New visual perspectives on fibonacci nummbers

EW VISUAL PERSPECTIVES (CM
FIBONACCI NUMBERS
Krassimir Atanassov
Vassia Atanassova
Anthony Shannon
John Turner

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NEW VISUAL PERSPECTIVES ON
FIBONACCI NUMBERS


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NEW VISUAL PERSPECTIVES ON
FIBONACCI NUMBERS

K T Atanassov
Bulgarian Academy of Sciences, Bulgaria

V Atanassova
University of Sofia, Bulgaria

A G Shannon
Warrane College, University of New South Wales, Australia

J C Turner
University of Waikato, New Zealand

(©World Scientific
U

New Jersey London • Singapore • Hong Kong


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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.

NEW VISUAL PERSPECTIVES ON FIBONACCI NUMBERS
Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd.
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Introduction

There are many books now which deal with Fibonacci numbers, either
explicitly or by way of examples. So why one more? What does this book
do that the others do not?
Firstly, the book covers new ground from the very beginning. It is
not isomorphic to any existing book. This new ground, we believe, will
appeal to the research mathematician who wishes to advance the ideas still
further, and to the recreational mathematician who wants to enjoy the
puzzles inherent in the visual approach.
And that is the second feature which differentiates this book from others.
There is a continuing emphasis on diagrams, both geometric and combinatorial, which act as a thread to tie disparate topics together - together,
that is, with the unifying theme of the Fibonacci recurrence relation and
various generalizations of it.
Experienced teachers know that there is great pedagogic value in getting
students to draw diagrams whenever possible. These, together with the elegant identities which have always characterized Fibonacci number results,
provide attractive visual perspectives. While diagrams and equations are
static, the process of working through the book is a dynamic one for the
reader, so that the reader begins to read in the same way as the discoverer
begins to discover.

V


VI

Introduction

The structure of this book follows from the efforts of the four authors
(both individually and collaboratively) to approach the theme from different starting points and with different styles, and so the four parts of the
book can be read in any order. Furthermore, some readers will wish to
focus on one or two parts only, whilst others will digest the whole book.
Like other books which deal with Fibonacci numbers, very little prior
mathematical knowledge is assumed other than the rudiments of algebra
and geometry, so that the book can be used as a source of enrichment material to stimulate that shrewd guessing which characterizes mathematical
thinking in number theory, and which makes many parts of number theory
both accessible and attractive to devotees, whether they be in high school
or graduate college.
All of the mathematical results given in this book have been discovered
or invented by the four authors. Some have already been published by
the authors in research papers; but here they have been developed and
inter-related in a new and expository manner for a wider audience. All
earlier publications are cited and referenced in the Bibliographies, to direct
research mathematicians to original sources.


Foreword
by A. F. Horadam

How can it be that Mathematics, being after all a product of human thought independent of experience, is so
admirably adapted to the objects of reality?
— A. Einstein.

It has been observed that three things in life are certain: death, taxes
and Fibonacci numbers. Of the first two there can be no doubt. Nor,
among its devotees in the worldwide Fibonacci community, can there be

little less than certainty about the third item.
Indeed, the explosive development of knowledge in the general region of
Fibonacci numbers and related mathematical topics in the last few decades
has been quite astonishing. This phenomenon is particularly striking when
one bears in mind just what little attention had been directed to these
numbers in the eight centuries since Fibonacci's lifetime, always excepting
the significant contributions of Lucas in the nineteenth century.
Coupled with this expanding volume of theoretical information about
Fibonacci-related matters there have been extensive ramifications in prac-


Vlll

Foreword

tical applications of theory to electrical networks, to computer science, and
to statistics, to name only a few special growth areas. So outreaching have
been the tentacles of Fibonacci-generated ideas that one ceases to be surprised when Fibonacci and Lucas entities appear seemingly as if by magic
when least expected.
Several worthy texts on the basic theory of Fibonacci and Lucas numbers already provide background for those desiring a beginner's knowledge
of these topics, along with more advanced details. Specialised research journals such as "The Fibonacci Quarterly", established in 1963, and "Notes
on Number Theory and Discrete Mathematics", begun as recently as 1996,
offer springboards for those diving into the deeper waters of the unknown.
What is distinctive about this text (and its title is most apt) is that it
presents in an attractive format some new ideas, developed by recognised
and experienced research workers, which readers should find compelling and
stimulating. Accompanying the explanations is a wealth of striking visual
images of varying complexity - geometrical figures, tree diagrams, fractals,
tessellations, tilings (including polyhedra) - together with extensions for
possible further research projects. A useful flow-chart suggests the connections between the number theoretic and geometric aspects of the material

in the text, which actually consists of four distinct, but not discrete, components reflecting the individualistic style, tastes, and commitment of each
author.
Beauty in Mathematics, it has been claimed, can be perceived, but not
explained. There is much of an aesthetic nature offered here for perception,
both material and physical, and we know, with Keats, that
A thing of beauty is a joy for ever:
Its loveliness increases; ...
Some germinal notions in the book which are ripe for exploitation and
development include: the generation of pairs of sequences of inter-linked
second order recurrence relations (with extensions and modifications); Fibonacci numbers and the honeycomb plane; the poetically designated goldpoint geometry associated with the golden ratio divisions of a line segment;
and tracksets. Inherent in this last concept is the interesting investigation
of the way in which group theory might have originated if Cayley had used
the idea of a trackset instead of tables of group operations.
An intriguing application of goldpoint tiling geometry relates to recre-


Foreword

IX

ational games such as chess. Indeed, there is something to be gleaned from
this book by most readers.
In any wide-ranging mathematical treatise it is essential not to neglect
the human aspect in research, since mathematical discoveries (e.g., zero^
the irrationals, infinity, Fermat's Last Theorem, non-Euclidean geometry,
Relativity theory) have originated, often with much travail and anguish, in
the human mind. They did not spring, in full bloom, as the ancient Greek
legend assures us that Athena sprang fully-armed from the head of the god
Zeus. Readers will find some of the warmth of human association in various
compartments of the material presented.

Moreover, those readers also looking for a broad and challenging outlook
in a book, rather than a narrow, purely mathematical treatment (however
effectively organised), will detect from time to time something of the music, the poetry, and the humour which Bertrand Russell asserted were so
important to an appreciation of higher mathematics.
A suitable concluding thought emanates from Newton's famous dictum:
... / seem to have been only like a boy playing on the
seashore, and diverting myself in now and them finding a smoother pebble or a prettier shell than ordinary,
while the great ocean of truth lay all undiscovered before
me.
While much has changed since the time of Newton, there are still many
glittering bright pebbles and bewitching, mysterious shells cast up by that
mighty ocean (of truth) for our discovery and enduring pleasure.

A. F. Horadam
The University of New England,
Armidale, Australia
October 2001


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Preface

This book presents new ideas in Fibonacci number theory and related topics, which have been discovered and developed by the authors in the past
decade. In each topic, a diagrammatic or geometric approach has predominated. The illustrations themselves form an integral part of the development of the ideas, and the book in turn unravels the illustrations
themselves.
There is a two-fold emphasis in the diagrams: partly to illustrate theory
and examples, and partly as motivation and springboard for the development of theory. In these ways the visual illustrations are tools of thought,
exemplifying or analogous to ideas developed by K. E. Iverson about mathematical symbols.*

The resulting visual perspectives comprise, in a sense, two sub-books
and two sub-sub-books! That is not to say that there are four separate and
unrelated monographs between the same covers. The two major parts, the
number theoretic and the geometric, and the four sections are distinct, but
there are many interrelations and connecting links between them.

'Iverson, Kenneth E. 1980: Notation as a Tool of Thought.
Association of Computing Machinery. Vol. 23(8), 444-465.
xi

Communications

of the


Preface

The following flowchart gives a simple overview of the book's structure
and contents.
FLOWCHART
PART A:
I
I
1
1

:

1

!

NUMBER THEORY PERSPECTIVES

A1: Coupled
Recurrence
Relations

i1

r
I A2: Number Trees |

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PART B:
B1: Fibonacci
Geometry

-

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GEOMETRIC PERSPECTIVES
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B2: Goldpoint
Geometry

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!1

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CH. 2

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1

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[

The reason for developing the book this way, rather than producing a
traditional text on number theory, is to preserve the styles of the originating
authors in the various parts rather than to homogenize the writing. The
'urgency' of the authors' work is thereby conserved.


Preface

xm

The topics in the book can be entered at different points for different
purposes. There are sections for various (though overlapping) audiences:
*
*
*
*
*

Enrichment work for high school students,
Background material for teacher education workshops,
Exercises for undergraduate majors,

Ideas for development by graduate students,
Topics for further research by professional mathematicians,
* Enjoyment for the interested amateur (in the original
sense of this word, which comes from the Latin amo:
I love).
There are several common notational threads built around the sequence
of Fibonacci numbers, Fn, defined by the second order homogeneous linear
recurrence relation
Fn = F n _i + F„_ 2 , n > 2,
with initial conditions Fi = F2 = 1.
The golden ratio is also a constant thread which links otherwise diverse
topics in the various sections. It is represented here by a = (1 + V5)/2 , and
it arises as the dominant solution of the auxiliary (or characteristic) polynomial equation x2 — x — 1 = 0, associated with the Fibonacci recurrence
relation.
In this book golden ratio gives rise in turn to new ideas relating golden
means to a variety of geometric objects, such as goldpoint rings, various
goldpoint fractals, and jigsaw tiles marked with goldpoints.
The geometric connections with number theory bring out some fundamental mathematical properties which are not always included in the modern school syllabus, yet they are very much part of the cultural heritage of
mathematics, which, presumably, is one reason for including mathematics
in a high school curriculum.
These fundamental properties are also an important component in the
development of conceptual frameworks which enable mathematicians to experiment, to guess shrewdly, to test their guesses and to see visual perspectives in symbols, formulas and diagrams.


XIV

Preface

Thus we have presented new extensions and inventions, all with visual
methods helping to drive them, in several areas of the rapidly expanding

field now known as Fibonacci mathematics. We hope that readers within a
wide range of mathematical abilities will find material of interest to them;
and that some will be motivated sufficiently to pick up and add to our
ideas.

K. T. ATANASSOV
Bulgarian Academy of Sciences, Sofia-1113, Bulgaria.
VASSIA ATANASSOVA
University of Sofia, Bulgaria.
A. G. SHANNON
Warrane College, University of New South Wales, Kensington, 1465
& KvB Institute of Technology, North Sydney, NSW, 2060, Australia.
J. C. TURNER
University of Waikato, Hamilton, New Zealand.

May 2002


Contents

Introduction
Foreword
Preface

v
vii
xi

PART A. N U M B E R T H E O R E T I C P E R S P E C T I V E S

Section 1. Coupled Recurrence Relations
1.
2.
3.
4.

Introductory remarks by the first author
The 2-Fibonacci sequences
Extensions of the concepts of 2-Fibonacci sequences
Other ideas for modification of the Fibonacci sequence
Bibliography

3
7
29
39
47

Section 2. Number Trees
1.
2.
3.
4.
5.

Introduction - Turner's Number Trees
Generalizations using tableaux
On Gray codes and coupled recurrence trees
Studies of node sums on number trees
Connections with Pascal-T triangles

Bibliography
XV

53
57
63
71
75
81


xvi

Contents

PART B. GEOMETRIC PERSPECTIVES

Section 1. Fibonacci Vector Geometry
1.
2.
3.
4.
5.
6.
7.

Introduction and elementary results
Vector sequences from linear recurrences
The Fibonacci honeycomb plane
Fibonacci and Lucas vector polygons

Trigonometry in the honeycomb plane
Vector sequences generated in planes
Fibonacci tracks, groups, and plus-minus sequences
Bibliography

85
95
107
119
123
137
153
177

Section 2. Goldpoint Geometry
1.
2.
3.
4.
5.
6.
7.

On goldpoints and golden-mean constructions
The goldpoint rings of a line-segment
Some fractals in goldpoint geometry
Triangles and squares marked with goldpoints
Plane tessellations with goldpoint triangles
Tessellations with goldpoint squares
Games with goldpoint tiles

Bibliography

Index

183
205
215
229
245
269
293
307

309


PART A: N U M B E R T H E O R E T I C
PERSPECTIVES

SECTION 1
COUPLED RECURRENCE RELATIONS
Krassimir Atanassov and Anthony

Shannon

Coupled differential equations are well-known and arise quite naturally in
applications, particularly in compartmental modelling [13]. Coupled difference equations or recurrence relations are less well known. They involve
two sequences (of integers) in which the elements of one sequence are part
of the generation of the other, and vice versa. At one level they are simple
generalizations of ordinary recursive sequences, and they yield the results

for those by just considering the two sequences to be identical. This can
be a merely trivial confirmation of results. At another level, they provide
visual patterns of relationships between the two coupled sequences which
naturally leads into 'Fibonacci geometry'. In another sense again, they can
be considered as the complementary picture of the intersections of linear
sequences [32] for which there are many unsolved problems [25].

l


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Chapter 1

Introductory Remarks by the First
Author

The germ of the idea which has been unfolded with my colleagues came to
me quite unexpectedly. A brief description of that event will help explain
the nature of this first Section of our book.
It was a stifling hot day in the summer of 1983. I had started to work
on Generalized Nets, and extension of Petri Nets, and I had searched for
examples of parallel processes which are essential in Generalized Nets. After
almost twenty years I can remember that day well. I was discussing my
problem with colleagues at the Physical Institute of the Bulgarian Academy
of Sciences, when one of them (she is an engineer) asked me: "Are there
real examples of parallelism in Mathematics?"
I was not ready for such a question then. Nor do I have a good answer
to it now! Nevertheless, as a young mathematician who naively thought

that Mathematics readily discloses its secrets, I answered that there are
obviously such examples; and I began to try to invent one on the spot.
I started by saying that the process of construction of the Fibonacci
numbers is a sequential process [1,2], and began to describe the sequence
and its properties. At that moment I suddenly thought of an extension of
the idea, which perhaps would give an example of a parallel process. I said:
"Consider two infinite sequences {an} and {bn}, which have given initial
values ai,d2 and 61,62, and which are generated for every natural number
n > 2 by the coupled equations:
O-n+2 —

bn+\ + bn

t>n+2

O-n+1 + an •

=

3


4

Coupled Recurrence

Relations

Is this a good example of parallelism in Mathematics?" The answer is
No, although at the time it seemed to satisfy my colleagues. It did not

satisfy me, however, because the process of computation of each sequence
can be realized sequentially, though this is not reflected in the results.
The problem of parallelism in Mathematics, and the 'example' I had
created, nagged away at me; and I continued to think about it that day. Towards evening I had invented more details, when Dimitar Sasselov, a friend
of mine (now at the Harvard-Smithsonian Astrophysical Centre) came to
my home. I asked him and my wife Lilija Atanassova (a fellow student and
colleague at the Bulgarian Academy of Sciences) to examine some of the
cases I had formulated.
This process was just an intellectual game for us. We ended our calculations, and then at that moment the following question was generated:
"Why do we waste time on all this?". These results are very obvious and
probably well-known. Up until this time I had not been seriously interested
in the Fibonacci numbers. In the next two months I interviewed all my colleagues - mathematicians - and read books on number theory, but nowhere
did I come across or find anything about such results. In the library of my
Institute I found the only volumes of The Fibonacci Quarterly in Bulgaria
and read everything which was available, but I did not find similar ideas
there either. Then I decided to send my results to Professor Gerald Bergum,
the Editor of the Quarterly. His answer was very encouraging.
This was the history of my first paper on the Fibonacci numbers [9].
The second one [4] is its modification. It was written some months after
the positive referee's report.
In the meantime, the first paper was published and three months after
this I obtained a letter from Professor Bergum with a request to referee
J.-Z. Lee and J.-S. Lee's paper [22] in which almost all the results of my
second paper and some other results were included. I gave a positive report
on their paper and wrote to Prof. Bergum that my results were weaker and
I offered him to throw them into the dustbin. However, he published first
my second paper and in the next issue J.-Z. Lee and J.-S. Lee's paper.
I write these words to underline the exceptional correctness of Professor
Bergum. Without him I would not have worked in the area of the Fibonacci
sequence at all. The next results [5; 10] were natural consequences of the

first ones. I sent some of them to Professor Aldo Peretti, who published


Introductory

Remarks by the First

Author

5

them in "Bulletin of Number Theory and Related Topics" and I am very
grateful to him for this (see [6; 7]).
The essentially new direction of this research, related to these new types
of the Fibonacci sequences, is related to my contacts with Professor Anthony Shannon. He wrote to me about the possibility for a graph representation of the Fibonacci sequence and I answered him with the question
about the possibility for analogical representation of the new sequences. In
two papers Anthony Shannon, John Turner and I showed this representation [26; 11].
In the last seven years other results related to extensions of the Fibonacci numbers were obtained. Some of them are continuations of the
first one, but the others are related to new directions of Fibonacci sequence
generalizations or other non-standard ideas.
Four years ago, I invited my friend the physicist Professor Peter Georgiev
from Varna to research the matrix representation of the new Fibonacci sequences. When he ended his research, I helped him to finalize it. Therefore,
my merit in writing of the series of (already 6) papers [14; 15; 16; 17; 18;
19] in press in "Bulletin of Number Theory and Related Topics" is in general in the beginning and end of the work, and only Peter's categorical
insistence made me his co-author. With these words I would like to underline his greater credit for the matrix representation of the new Fibonacci
sequences.
I must note also the research of V. Vidomenko [36], W. Spickerman, R.
Joyner and R. Creech [28; 29; 30; 3l], A. Shannon and R. Melham [27], S.
Ando S. and M. Hayashi [l] and M. Randic, D. A. Morales and O. Araujo
[24].

In this Section I would like to collect only those of my results related
to the Fibonacci sequence, which are connected to ideas for new generalizations for this sequence. For this reason, the results related to their
representations and applications will not be included here.


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Chapter 2

The 2-Fibonacci Sequences

In this chapter we first define and study four different ways to generate pairs
of integer sequences, using inter-linked second order recurrence equations.
2.1

The four 2-F-sequences

Let the arbitrary real numbers a, b, c, and d be given.
There are four different ways of constructing two sequences { a j } ~ 0 and
{&}£()• We shall call them 2-Fibonacci sequences (or 2-F-sequences). The
four schemes are the following:
a 0 = a, ft, = b, ai = c, ft = d
Ctn+2 = fti+1 + fti, n > 0
/?„ + 2 = O-n+l +C*n, n > 0

(2.1)

a0 = a, ft. — b, ai = c, ft = d
Qn+2 = OJn+1 + ft, " > 0

ft+2 = f t i + i + " « , n > 0

(2.2)

Q0 = a, ft. = &, a i = c, ft = d
a n +2 = fti+i + a „ , n > 0
ft+2 = an+i +ft», n > 0

(2.3)

7


Coupled Recurrence Relations

ao = a, Po = b, a\ = c, Pi = d
(2.4)

an+2 = Qn+1 + Qm n > 0
Pn+2 = Pn+1 + Pn, n > 0

Graphically, the (n+2)-th members of the different schemes are obtained
from the n-th and the (n + l)-th members as is shown in Figures 1-4.

an+i

Qn+2

Figure 1

ttn+l

Figure 2

Qn+l

Figure 3

Otn+2