Properties of Pruned, Binary, Planar Fractal Trees

Rita Gnizak Department of Mathematics and Computer Science, Fort Hays State University

1 Big whorls have little whorls, That feed on their velocity; And little whorls have lesser whorls, And so on to viscosity.

Lewis F. Richardson

2 Contents

1 Introduction 5

2 Background 5 2.1 Fractal Trees ...... 5 2.2 The Canopy ...... 7 2.3 Space-ﬁlling Trees ...... 9 2.4 Hausdorﬀ Dimension ...... 9 2.5 Pruned Trees ...... 10

3 Counting Forbidden Words 13

4 Calculating Dimension of Pruned Trees 15

5 Pruning Space-ﬁlling Trees 17

6 Further Study 23

7 Appendix 24 7.1 Appendix I: Mathematica Coding for Pruned Trees ...... 24

3 Abstract Symmetric, planar, binary branching trees have been extensively described by Benoit Mandelbrot and Michael Frame [7], however, little is known about the effects that pruning has on the properties of these trees. Investigation of self-contact, connectedness, fractal dimension, and the space-filling properties of pruned fractal trees has lead us to the creation of a new method for calculating fractal dimension of pruned trees as well as a proof that the space-filling property of the special 90◦ and 135◦ trees is lost when any finite sequence of a transformation is forbidden.

4 1 Introduction

In 1957, Alan W. Watts in his book The Way of Zen wrote, The Tao is a certain kind of order, and this kind of order is not quite what we call order when we arrange everything geometrically in boxes, or in rows. That is a very crude kind of order, but when you look at a plant it is perfectly obvious that the plant has order. We recognize at once that it is not a mess, but it is not symmetrical and it is not geometrical looking. The plant looks like a Chinese drawing, because they appreciated this kind of non-symmetrical order so much that it became an integral aspect of their painting. In the Chinese language this is called li, and the character for li means the marking in jade. It also means the grain in wood and the ﬁber in muscle. We could say, too, that clouds have li, marble has li, the human body has li. We all recognize it, and the artist copies it whether he is a landscape painter, a portrait painter, an abstract painter, or a non-objective painter. They all are trying to express the essence of li. The interesting thing is, that although we all know what it is, there is no way of deﬁning it.

Mathematics now defines li through fractal geometry. Fractal geometry is a discipline in mathematics concerned with infinitely de- tailed, self-similar objects that lack a derivative. Benoit Mandelbrot developed the discipline from studies of chaos involving fluctuations of the stock market, turbulent motion of fluids, and the irregular shorelines of the English coast [4]. He defines a fractal as “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole.” [6] As Mandelbrot’s definition suggests, a fractal tree is a tree whose branches are each a smaller version of the whole tree. Since Mandelbrot coined the term fractal in 1975, he and Michael Frame have extensively studied the properties of these trees, proving theorems and conjecturing on properties of space-filling trees, canopy shapes, and conditions for self-contacting trees [7]. Pruned fractal trees are fractal trees in which particular branches are removed from the symmetric, planar version. Self-contact ratios, fractal dimension, and connectedness are altered by the process of pruning. The implications of these effects on the properties of fractal trees must be investigated to further the understanding and applications of fractals.

2 Background 2.1 Fractal Trees A more rigorous deﬁnition of a symmetric, planar, binary, fractal tree is now described using two parameters, and a set of transformations in repetition.

5 Deﬁnition 1 Fractal Tree A fractal tree is denoted by T (θ, r), where r is the scaling ratio at each level and θ is the angle of rotation, both clock-wise and counter clock-wise, from the linear extension of the trunk.

Figure 1: Construction of a Planar, Binary Fractal Tree

Deﬁnition 2 Tree Transformations For a fractal tree T (θ, r), the corresponding aﬃne transformations are {F1,F2} as follows. x cos θ − sin θ x 0 F = r + 1 y sin θ cos θ y 1 x cos θ sin θ x 0 F = r + 2 y − sin θ cos θ y 1 Note that the trunk, or initial condition, has a length of one.

Deﬁnition 3 Iterated Function System An iterated function system (IFS) is a collection of contractive, aﬃne transformations.

One application of the tree transformations will map the tip of the trunk to each of two branches, the left branch generated by F1 and the right generated by F2. The trunk is of length one for simplicity, so that these ﬁrst two branches will have a length of r and the length of every branch to follow will be ri where i is the number of times the transformations have been applied, or the iteration. Note that to be considered an IFS by Deﬁnition 3, r must satisfy 0 < r < 1 so that contraction of the system is ensured. Notice also that the angle each branch is to be dropped from the linear extension of the trunk must lie within

6 (0, 180◦) to prevent collapse of the tree. Since little is known about asymmetric trees, only the eﬀects of pruning on symmetric trees is explored. That is, the same ratio, r, and angle, θ, are used in both F and F . The result of eight 1 2 iterations of the tree transformations with r = √1 , and θ = 45◦ is seen in 2 Figure 2.

Figure 2: T √1 , 45◦ 2

2.2 The Canopy Definition 4 Address Given a fractal tree T (r, θ) having a branch generated by the following sequence of tree transformations, applied from left to right, 0 F ◦F ◦F ◦...◦F a1 a2 a3 an 1 the address of the tip of the branch is the string of integers a1a2a3 . . . an for n ∈ [1, ∞) where aj ∈ {1, 2}. Each branch tip is inherently addressed by the sequence of tree transformations that generate it. Visually speaking, the address is the string of turns which consist of left, generated by F1 and denoted by a 1, and right, generated by F2 and denoted with a 2, that carry the viewer from the trunk of the fractal tree to the branch tip of discussion. It is important to notice that in Definition 4, the tree transformation Fa1 is the first transformation applied to the trunk, and

Fan is the last transformation applied. This inversion is necessary for ease of reference when addressing branch tips arrived at after numerous or inﬁnitely many iterations. Deﬁnition 5 Canopy The canopy of the fractal tree T (r, θ) is the attractor of the iterated function

7 system F1,F2. That is, the limit of the ﬁnite address via 0 lim Fa ◦Fa ◦Fa ◦ · · · ◦Fa n→∞ 1 2 3 n 1

Figure 3: The approximation of the canopy generated by the tree from Figure 2

Deﬁnition 6 Self Contact If two branches of a tree T (r, θ) intersect, the tree is said to be self-contacting. If only the tips of two distinct branches intersect, the tree is said to have tip-to-tip self-contact.

The tree branches are not actually the fractal, they are only means by which one arrives at the fractal canopy. The canopy itself is the primary area of interest when studying fractals, and attention is especially paid to tip-to-tip, self-contacting trees. Since symmetric fractal trees are inﬁnitely self-similar, one point of contact between two branches ensures inﬁnitely many contact points among branch tips [5]. For a given angle θ, there is one ratio r which ensures self-contact of the binary branching planar tree, as seen in Figure 4 [7].

Figure 4: Relation between θ in degrees and the ratio r necessary for tip-to-tip self-contact [7].

8 2.3 Space-filling Trees In planar, binary branching fractal trees, two special cases exist in which the canopy of the tree fills two-dimensional space [7]. The two peaks in Figure 4 correspond to these two trees, T (90◦, √1 ) and T (135◦, √1 ). These trees have 2 2 infinitely branched limits that fill a rectangle and triangle respectively.

◦ 1 Figure 5: The space-ﬁlling T (90 , 2 ) tree shown at 11 iterations

◦ 1 Figure 6: The space-ﬁlling T (135 , 2 ) tree shown at 15 iterations

2.4 Hausdorff Dimension Fractals are sets that are too complex to be described in Euclidean terms, which is why it is necessary to build trees that describe the relationship of each point to each of the other points in the canopy. A characteristic that aids in describing the complexity of these points, which is standard for many mathematical spaces, is the notion of dimension. Although there are many ways to calculate the dimension of a set, the Haus- dorff dimension (also known as the Hausdorff-Besicovitch dimension) is ideal when studying fractals since it allows for non-integer dimensions, as are necessary for describing fractals, and it coincides with the more familiar integer dimensions of well-behaved sets [1]. The Hausdorff Dimension is often called sim- ply fractal dimension because it distinguishes fractals from non-fractal shapes. For example, the Hausdorff dimension of any countable set is D = 0, a standard line or curve has a Hausdorff Dimension of D = 1, and any n-dimensional Eu-

9 clidean space has a Hausdorﬀ dimension of D = n. However, any object with a non-integer Hausdorﬀ dimension is a fractal [2].

Deﬁnition 7 Hausdorﬀ Dimension Given an IFS {f1, . . . , fn} with scaling ratios r1, . . . , rn, respectively, the self- similarity dimension is found by solving

n X D rj = 1 j=1

For example, the unpruned, symmetric, self-contacting 30◦ tree has a fractal dimension of

D D 1 = r1 + r2 1 = 2(.5734040752444177)D Log[.5] D = Log[.573404075244] D = 1.2463

Using this formula, it can also be shown that the unpruned T (90◦, √1 ) tree is 2 2-dimensional.

D D 1 = r1 + r2 1 D 1 = 2 √ 2 1 Log 2 D = h i Log √1 2 D = 2

Although the Hausdorﬀ Dimension is useful for describing fractal trees, many issues arise when using the equation to generate dimensions of pruned fractal trees. This is discussed further in Section 4.

2.5 Pruned Trees Deﬁnition 8 Pruned Addresses Given an alphabet {1, 2, 3,...N} with N symbols, a word of length k is termed forbidden if it contains the forbidden sequence xy and denoted Fk; otherwise, the word is termed allowable and denoted Ak. Allowable words are further distinguished between vulnerable Vk, and safe Sk words. A word is vulnerable if, upon an additional iteration of the tree transformations, it can give rise to the forbidden sequence; otherwise, it is safe. For example, if 12 is forbidden, then 2221 is vulnerable while 2222 is safe.

10 Deﬁnition 9 Pruned Tree When any sequence of tree transformations is forbidden, the tree is termed pruned and denoted Tf (θ, r) where f is the forbidden sequence. When f is long or inﬁnite, the pruned tree is denoted Tf n (θ, r) where n is the length of the forbidden sequence.

In binary branching fractal trees, the alphabet consists of two letters, 1 and 2, so there are 2k possible words of length k. Each distinct word of length k corresponds to one branch tip at the kth iteration of the IFS. When k is infinite, the sequence f k corresponds to one point in the canopy. If this sequence is pruned, then only one point is removed from the canopy; however, due to the self-similarity of fractals, if k is finite, the forbidden sequence f k prunes an infinite amount of points from the canopy. Notice that when x 6= y, each safe k-word gives rise to N − 1 safe (k + 1)- words and 1 vulnerable (k + 1)-word. Similarly, each vulnerable k-word gives rise to N − 2 safe (k + 1)-words, 1 vulnerable (k + 1)-word, and 1 forbidden (k+1)-word, and each forbidden k-word gives rise to N forbidden (k+1)-words. This will be useful when counting allowable words in Theorem 1. Directed graphs are useful to show which sequences of transformations are allowable given a forbidden sequence xy. The nodes correspond to the tree transformations and the arrow(s) point from a given node to every node which would generate an allowable sequence. Since the unpruned fractal tree is symmetric, forbidding 11 results in the mirror image of the tree with 22 forbidden, which can bee seen in Figure 8 and Figure 9. For this reason, opposite sequences are considered mirror images and only one is considered. Figure 7 is the directed graph corresponding to the forbidden sequence 12 (recall that 12 corresponds to F1 ◦ F2 and not F2 ◦ F1). Directed graphs become useful when generating new equations for calculating fractal dimension as in Section 4.

2 1

Figure 7: Directed Graph of 12 Pruned

11 Figure 8: T (45◦, √1 ) with 11 Pruned 2

Figure 9: T (45◦, √1 ) with 22 Pruned 2

12 3 Counting Forbidden Words

When studying pruned fractal trees, the count of allowable and forbidden branches at a given iteration for a given pruning becomes important in order to understand the eﬀects of pruning on the canopy. As discussed previously, forbidden and allowable words can be distinguished by the presence or absence of a forbidden sequence xy. Although N, the amount of symbols in the alphabet, is 2 for binary branching trees, in the following method for counting allowable words, N is left arbitrary. Theorem 1 The number of allowable k + 1 words when x = y is

Ak+1 = NSk + (N − 1)Sk−1 And when x 6= y, Ak+1 = Sk Proof. Recall from Deﬁnition 8 that

Sk = the number of safe words of length k Vk = the number of vulnerable words of length k Fk = the number of forbidden words of length k

Note that     S1 N − 1  V1  =  1  F1 0 Assume within the forbidden sequence xy, that x 6= y. Then since each safe k-word gives rise to N −1 safe (k +1)-words and 1 vulnerable (k + 1)-word, each vulnerable k-word gives rise to N − 2 safe (k + 1)-words, 1 vulnerable (k + 1)-word, and 1 forbidden (k + 1)-word, and each forbidden k-word gives rise to N forbidden (k + 1)-words,

Sk+1 = (N − 1)Sk + (N − 2)Vk

Vk+1 = Sk + Vk

Fk+1 = Vk + NFk Thus,           Sk+1 N − 1 N − 2 0 Sk S1 N − 1  Vk+1  =  1 1 0   Vk  where  V1  =  1  Fk+1 0 1 N Fk F1 0 Or more compactly,     Sk N − 1 k−1  Vk  = M  1  Fk 0

13 Where  N − 1 N − 2 0  M =  1 1 0  0 1 N When x = y, the matrix becomes,  N − 1 N − 1 0  M =  1 0 0  0 1 N

The number of forbidden k-words is then  N − 1  k−1 [0, 0, 1] · M  1  0 and the number of allowable k-words is  N − 1  k−1 [1, 1, 0] · M  1  0 where  N − 1 N − 1 0   N − 1 N − 2 0  M =  1 0 0  if x = y and M =  1 1 0  if x 6= y. 0 1 N 0 1 N

In the case that x = y,

Sk+1 = (N − 1)Sk + (N − 1)Vk

Vk+1 = Sk

Fk+1 = Vk + NFk

Since Vk = Sk−1, Sk+1 = (N − 1)Sk + (N − 1)Sk−1 Thus, the number of allowable (k + 1)-words is

Ak+1 = Sk+1+Vk+1 = (N−1)Sk+(N−1)Sk−1+Sk = NSk+(N−1)Sk−1 = N(Sk−Sk−1) In the case that x 6= y,

Sk+1 = (N − 1)Sk + (N − 2)Vk

Vk+1 = Sk + Vk

Fk+1 = Vk + NFk Thus, the number of allowable (k + 1)-words is

Ak+1 = Sk+1+Vk+1 = (N−1)Sk+(N−2)Vk+Sk+Vk = NSk+(N−1)Vk = N(Sk−Vk)

14 This method can be used in an identical fashion for forbidden sequences of length 3 and greater; that is, for three, with xxx, xxy, xyy, xyx, and xyz with the matrix M having another level of vulnerability represented in a 4×4 matrix. Since binary fractal trees have an alphabet of only two letters, the process is greatly simpliﬁed.

4 Calculating Dimension of Pruned Trees

Although the Hausdorﬀ Dimension (Deﬁnition 7) cannot be directly applied to calculate the dimension of pruned fractal trees, M. Frame and J. Lanski used the method of Rellick, Edgar, and Klapper to generate the dimension of IFS attractors with a sequence of two letters pruned [3, 9]. Both of these methods, however, fail to generate the dimensions of fractal trees with longer sequences pruned. The dimension of self-contacting trees can be calculated by generating equations from directed graphs and slightly altering the ratio used in the calculation of self-similarity dimension according to the length of the pruned sequence. It is important to notice, however, that since the equations do not rely on the angle θ, the results are only useful for describing trees that can be shown to self- contact and create a connected space.The following example shows the process for T (90◦, √1 ), which self-contacts and generates a connected canopy set [7]. 111 2 In order to generate a directed graph for a higher level pruning than of the example seen in Section 2.5, the nodes of the directed graph must represent all the possibilities of words having a length one less than that of the pruned sequence. The directed graph for the 111 pruning is shown in Figure 10.

11 22

21 12

Figure 10: Directed Graph of 111 Pruned

The directed graph of this pruning then yields the following equations, where A(j) is the set of nodes that are allowed to follow j in sequence.

15 A(11) = {21, 22} A(12) = {11, 12, 21, 22} A(21) = {12, 21, 22} A(22) = {11, 12, 21, 22}

Since A(12) = A(22) the above equations can be reduced to the following, where D is the dimension, t is a power of the ratio, and a represents A(11), b represents A(22), and c represents both A(12) and A(22). The change from the familiar r in the Hausdorﬀ dimension to some power of r, t, is necessary because when the nodes are made to represent a longer sequence, the addresses generated by them skip levels. Therefore, t represents rm−1, where m is the length of the pruned sequence, thus m − 1 is the size of the node.

aD = tbD + tcD (1) bD = tbD + 2cD (2) cD = taD + tbD + 2tcD (3)

The dimension is then obtained by setting the solution to the system of above equations equal to 1, as in the Hausdorﬀ dimension.

bD − tbD = 2tcD from 1 (4) cD = taD + bD from 4& 3 (5) 1 1 aD = tbD + bD − tbD from 1& 4 (6) 2 2 1 1 aD = ( t + )bD from 6 (7) 2 2 1 1 cD = ( t2 + t + 1)bD from (5)&(7) (8) 2 2 1 1 bD = tbD + 2t( t2 + t + 1)bD from (2)&(8) (9) 2 2 1 1 6 bD = t 6 bD + 2t( t2 + t + 1) 6 bD from 6 (10) 2 2 1 = t + t3 + t2 + 2t (11) 0 = t3 + t2 + 3t − 1 (12) 1 1 1 0 = ( )3D + ( )2D + 3( )D − 1 (13) 2 2 2 D = 1.175829 (14)

This process can be utilized on any level of pruning for appropriate trees, and some of the results are listed in Table 1.

16 Forbidden Address Canopy Dimension 12 0.00000 11 1.38848 112 1.38848 122 1.38848 111 1.75829 1112 1.75829 1122 1.75829 1222 1.75829 1211 1.80107 1121 1.80107 1221 1.80107 1212 1.82638 1111 1.89355

Table 1: Dimensions of Various Prunings for Self-Contacting trees where r = √1 2

5 Pruning Space-ﬁlling Trees

Table 1 is set up so as to make a curious trait obvious: the longer the sequence pruned, the closer the fractal dimension gets to 2. It becomes natural, then to ask if there exists a pruning which will still allow the tree to fill two-dimensional space. This question becomes even more apparent when looking at the effects of increasing lengths of pruned sequences on the space-filling T (135◦, √1 ) tree 2 as seen in Figure 11. The following multi-step proof addresses this question.

Figure 11: T (135◦, √1 ) with 11 pruned (left) and 111 pruned (right) 2

◦ 1 Theorem 2 Let T m (90 , √ ) be the pruned tree where m is the length of the 1 2 ◦ 1 forbidden sequence. For m ≥ 1, T m (90 , √ ) is not space-ﬁlling. 1 2 Proof.

1. The removal of a branch at level i removes from the canopy of T (90◦, √1 ) 2

17 a rectangle of area √ 1i A = 4 2 ∗ i 2 Proof. Let the length of the removed branch at i, ri, be the length of the trunk of the removed tree. i Let Ai be the area of the canopy of the removed tree with trunk length r . Since the canopy of T (90◦, √1 ) ﬁlls a rectangle [7], 2

Ai = li ∗ wi

Because each iteration causes the branches to form 90◦ angles with the previous branch, every other branch beginning with the length of the second iteration falls parallel to the trunk; Thus,

i+2 i+4 i+6 i+8 li = 2(r + r + r + r + ...).

Similarly, every other branch beginning with the length of the ﬁrst iteration falls perpendicular to the trunk such that

i+1 i+3 i+5 i+7 wi = 2(r + r + r + r + ...).

So,

i+2 i+4 i+6 i+8 i+1 i+3 i+5 i+7 Ai = 2(r + r + r + r + ...) ∗ 2(r + r + r + r + ...)

Or, ∞ !2 2i 3 X 2n Ai = 4 ∗ r ∗ r r n=0 Simpliﬁed with r = √1 , 2

√ 1i A = 4 2 ∗ i 2

2. For a 1m pruned tree, the number of removed rectangles at the ith iteration of the IFS is the ith m-bonacci number. Proof. Let a(i) be the number of allowed branches at level i, and b(i) be the number of missing branches, and thus shapes, at level i. Since the tree is binary, the number of missing shapes at level i is the diﬀerence between twice the allowed branches at the previous level and the allowed branches at the current level, or

b(i) = 2 ∗ a(i − 1) − a(i)

18 Which can be used to show that,

b(i) + b(i − 1) + ... + b(i − (m − 1))

= (2 ∗ a(i − 1) − a(i))+(2 ∗ a(i − 2) − a(i − 1))+...+(2 ∗ a(i − m) − a(i − (m − 1))) = 2∗[a(i − 1) + a(i − 2) + ... + a(i − m)]−[a(i) + a(i − 1) + ... + a(i − (m − 1)] And since a(i) = (a(i − 1) + a(i − 2) + ... + a(i − m)) by Theorem 1,

b(i) + b(i − 1) + ... + b(i − (m − 1)) = 2 ∗ a(i) − a(i + 1)

Thus, b(i) + b(i − 1) + ... + b(i − (m − 1)) = b(i + 1) √ 3. The total area removed is 4 2. Proof. th Let AT be the total area removed and Fm,i be the i m-bonacci number. Then, ∞ X AT = Ai ∗ Fm,i i=1 √ 1 i And when Ai = 4 2 ∗ 2 ,

∞ i √ X 1 A = 4 2 ∗ ∗ F T 2 m,i i=1 Let ∞ i X 1 S = ∗ F 2 m,i i=1 Then, √ AT = 4 2 ∗ S Where

m−1 i ! m ∞ i ! X 1 1 X 1 S = ∗ F + ∗ F + ∗ F 2 m,i 2 m,m 2 m,i i=1 i=m+1

Note that by deﬁnition of m-bonacci numbers,

m−1 i X 1 ∗ F = 0 2 m,i i=1 1m 1m ∗ F = 2 m,m 2

19 And, ∞ i ∞ i X 1 X 1 ∗ F = ∗ (F + F + F + ... + F ) 2 m,i 2 m,i−1 m,i−2 m,i−3 m,i−m i=m+1 i=m+1 ∞ j+1 X 1 = ∗ F + 2 m,j j=m ∞ j+2 X 1 ∗ F + 2 m,j j=m−1 ∞ j+3 X 1 ∗ F + 2 m,j j=m−2 . . ∞ j+m X 1 ∗ F 2 m,j j=1

∞ j X 1 = 1 ∗ ∗ F + 2 2 m,j j=m ∞ j 2 X 1 1 ∗ ∗ F + 2 2 m,j j=m−1 ∞ j 3 X 1 1 ∗ ∗ F + 2 2 m,j j=m−2 . . ∞ j m X 1 1 ∗ ∗ F 2 2 m,j j=1

 m−1 j  X 1 = 1 ∗ S − ∗ F + 2  2 m,j j=1   m−2 j 2 X 1 1 ∗ S − ∗ F + 2  2 m,j j=1   m−3 j 3 X 1 1 ∗ S − ∗ F + 2  2 m,j j=1 . . 1 m 2 ∗ (S)

20 ! 1 1 12 1m−1 = ∗ S ∗ 1 + + + ... + 2 2 2 2

m−1 k 1 X 1 = ∗ S ∗ 2 2 k=0 1m = S ∗ 1 − 2

Thus, 1m 1m S = 0 + + S ∗ 1 − 2 2 Simplifying,

1m 1m S − S ∗ 1 − = 2 2 1m 1m S ∗ = 2 2 S = 1

Therefore, √ AT = 4 2

4. The area of the unpruned tree is equal to the total area removed. Proof. √ ◦ 1 i The area of the unpruned tree, T (90 , r), is Ai = 4 2 ∗ 2 with i = 0

√ 10 A = 4 2 ∗ 0 2 √ = 4 2

Therefore, the canopy is no longer space-ﬁlling.

◦ 1 Theorem 3 Let T m (90 , √ ) be the pruned tree where m is the length of the 1 2 forbidden sequence. For m > 1, the canopy is a continuous curve. Proof. 1. 1222(12)∞ and 2222(21)∞ are preserved for any m > 1. Neither address involves two or more repetitions of F1 following one another in sequence, therefore, they are allowed.

2. Self contact occurs between 1222(12)∞ and 2222(21)∞ given r = √1 [7] 2 Therefore, the canopy is a continuous curve for m > 1.

21 This process also shows that the 135◦ tree loses all area, but remains a continuous curve with any m ≥ 1 for a 1m pruned tree.

◦ 1 Theorem 4 Let T m (135 , √ ) be the pruned tree where m is the length of the 1 2 ◦ 1 forbidden sequence. For m ≥ 1, T m (135 , √ ) is not space-ﬁlling. 1 2 Proof. 1. The forbidding of a sequence at level i removes from the canopy a triangle of area 1i A = i 2

The canopy of T (135◦, √1 ) ﬁlls a triangle, so, A = 1 ∗ b ∗ h where the 2 i 2 i i i−2 i base of the removed triangle,bi, is r and the height,hi, is r . So, 1 A = ∗ ri−2 ∗ ri i 2

Simplifying with r = √1 , 2 1i A = i 2

2. For a 1m pruned tree, the number of removed triangles at the ith iteration of the IFS is the ith m-bonacci number. Allowable and removed branches do not depend on the angle θ, therefore ◦ 1 this is the same as seen for T m (90 , √ ). 1 2 3. The total area removed is 1. Again, ∞ X AT = Ai ∗ Fm,i i=1 But for T (135◦, √1 ), 2 1i A = i 2 So, ∞ i X 1 A = ∗ S T 2 i=1 ∞ i X 1 As shown previously, S = 1 and = 1 2 i=1 Thus, AT = 1

22 4. The area of the unpruned tree is equal to the total area removed. The area ◦ 1 i of the unpruned tree, T (135 , r), is Ai = 2 with i = 0

10 A = 0 2 = 1

Therefore, the canopy is no longer space-ﬁlling.

◦ 1 Theorem 5 Let T m (135 , √ ) be the pruned tree where m is the length of the 1 2 forbidden sequence. For m > 1, the canopy is a continuous curve. Proof.

1. 122(12)∞ and 222(21)∞ are preserved for any m > 1. Neither address involves two or more repetitions of F1 following one another in sequence, therefore, they are allowed.

2. Self contact occurs between 122(12)∞ and 222(21)∞ given r = √1 [7] 2 Therefore, the canopy is a continuous curve for m > 1.

6 Further Study

Possibilities for furthering this research are immense. Potential areas of development include generalizing properties of asymmetric trees, n-branched trees, and n-dimensional trees as well as the effects of pruning one or multiple sequences from these variations. The proof given for the loss of the space- filling property of pruned trees could be further investigated for any number or combination of prunings. Additionally, the use of the Hausdorff dimension in determining the fractal dimension of these pruned trees could be more rigorously pursued. The currently described method still has many flaws, including its inability to accomidate for prunings with different parity or for variations of θ that generate disconnected canopies. Similarly, a generalized proof of the connectedness of a canopy under any given pruning has yet to be discovered.

23 7 Appendix 7.1 Appendix I: Mathematica Coding for Pruned Trees The following code was created due in large part to Bolanle Salam and Dr. David A. Brown.

r1 = .6; r2 = .6; theta = Pi 4; rot = RotationTransform theta ; negrot = RotationTransform -theta ; tf1 x_ := r1 * rot x + @0, 1 ;D tf2 x_ := r2 * negrot x + @0, 1 ; D tree2difs@ D = tf1, tf2@ D ;8 < maketuples@ D alphabet_@, LD_ 8:= Tuples< alphabet, L ; 8 < // Change your length and/or number of branches here. (Underneath) If the number of branches changes you must update the @ D @ D beginning of the program to reflect the new branch.

In[102]:= listCombo = maketuples 1, 2 , 8

makeprunedlist alphabet_, L_, forbidden_ := Module , @8 < D listCombo = maketuples alphabet, L ; *this makes the tuples* stringCombo@= ; D @8< For i = 1, i £ Length listCombo@ , iD++,H L 8< stringCombo@ = Append@ stringComboD , ToString FromDigits listCombo i 8 ; @ @ @ @@ DDDDD stringForbidden< = ; *this For loop converts forbidden list to strings* DFor v = 1, v £ Length forbidden , v++, 8< H L stringForbidden@ = Append@ stringForbiddenD , ToString forbidden v 8 ; @ @ @@ DDDD prunedlist< = ; *this loop creates the pruned list* DFor p = 1, p £ Length stringCombo , p++, 8< H L If@ StringFreeQ stringCombo@ pD , stringForbidden , 8 prunedlist = Append prunedlist, stringCombo p @ @ @@ DD D ; @ @@ DDDD prunedlist< D; prunedcanopy address_, ifs_ := ModuleD transf, points , n = Length IntegerDigits Max address ; points = ;@ D For k@=81, k £ Length address< , k++@, @ @ DDD If Length8< IntegerDigits address k n, transf = ; @ transf = IntegerDigits@ Daddress k ; 8 @points@= Append points@, ComposeList@@ DDDDifs Reverse8 transf8< , 0, 1 Last ; ; Show Graphics PointSize .001@ , Point@@ DpointsDD @ @ @@ @ DDD 8

24 // Ensure the branch and length are also the same in the below code. Place forbbiden sequence within end bracket.

In[103]:= address = ToExpression makeprunedlist 1, 2 , 8, 121

In[104]:= prunedtree address_, ifs_ := Module , @ @8 < 8

@ D

Out[105]=

In[106]:= prunedcanopy address, tree2difs

@ D

Out[106]=

25 References

[1] Barnsley, M., Fractals Everywhere! Addison Wesley, Massachusetts, Aca- demic Press Professional, 1993. [2] Brown, D.A., Personal Communication, 2010. [3] Frame, M.,& Lanski, J., When is a recurrent IFS attractor a standard IFS attractor? Fractals, 7(3), 257-266, 1999.

[4] Peitgen H.O. , Jurgens H. , and Saupe D., Chaos and Fractals: New Fron- tiers of Science 1992. [5] Lock, E.F.,& Frongillo R. Symmetric fractal trees in three dimensions. Chaos, Solitons,& Fractals, 32(2), 284-295.

[6] Mandelbrot, B.B. The fractal geometry of nature. New York: W.H. Free- man, 1983. [7] Mandelbrot, B.B.,& Frame, M. The canopy and shortest path in a self- contacting fractal tree. Math. Intelligencer, 21(2), 18-27, 1999.

[8] Mauldin, R.D.,& Williams, S.C. Hausdorﬀ dimension in graph directed constructions. Trans. Amer. Math. Soc., 309(2), 811-829. 1988. [9] Rellick, L. M., Edgar, G A.,& Klapper, M. H. Calculating the Hausdorﬀ dimension of tree structures. J. Statist. Phys., 64(1-2), 77-85. 1991. [10] Rudin,W. Principles of mathematical analysis. McGraw-Hill Science Engi- neering Math, 1976.