 1ffiffi 0707 arccosð 1 ffiffi Þ 01 4 arccosð1 ffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi Þ 1þ þ4 ffiffi 058 qffiffiffiffiffiffiffiffiffiffiffiffiffi 4 5 arccos ffiffi ffiffiffi ffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ arccosð Þ Table Paths and equations that determine self-contact in branch skew trees u Interval Path Equation determining self-contact ½arccosð ffiffi 1 Þ; arccosð ffiffi 1 ÞŠ 11 8+r (1 + (4 5r)r)+1r (1 + r)cosu +r 4 cos(u)=0 ½arccosð ffiffi 1 Þ; arccosð 1 ÞŠ 11 +r (4 + r)+r cosu =0 ½arccosð 1 Þ; aš r +0r +14r +( 48r +48r +7r )cosu +( 1r +r )cos(u) r cos(u)=0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½a; arccosð 1 ÞŠ 111 þ cos uþ þ7 cos u 54 cosðuþ r ¼ 5þ1 cos u cosðuþ ½arccosð 1 Þ; arccosð ffiffi 5 ÞŠ 11 r ¼ 1 cos u ffiffi arctan ffiffi! 1 7 cos þ ffiffi arctan ffiffi!!! sin : þ To find a fractal tree that fills sace, we must first define exactly what we are looking for Below are definitions of dimension and of sace-filling Definition 1 (Hausdorff dimension) The Hausdorff dimension d of a fractal tree T ¼ðr; u; bþ or T ¼ðr; u; bþ S is given by the formula

9 RM Frongillo et al / Chaos, Solitons and Fractals (007) d ¼ ln b : ln 1 r Definition (Sace-filling) A fractal tree T fills n-dimensional sace if Y n i¼1 ½a i ; b i ŠtiðTÞ for a i < b i, and Q denotes the cartesian roduct These definitions are not as similar as one might think It is not enough for a tree to simly be n-dimensional for it to fill n-dimensional sace; it must also contain a closed box in n-dimensional sace The reason for this is that if the tree overlas itself significantly, the Hausdorff dimension would count many oints more than once (by self-similarity, infinitely many) Fig 8 shows a skew, -branch tree that illustrates this difference The tree T we are looking for is also a skew, -branch fractal tree, with the same scaling ratio of 1= ffiffi, since we want the Hausdorff dimension to be The arameter that we must change is u It turns out that T fills sace when u is a right angle, so T ¼ð1= ffiffi ; ; Þ S T has the following simle transformations: 0 0 r T 1 ð~xþ ¼4 r 0 05~x þ r r T ð~xþ ¼4 r 0 0 5~x þ 4 0 5: 0 r 0 1 Fig shows T and its canoy, which seems to fill u a box To rove this, however, we need to start by roving roerties of a simle set of sequences Definition (The L set) Let L be the set given by the sequence 1 þ 1 þ 1 4 þ with every ossible combination of signs That is, ( ) L ¼ X1 f 1; 1g : n Lemma 4 (Proerties of L) (a) { l l L} =L (b) {1 + l/ l L} [ { 1 + l/ l L} =L Fig 8 A -dimensional tree with arameters ð1= ffiffi ; ; Þ 18 S, and its ti set (to 15 iterations) The tree certainly does not fill -sace, though it may fill a -dimensional surface

10 RM Frongillo et al / Chaos, Solitons and Fractals (007) 84 5 Proof (a) For any l L, l is given by switching the sign of each Thus, l L Since negation is invertible ( ( l)=l for all l L), we have trivially that L ={ l l L} =L (b) Performing the union directly, we have ( ) ( ) f1 þ l=jl Lg[f 1þl=jl Lg ¼ 1 þ X1 n f 1; 1g [ 1 þ X1 n f 1; 1g n¼1 n¼1 ( ) ( ) ¼ r 0 þ X1 f 1; 1g ¼ X1 f 1; 1g ¼ L Though it may not be aarent, L is just a closed interval on the real line Lemma 5 (Equivalent exression of L) L =[,] n¼1 n Proof Let x be any real number in the interval [0,] and let b 0 Æ b 1 b b be a binary reresentation of x We can also write x = s 0 s 1 s s or x ¼ s 0 s 1 s s m 1 s m, where each s i = 00001, and s i ¼ 000 (with an imlicit decimal oint after the first digit of s 0 or s 0 ) Note that it is ossible for s i to be the string consisting of a solitary 1 We now find an element l of L such that l = x For each s i ¼ b J i b J iþ1 b J iþk i set r J i ¼ 1 and r J iþ1 r J iþk i ¼ 1, where K i = J i+1 J i 1 If in the second case, and the reeating zeros start at osition m, set r J m ¼ 1 and r J m þk ¼ 1 for k P 1 We check that these assignments are equivalent at each ste, first for s i : JX iþk i JX iþk i and also for s m : X 1 b n n ¼ 1 0 J i 1 þ 0 þ 0 4 þþ 0 Ki 1 þ 1 ¼ 1 1 ¼ 1 Ki J i Ki n ¼ 1 1 J i ¼ 1 1 ¼ 1 Ki J i Ki ; J iþki b n n¼j m n ¼ X1 0 ¼ 0; We now have the following: x ¼ s 0 s 1 s ¼ X1 X 1 n¼j m n ¼ 1 1 X1 J i n¼1! 1 n ¼ 1 J ð0þ ¼0: i J iþki x ¼ s 0 s m 1 s m ¼ X1 b n n ¼ X1 i¼0 JX iþk i b n n ¼ X1 b n n ¼ Xm 1 i¼0 JX iþk i Fig A sace-filling tree i¼0 JX iþk i b n n ¼ Xm 1 i¼0 n ¼ X1 JX iþk i n ¼ X1 n ¼ l n ¼ l: Clearly, for x [,0], we can use the above construction to find l = x and simly negate every as discussed in Lemma 4a, so that l = x Thus [,] L n h

11 RM Frongillo et al / Chaos, Solitons and Fractals (007) 84 5 Finally, for any l L, jlj ¼ X1 X1 n n ¼ X1 so L [,] Therefore, L =[,] 1 n ¼ ; h With this background in lace, we can show that T fills sace Theorem (Sace-filling tree) Let S * be the ti set of the fractal tree T with u = / and r ¼ 1= ffiffi Then S ¼½ ffiffi ffiffi ffiffi ffiffi 4 ; 4 Š½ ; Š½0; Š: Proof By Definition 5, the ti set S * is defined as the attractor of the iterated function system {T 1,T } This we can check by showing that S * is a fixed oint of the set-valued function F(S)=T 1 (S) [ T (S), or in other words, that F(S * )=S * We first show that 8 rl x S ¼ 4 r 7 l y 5 l x ; l y ; l z L 1 þ r l z ; which we can do directly using Lemma 4: 8 8 rð1 þ r l z Þ rð1 þ r l z Þ F ðs Þ¼ 4 rðrl x Þ 7 5 l a L [ rðrl x Þ l a L ð1þ 1 þ rðr l y Þ 1 þ rðr l y Þ 8 rð1 þ r 8 l x Þ rð 1 r l x Þ ¼ 4 r 7 l y 5 l a L 1 þ r [ r 7 4 l y 5 l a L ðþ l z 1 þ r l z 8 r 1 þ 1 l 8 x r 1 þ 1 l x ¼ 4 r l y 7 5 l a L [ r l y l a L ðþ 1 þ r l z 1 þ r l z 8 rl x ¼ 4 r 7 l y 5 l x ; l y ; l z L 1 þ r l ¼ S ; ð4þ z where () was obtained by renaming, () by Lemma 4a and the definition of r, and (4) by Lemma 4b Finally, by Lemma 5, we have h S ¼½ r; rš½ r ; r Š½1 r ; 1 þ r Š¼ ffiffi ffiffi h 4 ; 4 i ffiffi ffiffii ; ½0; Š: We can also find a tree T n that fills n-dimensional sace First, we must generalize the notion of a fractal tree Definition 7 (Higher dimensional fractal tree) We denote an n-dimensional fractal tree by T ¼ðr; u ; u ; ; u n 1 ; bþ, or () S, where r and b are as defined reviously, and u i is the angle of rotation in the lane of coordinates i and i +1 It may be confusing for {u i } to start with i =, but this is because b still determines the h values, which are rotations in the lane of coordinates 1 and Definition 8 (Generalized tree transformations) For an n-dimensional fractal tree T ¼ðr; u ; u ; ; u n 1 ; bþ, the corresonding affine transformations are T j ð~xþ ¼R 1; ðh j ÞR ; ðu ÞR n 1;n ðu n 1 Þþ~e n where h j =(j 1)/b, orðj 1 Þ=b if T is skew, and 1 j b Using these definitions, we can find a tree T n that fills n-dimensional sace

12 4 RM Frongillo et al / Chaos, Solitons and Fractals (007) 84 5 Theorem (Sace-filling tree in n dimensions) Let T n ¼ð n 1 ffiffi ; ; ; ; Þ S Then tiðt n Þ¼ Yn 1 h ffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffi i! n n i n ; n i ½0; Š: i¼1 Proof First, we must figure out the transformations for T n This is not hard due to all the right angles: R i;iþ1 I i ¼ ; ð5þ 0 0 I n i 1 so we have that T 1 ð~xþ ¼r 0 1 þ ðþ T ð~xþ ¼r þ : ð7þ The roof from this oint on is similar to that of Theorem We again define S n ¼ tiðt nþ and exress S n in terms of elements of L: 8 rl 1 r l S n ¼ l i L : l n þ l n We can again show this directly with Lemma 4: 8 rrð1 þ l n Þ rrðrl 1 Þ rðr l Þ F ðs n Þ¼ l i L; r f 1; 1g 4 rð 7 l n Þ 5 1 þ rð 1 l n 1 Þ 8 r r þ 1 l n r l 1 ¼ l i L; r f 1; 1g l n 5 1 þ l n 1 ð8þ ðþ

13 8 rðl 1 Þ r l ¼ l i L; ¼ S n r f 1; 1g l n þ l n where () was obtained by Lemma 4a and the definition of r, and 4 by Lemma 4b and then renaming the l i We then have, by Lemma 5,! S ¼ Yn 1 ½ r i ; r i Š ½0; Š ¼ Yn 1 h ffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffi i! n n i n ; n i ½0; Š: References i¼1 RM Frongillo et al / Chaos, Solitons and Fractals (007) i¼1 [1] Petko JS, Werner DH Miniature reconfigurable three-dimensional fractal tree antennas IEEE Trans Antennas Proag 004;5(8):145 5 [] Hart SR Equilibration during mantle melting: a fractal tree model Proc Natl Acad Sci USA 1;0(4): [] Mandelbrot BB, Frame M The canoy and shortest ath in a self-contacting fractal tree exactly what tangles the branches can get into Math Intelligencer 1;1():18 [4] Mandelbrot BB The fractal geometry of nature New York: Freeman; 18 (004) [5] Aris R Diffusion and reaction in a Mandelbrot lung Chaos, Solitons & Fractals 11;1():58 [] Sivakumar B Chaos theory in geohysics: ast resent and future Chaos, Solitons & Fractals 004;1():441 [7] Gabryś E, Rybaczuk M, Keßdzia A Fractal models of circulatory system Chaos, Solitons & Fractals 005;4(): [8] Abdusalam HA, Ahmed E Modeling the failure of a fractal tree and Julia sets Chaos, Solitons & Fractals 000;11(1):075 8 [] Abdusalam HA A robabilistic aroach to the site ercolation roblems using fractal tree Chaos, Solitons & Fractals 000;11(): [10] El Naschie MS Silver mean Hausdorff dimension and cantor sets Chaos, Solitons & Fractals 14;4(10): [11] El Naschie MS Dimensions and cantor sectra Chaos, Solitons & Fractals 14;4(11):11 [1] Castro C Fractal strings as an alternative justification for El Naschie s cantorian sacetime and the fine structure constant Chaos, Solitons & Fractals 00;14(): ð10þ