Properties of Pruned, Binary, Planar Fractal Trees

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 re- moved 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 fF1;F2g 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 = p1 , and θ = 45◦ is seen in 2 Figure 2. Figure 2: T p1 ; 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 2 [1; 1) where aj 2 f1; 2g. 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 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◦; p1 ) and T (135◦; p1 ). 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 mathe- matical 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.

Load more