Fibonacci and Lucas Numbers, and the Golden Section

200 Pages·1989·10.907 MB·English

by  Steven Vajda

MATHEMATICS AND
ELLHIOSR WOOSDE RIIENS
MATHEMATICS AND ITS APPLICATIONS
SerieEsdi toGr.M: . B ELPLr,o feosfMs aotrh ematics,
KingC'osl l(eKgOeC U)n,i veorfsL iotnyd on
StatisOtpiecrsa,t Rieosneaalar ncdCh o mputational
MathemaEtdiitcoBsr.W : .C ONOLLEYm,e riPtruosf essor
ofM athemaQtuieceMsna, r Cyo lleUgnei,v eorfs ity
London
FIBONA&C LCUIC ANSU MBERS,
ANDT HEG OLDESNE CTION:
TheoarnydA pplications
ProfeSs.Vs AoJrD DAe,p artomfMe antth emaUtniicvse,r ­
siotfyS ussex
Thiism portteaxbntet g iwnista hno vervoifpe rwo blems
inw hicFhi bonnaucmcbie arrsi asnedt hepnr oceteod s
devetlhoepmi arn iyn tripgruoipnegrT thiLeeu sc.na usm ­
berasr cel orseel atives.
Thef irrsetc oradpepdl icoafFt iiboonn naucmcbie rres­
latteoasm odeflot rh ger owotfahr abpboiptu labtuito n,
theayr ael swoe lkln owinnp lamnotr pholCoogmyp.u ter
scienwtiirlseltc so gtnhiesoiecr c urraestn hcbeea sfiosr
efficsieeanrtc phr oceduwrheisl,Fe i bonacci-type
mechaniasrmoesf tuesne fdo prs eudorannudmobme r
generaitnic oonm putseirm ulaatnidoM onn tCea rlo
methobdost,hh a viwnigd aep plicianOt pieorna tional
reseaSrtcaht,i asntdci ocmsp utatmiaotnhaemla tics.
TheG oldSeenc tisotnu,d iinea dn tiqiusia mt yys,t ical
princailplleege exdplrye saseisntghh eatrimco knnyo,w n
sintchete i moefst haen ciGernete Ikhtsa .as c locsoen ­
nectwiiotnth h Fei bonmaeccchia nism.
Morree centthFleiy b onparcicnic oiffp olrem iann ge w
numbbeyra ppropcroimabtien aotfpi roenv inouumsb ers
habse eenx tentdoye ide sledq uenwcietsshu rprainsdi ng
sometimmyesst ifpyrionpge rTthieesaser.t e h seo -called
Meta-Fibsoenqaucecnic es.
A discusosfMi eotna -Fibnounmabcecaril s a,r guenl­y
explotreerdr iwthoircsyhof aorf feornslg yl impisnetso
ani ntrifgiueiplnrdge, c eael dosoa ktt hGeo ldSeenc tion
int hpel aanneds pacPel.a tsoonliiacdr dsee scrainbde d
someo ft heliersf sa milfieaart uarreeesx hibiTtheed .
theorectoinctaelox ftt hibso okf itisn taol gebra,
geometprryo,b abtihleioctroyym ,p utaatsipoenacantlds
thceo mbinastiodorefin aulm btehre ory.
Readership: Mathematics at undergraduate and graduate levels,
recreational mathematics.

FIBONACCI & LUCAS NUMBERS,
AND THE GOLDEN SECTION
Theory and Applications
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FIBONACCI &
LUCAS NUMBERS,
AND THE
GOLDEN SECTION
TheoarnAydp plications
S.V AJDA,
Dr. Phil., D. Tech. h.c.
Department of Mathematics, University of Sussex
ELLIS HORWOOD LIMITED
Publishers · Chichester
Halsted Press: a division of
JOHN WILEY & SONS
New York ·Chichester· Brisbane• Toronto
First published in 1989 by
ELLIS HORWOOD LIMITED
Market Cross House, Cooper Street,
Chichester, West Sussex, P019 lEB, England
The publisher's colophon is reproduced from James Gillison's drawing of the ancient Market Cross,
Chichester.
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©
1989 S. Vajda/Ellis Horwood Limited
BVraitisjh dLSiabtr,aer yv /Cea9tnal0o,gu/ in-g in Publication Data
Fibon&a Lcuccina usm beanrtdsh geo ldseenc tion
I.F ibonnaucmcbie rs
IT.i tlIeIS .e ries
512'.72
Library of Congress data arnilable
ISB0-N- 7458-(0E7l1Hl5oi-rs1w oLoidm ited)
ISB0-N- 470--2(1H5a0l8sP-tr9ee ds s)
TypeisnTe itm heysE llHiosr woLoidm ited
PrinitnGe rde Barti thayHi anr tnBooldlmsi.n
COPYRIGHT NOTICE
AlRli ghRtess erNvoep da.ro tft hpiusb licmaatyhi ero enp rodsutcoeridena,d r etrsiyesvatoler m ,
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withtohupete rmiosfEs lihoHsno rwoLoidm iMtaerdk.Ce rtoH sosu sCeo,o pSetrr eCehti,c heWsetsetr ,
SussEenxg,l and.
Table of contents
Preface ............................................. 7
I. Introduction .. . . . . . . . . .. . . . . . . . . .. .. .. .. . .. . . .. . . . . 9
II. Background .......................................1 7
III. Relationships .......................................2 4
IV. Fibonacci numbers and the Golden Section ...................5 1
V. Fibonacci Series .....................................6 2
VI. Divisibility properties .................................7 3
VII.
Congruences and uniformity .............................8 8
VIII.
Continued fractions and convergents .......................1 01
IX. Fibonacci representation ..............................1 08
X. Search and games ...................................1 16
XI. Hyperbolic functions and Fibonacci numbers .................1 24
XII. Meta-Fibonacci sequences (a letter from B.W .C onolly) ..........1 27
XIII. The Golden Section in the plane .........................1 40
XIV.
The Golden Section in three-dimensional space ................1 49
Appendix ...........................................1 55
List of formulae .......................................1 76
References ..........................................1 85
n n
Table of F and L and their prime factors ......................1 88
Index .............................................1 90