José L. Ramírez, Gustavo N. Rubiano

## Properties and Generalizations of the Fibonacci Word Fractal NB   CDF   PDF

### Exploring Fractal Curves

This article implements some combinatorial properties of the Fibonacci word and generalizations that can be generated from the iteration of a morphism between languages. Some graphic properties of the fractal curve are associated with these words; the curves can be generated from drawing rules similar to those used in L-systems. Simple changes to the programs generate other interesting curves.

### 1. Introduction

The infinite Fibonacci word,

is certainly one of the most studied words in the field of combinatorics on words [14]. It is the archetype of a Sturmian word [5]. The word can be associated with a fractal curve with combinatorial properties [67].

This article implements Mathematica programs to generate curves from and a set of drawing rules. These rules are similar to those used in L-systems.

The outline of this article is as follows. Section 2 recalls some definitions and ideas of combinatorics on words. Section 3 introduces the Fibonacci word, its fractal curve, and a family of words whose limit is the Fibonacci word fractal. Finally, Section 4 generalizes the Fibonacci word and its Fibonacci word fractal.

### 2. Definitions and Notation

The terminology and notation are mainly those of [5] and [8]. Let be a finite alphabet, whose elements are called symbols. A word over is a finite sequence of symbols from . The set of all words over , that is, the free monoid generated by , is denoted by . The identity element of is called the empty word. For any word , denotes its length, that is, the number of symbols occurring in . The length of is taken to be zero. If and , then denotes the number of occurrences of in .

For two words and in , denote by the concatenation of the two words, that is, . If , then ; moreover, by denote the word ( times). A word is a subword (or factor) of if there exist such that . If , then and is called a prefix of ; if , then and is called a suffix of .

The reversal of a word is the word and . A word is a palindrome if .

An infinite word over is a map , written as . The set of all infinite words over is denoted by .

Example 1

The word , where if is a prime number and otherwise, is an example of an infinite word. The word is called the characteristic sequence of the prime numbers. Here are the first 50 terms of .

Definition 1

Let and be alphabets. A morphism is a map such that, for all , .

There is a special class of words with many remarkable properties, the so-called Sturmian words. These words admit several equivalent definitions (see, e.g. [5], [8]).

Definition 2

Let . Let , the complexity function of , be the map that counts, for all integer , the number of subwords of length in . An infinite word is a Sturmian word if for all integer .

For example, .

Since for any Sturmian word, , Sturmian words have to be over two symbols. The word in example 1 is not a Sturmian word because .

Given two real numbers , with irrational and , , define the infinite word as . The numbers and are the slope and the intercept, respectively. This word is called mechanical. The mechanical words are equivalent to Sturmian words [5]. As a special case, gives the characteristic words.

Definition 3

Let be irrational, . For , define and ; then is called a characteristic word with slope .

On the other hand, note that every irrational has a unique continued fraction expansion

where each is a positive integer. Let be an irrational number with and for . To the directive sequence , associate a sequence of words defined by , , , .

Such a sequence of words is called a standard sequence. This sequence is related to characteristic words in the following way. Observe that, for any , is a prefix of , which gives meaning to as an infinite word. In fact, one can prove that each is a prefix of for all and [5].

### 3. Fibonacci Word and Its Fractal Curve

Definition 4

Fibonacci words are words over defined inductively as follows: , , and , for . The words are referred to as the finite Fibonacci words. The limit
is called the Fibonacci word.

It is clear that , where is the Fibonacci number, recalling that the Fibonacci number is defined by the recurrence relation for all integer and with initial values . The infinite Fibonacci word is a Sturmian word [5]; exactly, , where is the golden ratio.

Here are the first 50 terms of .

Definition 5

The Fibonacci morphism is defined by and .

The Fibonacci word satisfies and for all .

Here are the first nine finite Fibonacci words.

Definition 6

Let be the map such that deletes the last two symbols.

The following proposition summarizes some basic properties about the Fibonacci word.

Proposition 1

The Fibonacci word and the finite Fibonacci words satisfy:
1. The words 11 and 000 are not subwords of the Fibonacci word.
2. Let be the last two symbols of . For , if is even and if is odd.
3. The concatenation of two successive Fibonacci words is almost commutative; that is, and have a common prefix of length , for all .
4. is a palindrome for all .
5. For all , , where ; that is, exchanges the two last symbols of .

#### The Fibonacci Word Fractal

The Fibonacci word can be associated with a curve using a drawing rule. A particular action follows on the symbol read (this is the same idea as that used in L-systems [9]). In this case, the drawing rule is called “the odd-even drawing rule” [7].

Table 1. The odd-even drawing rule.

Definition 7

The Fibonacci curve, denoted by , is the result of applying the odd-even drawing rule to the word . The Fibonacci word fractal is defined as

The program LShow is adapted from [10] to generate L-systems.

Figure 1 shows an L-system interpretation of the odd-even drawing rule.

Figure 1. Interpretation of the odd-even drawing rule.

Here are the curves for .

The next proposition about properties of the curves and comes directly from the properties of the Fibonacci word from Proposition 1. More properties can be found in [7].

Proposition 2

The Fibonacci word fractal and the curve have the following properties:
1. is composed only of segments of length 1 or 2.
2. The number of turns in the curve is the Fibonacci number .
3. The curve is similar to the curve .
4. The curve is symmetric.
5. The curve is composed of five curves: , where is the result of applying the odd-even drawing rule to the word .

The next figure shows the curve and the five curves; here .

#### Some Variations

The Fibonacci word and other words can be derived from the dense Fibonacci word, which was introduced in [7].

Definition 8

The dense Fibonacci word comes from the Fibonacci word by applying the morphism
so that .

Given a drawing rule, the global angle is the sum of the successive angles generated by the word through the rule. With the natural drawing rule, , , , then .

For a drawing rule, the resulting angle of a word is the function that gives the global angle. A morphism preserves the resulting angle if for any word , ; moreover, a morphism inverts the resulting angle if for any word , .

The dense Fibonacci word is strongly linked to the Fibonacci word fractal because can generate a whole family of curves whose limit is the Fibonacci word fractal [7]. All that is needed is to apply a morphism to that preserves or inverts the resulting angle.

Here are some examples.

Here are some examples with other angles.

### 4. Generalized Fibonacci Words and Fibonacci Word Fractals

This section introduces a generalization of the Fibonacci word and the Fibonacci word fractal [11].

Definition 9

The -Fibonacci words are words over defined inductively by , , and , for and . The infinite word

is called the -Fibonacci word.

The 2-Fibonacci word is the classical Fibonacci word. Here are the first six -Fibonacci words.

The following proposition relates the Fibonacci word to .

Proposition 3

Let be the morphism defined by and , ; then
for all .

Definition 10

The -Fibonacci number is defined recursively by , , and , for all and .

The -Fibonacci numbers are the Fibonacci numbers and the -Fibonacci numbers are the Fibonacci numbers shifted by one. The following table shows the first terms in the sequences and their reference numbers in the On-Line Encyclopedia of Sequences (OIES) [12].

Proposition 4

The -Fibonacci word and the -Fibonacci word satisfy the following:
1. The word 11 is not a subword of the -Fibonacci word, .
2. Let be the last two symbols of . For , if is even and if is odd, .
3. The concatenation of two successive -Fibonacci words is almost commutative; that is, and have a common prefix of length for all and .
4. is a palindrome for all .
5. For all , , where .

Theorem 1

Let be an irrational number, with a positive integer; then .

For the proof, see [11]. This theorem implies that -Fibonacci words are Sturmian words.

Note that

where is the golden ratio.

#### The -Fibonacci Word Fractal

Definition 11

The Fibonacci curve, denoted by , is the result of applying the odd-even drawing rule to the word . The -Fibonacci word fractal is defined as

Here are the curves for .

Proposition 5

The -Fibonacci word fractal and the curve have the following properties:
1. The Fibonacci fractal is composed only of segments of length 1 or 2.
2. The curve is similar to the curve .
3. The curve is composed of five curves: .
4. The curve is symmetric.
5. The scale factor between and is .
##### Other Characteristic Words

This section applies the above ideas to generate new curves from characteristic words (see
Definition 3).

Conjecture 1

If , then the curve displays the Fibonacci word fractal pattern.

Here are seven examples.

### Acknowledgments

The first author was partially supported by Universidad Sergio Arboleda under grant number USA-II-2012-14. The authors would like to thank Borut Jurčič-Zlobec from Ljubljana University for his help during the development of this article.

### References

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### About the Authors

José L. Ramírez
Instituto de Matemáticas y sus Aplicaciones