Hausdorff Dimension of Generalized Fibonacci Word Fractals
Fibonacci word fractals are a class of fractals that have been studied recently, though the word they are generated from is more widely studied in combinatorics. The Fibonacci word can be used to draw a curve which possesses self-similarities determined by the recursive structure of the word. The Hausdorff dimension of the scaling limit of the finite Fibonacci word curves is computed for -Fibonacci curves and any drawing angle between and .
Motivated by the work of Monnerot-Dumaine , we analyze a generalized class of Fibonacci fractals for their Hausdorff dimension. The two parameters we consider are the drawing angle as introduced by Monnerot-Dumaine and the use of the Fibonacci words as introduced by Ramírez et al [10, 11, 12]. The necessity of this analysis is that the dimensions estimated in the previous works are for the self-similarity dimension despite occasional references to Hausdorff dimension. The equality of these two dimensions often holds but is not automatic. Equality holds in the familiar case when there exists an iterated function system satisfying the open set condition whose attractor is the fractal in question. Many of the already mentioned papers and those of Blondin-Massé  also include a study of the Fibonacci snowflake fractal which we do not consider in the present paper.
The basic construction of the Fibonacci fractals is based on three operations: one is a recursive construction of a string of symbols, the second is the processing of a finite string into a curve, and the third is taking a scaling limit of the finite curves to a fractal limit. The construction of the recursively defined words is what gives the name Fibonacci to these fractals. Fibonacci words, as strings of and have been extensively studied since at least the 1980’s. See for a few examples [1, 3]. The processing of the finite words into curves is inspired, though not identical, to a similar process in systems .
Much of the notation will look similar, but mean very different things. Any given finite Fibonacci word is written with notation . More on what and represent later. A finite Fibonacci curve is denoted as . A Fibonacci number itself is shown as . It should be noted the difference in the case of the letter and the font in determining the meaning of a symbol. Additionally we denote the fractal associated to an Fibonacci word with drawing angle as .
We begin by first stating the definitions of the Fibonacci and i-Fibonacci words along with some of their particularly useful properties (Section 2). We then draw the corresponding curves and explore their properties as well (Section 3). Finally we demonstrate the existence of the fractals and most importantly calculate the Hausdorff dimension of any given i-Fibonacci fractal with drawing angle (Section 5). The final theorem of the paper will be the continuity of the fractal construction as a function of the drawing angle, Proposition 12 and Theorem 6.
We thank Zachary Littrell for his aid in reviewing our code and Spencer Hamblen for his reoccurring timely assistance. McDaniel College kindly provided support for this project through a Student-Faculty Collaboration Grant.
2. Fibonacci Words
The (classical) Fibonacci word is the word defined over the alphabet that is generated by the concatenation rule with initial values and . Later we will discuss the infinite word . More information on the sequence can be found in the On-line Encyclopedia of Integer Sequences for sequence number A003849 . Since the first digits of are fixed it is possible to define as the infinite Fibonacci word. We however show the existence of metrically in Proposition 3.
The Fibonacci word is constructed according the the same rules as a classical Fibonacci word however the initial values are and . We use to represent a string of ’s of length . Table 1 lists the first five Fibonacci words for , the standard Fibonacci words, and
While concatenation is the basic construction technique for Fibonacci words we will define a substitution rule that will produce from as well. The main utility is that the application of a substitution rule is a local operation while concatenation is global operation. This perspective will be more useful in the discussion of the curves in the next section. It should be noted that while our substitution is similar to the substitution used in , it is not the same.
Define to be the substitution rule:
over words comprised of blocks of the form and .
Since transforms a sequence of blocks and into another sequence of blocks of and the transformation can be iterated.
Let be an Fibonacci word then .
Since is unambiguous in how it is written as a single block of we will always assume that is broken down into these blocks in the way that it arises as the output of this process.
We proceed by induction. Note that and . Similarly that can be checked manually. Suppose the proposition holds for and and we will show it holds for . Since acts locally by replacing blocks and can be split into two sub-collections of blocks the action of can be split over and so that
The following properties of the Fibonacci words will be useful later. They are cited here without proof.
[12, Proposition 5] The following properties hold for all i-Fibonacci words .
The subword ”11” can never be found in an i-Fibonacci word with .
The word can be decomposed as where is a palindrome and depend on the parity of only.
For even , and for odd ,
The i-Fibonacci word has a five-partite structure
where (i.e. the last two letters are swapped).
The infinite i-Fibonacci words exist for any given i and are .
The limit is taken in the sense of the standard adic metric on the space of finite and infinite words. That is where is the number of initial symbols that and share. By the definition of as concatenations, for each the sequence is Cauchy in this metric. Thus there is a limit word that is denoted ∎
It was mentioned earlier that the infinite Fibonacci words could also be considered as fixed points of the concatenation rule. This metric proof of the existence is equivalent to that argument by the definition of the metric measuring distance between two words by measuring the length of the longest shared prefix of those two words.
3. Fibonacci Curves
The Fibonacci word curve is generated by taking the corresponding Fibonacci word and applying the following drawing rule. The Fibonacci curves introduced in  are just the case when .
Drawing Rule: Let be the ’th character of and perform the following procedure over each character .
Set initial direction,
Choose drawing angle .
Draw a segment in the current direction.
If is 0 then
If is even, turn left, i.e. add to the current direction.
If is odd, turn right, i.e. subtract from the current direction.
If is 1, do not change direction.
Repeat from step 3 to step 5 for to
It should be noted that the instructions “turn right” or “turn left” mean to change the angle at which the next segment will be drawn. When the end of a word is reached by the drawing rule, there will be an angle pointing to where the next segment would be drawn and this final direction will be called the net angle of a word.
Let be a function on finite words on the alphabet . Set , this value is the “initial angle.” Then where
if and is even,
if and is odd.
Following from the properties of the i-Fibonacci words are the corresponding properties for the curves.
[12, Proposition 6] The following properties hold for all curves .
There exist only segments of length 1 or 2 in the curve.
The curve is similar to the curve with the same shape, but different number of segments.
The scaling factor between curves and is .
Similar to the above word property, the curve has a five-partite structure written
where is the curve corresponding to the word .
It is assumed that the lengths of the line segments is “unit length.” In Section 5 scaling ratios between and will be determined. They will depend on , , and also . The scaling ratios will then allow us to take a scaling limit to obtain a Fibonacci fractal. For the time being though, the specific length of the line segments is left unspecified.
Next we want a function that determines what the end behavior of any given subword is in terms of its angle.
Let be a word composed of the blocks and . Then Furthermore, .
Consider the third substitution on each of two basic blocks:
Thus we can see by processing the words according to the drawing rule that . Similarly . By the additivity of the angle function over concatenation of words when is applied another three times to the sign of the overall angle is again negated. Thus and the same for as well. ∎
From Table 3 it is clear that as increases the curve pass through several global shapes. It is clear that their local structure is similar but for the global structure to be the same and for the orientation of the curve to be the same again it is necessary to consider only when . Notice that Proposition 5 gives another reason for considering every sixth curve as well. For purely historical reasons from  we consider for even . For odd we consider as it is visually the most similar. Equivalent analyses could be conducted for any choice of . The following proposition is the first that makes reference to this choice.
When is even and when is odd .
By Proposition 5 . So for even and for odd . ∎
Finally, we verify that there exists no overlap between any two blocks of the five-partite structure for a later proof of Proposition 10.
The bounding box of a given curve is the smallest rectangle enclosing the curve.
Refer to Figure 1 depicting where the dotted red rectangle is the bounding box formed by vertices A, B, C, and D.
Consider for even and for odd . Then the first and last vertices of these curves are co-located with two adjacent vertices of the bounding box.
Consider the even case. Curves of the form have first and last vertices on a line in and we wish to show that no other segments cross this line. It is true for because for any even it consists of a diagonal walk up, a double segment, and a diagonal walk down of equal length providing a net right turn. This gives that the net change in the vertical coordinate is zero. Notice that, with this description of , it is entirely contained inside of its bounding box. It is true for because it is the assemblage of four copies of and an in the shape shown in Figure 2.
Recall that each curve is made up of curves and (from the five-partite structure). Assuming that the claim is true for and , then it holds for their assemblage because of their arrangement in the five-partite structure.
Finally, since the case for odd introduces only a rotation of and rotation does not change these properties, it holds for this case as well. ∎
The bounding boxes of each individual sub-curve of from the five-partite structure have disjoint interiors.
From the five-partite structure, we know that the last vertex of one sub-curve is the first vertex of the next sub-curve since the sub-curves are connected and form a continuous path. From Proposition 7, we know that the first and last vertices of a sub-curve are also vertices of the bounding box. This means that the bounding boxes of the two sub-curves that share a vertex also share that same vertex. Due to the fact that we draw with an angle and that this vertex is specifically the first or last vertex, no segment from one bounding box, that is one sub-curve, will ever intersect with the interior of the adjacent sub-curve’s bounding box.
It remains to be shown that non-adjacent sub-curves also have bounding boxes with disjoint interiors. Specifically, we need to show that that the bounding boxes for the first and last sub-curves do not intersect. Because of the five-partite structure, we know that the first and last boxes are the same as the second and fourth boxes, but the second and fourth are at a rotation of . From Proposition 9, we know that these boxes are taller than they are wide, the distance created by the middle three boxes is greater than the combined width of the first and last boxes, thus, causing no overlap. ∎
The first visual impressions from Figure 3 is that as changes from even to odd and back again that to pick a particular “Fibonacci fractal” is not a canonical choice. We follow  and choose for even as the approximating curves to a fractal. For odd we choose because it is only varies from the first choice by a rotation.
4. Fractal Geometry
In this section we state the definitions and constructions necessary for the computation of the Hausdorff dimension of a fractal given by an iterated function system (IFS) satisfying the open set condition. We start with the definition of Hausdorff dimension. The definitions and theorems in this section are those of .
The Hausdorff dimension, of a subset is
where the Hausdorff measure is
While a theoretically satisfying definition of dimension it is a difficult one to compute. But for a particular class of self-similar fractals its computation can often be done via a simpler route.
An Iterated Function System (IFS) is an indexed collection of mappings which in this case are from onto itself. We also make the assumption that the mappings are contracting similarities. An IFS defines a set function by
A set is called self-similar if . Since similarities have a contraction ratio we can define a notion of dimension called variously the fractal dimension, self-similarity dimension, or similar by where is the solution to
Needless to say the computation of the self-similarity dimension from an IFS is much simpler than the direct computation of the Hausdorff dimension. However the two dimensions often take the same value for ”nice” self-similar fractals.
From , an iterated function system satisfies the open set condition if
where is some non-empty, bounded and open set, and the are the functions from the IFS.
[4, Theorem 9.1] For an IFS there exists a unique attractor F, if the contractions on satisfy
with and for each .
In  we are given a relationship between the self-similarity and Hausdorff dimensions.
[4, Theorem 9.3] If the IFS satisfies the open set condition, the Hausdorff dimension is given by solving for in the equation
where is the scaling ratio of the corresponding from the IFS.
Because the Fibonacci curves are not nativley generated by an IFS we also need a notion in which to say sequences of curves have limits other than being the fixed set of an IFS. To this end we use the Hausdorff metric.
Let be the collection of non-empty compact subsets of . Define for
5. Fibonacci Fractals
In this section we are concerned with showing the existence and Hausdorff dimension of the Fibonacci fractals. The fractal itself is a scaling limit of the curves along the subsequence for even and for odd . But to create a self-similar description of the fractal similar to the self-similar/five-partite property of the Fibonacci words we have to include and curves as well. Let the -Fibonacci curves be drawn with line segments of length one.
Consider the distance between the initial and final point of a Fibonacci curve. The ratio of this length between and is the scaling ratio.
For drawing angle , the scaling ratio of the curve is
Consider the trapezoid in Figure 5 describing the proportions of the curve and label the wider base of the trapezoid to have length . Similarly, label the width of each of the sub-curves to have length 1, and finally the sub-curve to have length . Here, is the scaling ratio from a previous curve to a larger curve. By the properties of trapezoids, the width of the larger base, , can also be written as
Multiply through by to obtain a quadratic and solve via the quadratic formula:
and because we are concerned with the scaling ratio from a larger curve to a smaller curve, we raise to the and simplify to get our scaling ratio. ∎
Let be the Euclidean distance between the starting and ending vertices of and the height of the curve in the direction perpendicular to the line connecting the first and last vertices of the curve. Notice that both and depend on the drawing angle, , and . The aspect ratio is defined as .
Let or depending as is even or odd respectively. The scaling limit is achieved by taking the length of the individual line segments to be , thus the width of the scaled curves is always . Theorem 4 proves that the scaling limit exists as a limit in the Hausdorff metric space. The following proposition about the aspect ratio as increases will be a central piece to the proof of Theorems 4 and 6.
Consider the two relations between the “height” and “width” of the :
By using the width formula repeatedly in the height formula we can arrive at a more useful formula for the height. That is
where and are the roots of the polynomial . That is
It is important to observe that for and when . Thus we can compute
The statement of the proposition arises from the relation between and . ∎
There are two central existence problems that need to be addressed about Fibonacci fractals. The first is the existence of a scaling limit of the Fibonacci curves for a given and (sub)-sequence of . The second is if it is possible to represent this limit curve as a self-similar set with a particular IFS. Doing so will enable the standard argument for computing Hausdorff dimension by way of computing the self-similarity (fractal) dimension.
For drawing angle there exists a scaling limit along a subsequence of the Fibonacci curves. For even , the scaling limit of exists, for odd the scaling limit of exists. The scaling factors are and respectively for the parity of .
The choice of scaling to have width is to maintain consistency with the scaling used in . Any other value would of course make no difference in the following proof, however the aspect ratio would still limit to .
Consider the case when is even. The odd case is similar. We compute the Hausdorff distance between scaled copies of and to show that the sequence is Cauchy and thus has a limit, . First is composed of copies of and a single copy of , scaled so that so
which for large enough is within any given of by Proposition 9.
Thus each of the five parts of has a bounding box with a Hausdorff distance of of the corresponding bounding box of . This is because with the widths fixed to the appropriate length the bounding boxes at each level share at least one common vertex and orientation with only their heights varying.
Now consider the portions of the curves within one of those five bounding boxes. Consider the first and last vertex of . As are recursively constructed and scaled the images of the two points has a mesh size that is no more than which goes to zero as grows. Since the difference between and between two adjacent images of the first and last points is that while has an connecting them has a . That is a bounded number of extra edges and so the distance traveled by the excursion is a bounded multiple of . For large enough this will be less than . Thus the Hausdorff distance between the scaled copies of and is less than any given for large enough. Since the Hausdorff metric space is complete there exists a scaling limit set. ∎
For even let the IFS be the five contractive similarities that map the large trapezoid onto the five similar trapezoids in Figures 2 and 6 such that the image of the axis falls onto the dark line. For odd , use the same five maps composed with a clockwise rotation of . The maps are referred to as , .
The following lemma follows directly from this definition of the IFSs by choosing to be the interior of the largest trapezoid in Figure 6. In Proposition 8 it was shown that the Fibonacci curves themselves satisfy the open set condition so that it is expected that an IFS we choose should as well.
For both parities of , the IFSs described both satisfy the open set condition.
There exists a unique attractor for both of the above IFS for .
There is no a priori guarantee that is equal to the attractor of the appropriate IFS. However it is.
The attractor of the IFS corresponding to the parity of and value of is the equal to .
Fix an and . Let be the scaling limit and apply the IFS to it, i.e. consider . Because the scaling limit is already invariant under the scaling process, replacing the curve with five scaled copies of the curve so that the width remains we can use Theorem 3 to say that the scaled copies are scaled by exactly and . Again since the scaling limit is as we said invariant under the scaling operation we note that Proposition 9 that the aspect ratio remains unchanged. Thus the scaling operation acts on as an IFS with the five maps, the same scaling ratios, and general “U-shaped” pattern as the IFS defined above.
To show that they are the same IFS we refer back to Propositions 5 and 6 which together imply that since has initial point at the origin that the five sub-curves are in exactly the same alignment as shown in Figure 6. The previous paragraph shows they are the same size as well. Thus is invariant under the IFS, so by the uniqueness of the attractor is the attractor. ∎
It should be noted what happens when . In this case the apsect ratio is and the scaling limit becomes a line segment in the direction of length . The fractal degenerates to a line.
With these results, we now know enough to properly calculate the Hausdorff dimension of the attractor of our IFS which in turn is the same dimension as that of the curve .
We are now in a position to state the first significant result of the paper, the Hausdorff dimension as a function of drawing angle. It is noteworthy that the dimension, in fact the fractals themselves do not depend on in any way. Thus the value of is only of combinatorial interest.
For drawing angle the Hausdorff dimension of the fractal is given by
As the drawing angle, , goes to zero, the Hausdorff dimension of goes to one.
The last observation is that not only is the Hausdorff dimension of the Fibonacci fractals continuous in the drawing angle but also the fractals themselves. The continuity of the fractals as grows above is conjectured but in that case the IFS has many overlaps that the continuity of the dimension function is not obvious.
The function that maps to the fractal for any is continuous from above at as a map from the interval into the Hausdorff metric space.
Consider the points and , denote them by the set . These are the first and last points mentions in the proof of Theorem 4. By Theorem 5 can be represented by the IFS instead of the scaling limit construction. Let be the image of under many applications of the iterated function system. As goes to infinity the distance between and goes to zero. It thus suffices to show continuity in the Hausdorff metric of as a function of for all .
Consider the set . As a non-empty compact subset of it is clearly a Hausdorff-metric continuous function of . Since self-similar IFSs are Hausdorff-metric continuous functions themselves we have that depends on continuously as it is a composition of continuous maps. ∎
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