The Golden Ratio and Fibonacci Numbers

The following sections are included:

• PREFACE

• CONTENTS

The following sections are included:

• Golden rectangles

• Art

• The great pyramid

• Pi

The golden ratio appears in some very fundamental relationships involving numbers from which many of its properties can be derived. One of the most basic occurrences of the golden ratio, and one of the most intriguing, involves the properties of numerical sequences. A numerical sequence is an ordered set of numbers which is generated by a well defined algorithm. One of the simplest methods of producing a numerical sequence is by the use of one or more seed values and an appropriate recursion relation …

The golden ratio plays an important role in the dimensions of many geometric figures, in both two and three dimensions. The simplest appearance of τ in geometry occurs in two dimensional figures, and it is here that the close affinity of the golden ratio with fivefold symmetry is first apparent. Among the most aesthetically appealing two dimensional shapes are the regular polygons. These are figures which have all edges equal and all interior angles equal and less than 180° (see e.g. Dunlap 1997). The simplest of these is the equilateral triangle, with three edges. In general a regular n-gon has n edges and interior angles which are given by the relation

In two dimensions the number of regular n-gons with interior angles defined by Eq. (3.1) is infinite. In three dimensions the regular polyhedra may be defined in an analogous way; All faces are the same regular polygon and each vertex is convex (as viewed from outside the figure). There are precisely five such figures and these are known as the Platonic solids. It is, perhaps, curious that only five such solids exist but a simple proof of this fact may be given in the following manner. Each face of the regular polyhedron is a regular polygon with n edges. From the discussion in the previous chapter it is known that values of n which are permitted are the integers

The Italian mathematician Leonardo de Pisa was born in Pisa around 1175 AD. He is commonly known as Fibonacci which is a shortened form of Filius Bonaccio (son of Bonaccio). His father, Bonaccio, was a customs inspector in the city of Bugia on the north coast of Africa (presently Bougie in Algeria) and as a result, Fibonacci was educated by the Mohammedans of Barbary. He was taught the Arabic system of numbers and in the early thirteenth century returned to Italy to publish the book Liber Abaci (Book of the Abacus) in 1202 (Leonardo di Pisa 1857). This book introduced the Arabic system of numbers to Europe and is responsible for Fibonacci's reputation as the most accomplished mathematician of the middle ages. The book also posed a problem involving the progeny of a single pair of rabbits which is the basis of the Fibonacci sequence (or Fibonacci series). It was, however, Edouard Lucas, whose contribution to this area of mathematics will be discussed in detail in the next chapter, who rediscovered the Fibonacci sequence in the late nineteenth century, and properly attributed it to its original founder …

The discussion in the previous chapter demonstrated the relationship of the golden ratio to the additive sequence known as the Fibonacci sequence. This is generated by the additive recursion relation (Eq. (2.1)) using the seed values F₀ = 0 and F₁ = 1. Numerous other additive sequences can be formed by using different seed values. One, given by Eq. (2.14), has already been discussed. In the present chapter some additional possibilities for integer seed values will be considered. Using integers which are any consecutive terms in a Fibonacci sequence as seed values (e.g. 1,2; 2,3; 3,5 etc.) will merely yield a Fibonacci sequence with the values of the indices shifted. Using seed values which are multiples of Fibonacci numbers (e.g. 0,2; 0,3; 2,2; 2,4; 6,10 etc.) will merely produce a Fibonacci sequence with all terms multiplied by a constant factor. Most of what has been discussed in the previous chapter will apply to such sequences, particularly the fact that the ratio F_n/F_n-1 will approach τ in the limit of large n. Certain choices of seed values will, however, yield additive sequences which are distinctly different from the Fibonacci sequence. One such possibility was studied extensively in the late nineteenth century by the French mathematician Edouard Lucas who published the results of these investigations in 1877. He considered the next smallest seed values L₀ = 2 and L₁ = 1. These values will generate the additive sequence

Any irrational number, I, can be expressed in terms of an infinite number of integers, a₀, a₁, a₂ … in the form …

The representation theorems discussed in this chapter are based on the fact that a generalized Fibonacci sequence forms a complete set. A sequence of integers is said to be complete if any positive integer can be expressed as a sum of a finite number of the terms in the sequence and each term is used, at most, once. Before dealing with generalized Fibonacci sequences in this respect, a commonly known complete set will provide an informative example of representation theorems …

A number line, say of length 1 unit from [0,1), my be divided into two segments by placing a point on the line at a location determined by the value of a real number, R. The terminology [0,1) indicates a line (or set) which includes the point 0 but excludes the point 1. If R ≥ 1 then it may be plotted on the line by using the fractional part of R,fract(R). The line may be divided into n+l segments by plotting the n values given by

In Chapter 5 the distinction between a periodic sequence as in Eq. (5.17) and a quasiperiodic sequence as in Eq. (5.16) was discussed. The sequences given in these equations may be displayed in a one dimensional graphical form by representing an A by a long line segment and a b by a shorter line segment as illustrated in Fig. 10.1. If a set of points is used to divide a line into long and short segments in either a periodic arrangement or in a quasiperiodic arrangement then these points form either a periodic or quasiperiodic array (sometimes called a lattice). Thus Figs. 10.1a and 10.1b may be referred to as one dimensional periodic and quasiperiodic lattices, respectively. The simplest kind of periodic array in one dimension is a sequence of evenly spaced points as shown in Fig. 10.2a. This may be viewed as a sequence of line segments of equal length …

The ideas concerning the generation of quasiperiodic structures as described in the previous chapter may be extended to higher dimensions. In the present chapter the concept of a two dimensional Fibonacci lattice is explored. Historically speaking, the first detailed work on two dimensional Fibonacci lattices was published in 1974 by Roger Penrose (1974), although certain ideas concerning these types of structures have been in existence for many years. It was, however, Penrose who first studied and properly appreciated the mathematical concepts present in these two dimensional Fibonacci lattices and it is, therefore, appropriate that they are generally referred to as Penrose tilings. In general the word tiling is used to describe a structure which is comprised of one or more types of cells or tiles which can be used to fill space. It is most common to think of a tiling in two dimensions, such as the arrangement of floor tiles, although in a mathematical sense the concept is valid in any number of dimensions. In this case the tiling consists of a finite number of different two dimensional figures which can be used to cover a plane without overlap. As expected, the Penrose tilings exhibit quasiperiodic rather than periodic order. There is extensive literature on both periodic and quasiperiodic tilings and many of the properties of these structures are beyond the scope of the present book and the present discussion will cover only those topics which are of relevance to the golden ratio. However, a basic introduction to the mathematics of tilings may be found in Grundbaum and Shepherd (1987) and in Dunlap (1990) …

An extension of the concepts discussed in the previous chapter to three dimensions describes the unique structure of a newly discovered class of metallic materials. Before discussing the structure of these novel materials, it is useful to consider the models for describing the microscopic structure of atoms in conventional materials …

The following sections are included:

• Biological symmetry

• Optimal spacing

• Fibonacci spirals

The following sections are included:

• APPENDIX I: CONSTRUCTION OF THE REGULAR PENTAGON

• APPENDIX II: THE FIRST 100 FIBONACCI AND LUCAS NUMBERS

• APPENDIX III: RELATIONSHIPS INVOLVING THE GOLDEN RATIO AND GENERALIZED FIBONACCI NUMBERS

• REFERENCES

• INDEX