Fractals

Oliver Linton

Description

What are fractals? Why are they such fun? How do you make one? Why is a dripping tap not as random as it seems? What is chaos? Is the Mandelbrot Set really the most complex object in mathematics? In this beautifully illustrated book, fractal-hunter Oliver Linton takes us on a fascinating journey into the mathematics of fractals and chaos, diving into many kinds of self- similar structures to reveal some of the most recently discovered and intriguing patterns in science and nature. "Fascinating" FINANCIAL TIMES. "Beautiful" LONDON REVIEW OF BOOKS. "Rich and Artful" THE LANCET. "Genuinely mind-expanding" FORTEAN TIMES. "Excellent" NEW SCIENTIST. "Stunning" NEW YORK TIMES. Small books, big ideas.

Sujets

• Sciences et techniques
• Mathematics

Informations

• Publié par: Wooden Books
• Date de parution: 20 mai 2021
• Nombre de lectures: 0
• EAN13: 9781912706136
• Langue: English
• Poids de l'ouvrage: 5 Mo

Extrait

A b o ve : R o ma ne s c o b r o c c o l i . Th i s a sto n i sh i n g e geta l e ha s a wo nd e r f u l l y s e l f - s i m i l a s tru c t u r e, t yp i c a l of f r a a l s , w i t h f l o r et te s h i c a r e sma l l e e r s i o n s of t he ho l e.
First published 20 1 9 eBook edition © W ooden Books Ltd 20 1 8 Published b y W ooden Books Ltd. Glastonbury , Somerset. British Libr ary Ca taloguing in Publica tion Da ta Linton, O. F r a ct als, O n t he Ed g e o f Cha o s A CIP ca talogue r ecor d f or this book ma y be obtained fr om the British Libr ary . eBook ISBN: 978-1-9 1 2706-1 3-6 Ph y sical ISBN : 978-1-90426 3-98-2 All rig hts r eserv ed. For permission to r epr oduce an y p art of this amazing lit tle book please contact the publishers. Designed and typeset in Glastonbury , UK . Con v erted and optimised f or digital displa y b y the eBook P artnership, UK.
Oliver Linton
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C O NTENT S Intr oduction 1 Fr actals in N a tur e 2 The K och Sno wflak e 4 Hausdorf f Dimensions 6 L-S y stems 8 Sp ace-Filling Curv es 10 Carpets and Spong es 12 For d Cir cles 14 The Chaos Game 16 Iter a ted Functions 18 The Barnsle y Fern 20 Hop along Fr actals 22 The Logistic Map 24 A t tr actors and R epellors 26 Islands of Stability 28 Chaos in the R eal W orld 30 Chaos in the Solar S y stem 32 Julia Sets 34 The Escape Alg orithm 36 Fr om Julias to Mandelbr ot 38 The Mandelbr ot Map 40 Zooming In 42 Labelling the Lobes 44 Ax ons and S ynapses 46 Iter a tiv e Orbits 48 Mor e Far e y Magic 50 The Antenna 52 Or der and Chaos 54 N e wton Raphson Fr actals 56 Appendix: Fr actal Ma thema tics 58
A b o ve : e i n te ri o d e c o a t i o n of t he d o me of t he S heik h L otf o a M o s q u e i n I s f a a n , I a n u s e s f a a l te c n i q u e s of ep et i t i o n a nd s c a l i n g to m i o f a a l s f ou nd i n na t u e.
1 INTR O DUC TI O N Clouds ar e not sp her es, mount ains ar e not cones, coast lines ar e not cir c les, and bar k is not smoot h, nor d oes lightnin g tr av e l in a str aight line. Benoit Mandelbr ot: The Fr actal Geometry of N a tur e (1 982)
F 20 0 0 s , ma thema tician , cienti t and philo opher , blinded b y the pr eci ion of Euclidean g eometry , a umed tha t e v erything in the w orld ar ound u could be built up fr om pher e , cone , cir cle , mooth plane and tr aig ht line . The y w er e not entir ely wr ong: much can be learned b y modelling a tom a pher e , f ace a multif aceted polyhedr a and hurricane wind a tr aig ht or cir cular . The r ea on f or thi i ma thema tical econom y . A pher e i completely de cribed b y a ing le number— it r adiu ; a triang le b y thr ee— the length of it thr ee ide . E v en a hurricane i larg ely de cribed b y ju t tw o number —it peed of r ota tion a t a char acteri tic diameter . But to de cribe a cloud or a co a tline in detail r equir e million of number . Wha t w ould be the point? By the time y ou had writ ten do wn all tho e number , the cloud w ould ha v e long ince v ani hed. Co a tline , ho w e v er , ar e mor e permanent and mor e important too. Ha v e y ou e v er w onder ed ho w man y number ar e needed to pecify the map in y our Sa tN a v? The an w er i liter ally billion . But in 1 982, a brilliant P oli h born ma thema tician named Benoit Mandelbr ot ho w ed the w orld tha t it w a po ible— ometime a t lea t— to de cribe complex tructur e lik e cloud and co a tline a ea ily a pher e and line , and fr actal g eometry w a born. Thi book a t tempt to de cribe the r e v olution in ma thema tic and art which f ollo w ed.
2
Fr
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Na
ture and little fleas have lesser fleas Ho w can y ou describe a tr ee ? One w a y is to sa y tha t a tr ee is a stick with tw o or mor e smaller tr ees sticking out of it. This ma y not be v ery poetic, but once y ou ha v e specified ho w man y br anches the tr ee must ha v e and wha t the ang les the br anches mak e with the stem it is sufficiently accur a te to enable a computer to dr a w a quite r ealistic one ( be low left ). Lik e wise a nautilus shell ma y be described as a smaller nautilus shell with an extr a segment added ( opp o sit e low er left ). In each case the whole is defined in terms of a smaller p art of it. Wh y ar e so man y structur es in N a tur e fr actal? One r eason is tha t fr actals of ten arise spontaneously out of the r epea ted applica tion of r ela tiv ely simple rules ( s ee riv er d e lt a opp o sit e ). N a tur e also uses fr actals f or r easons of econom y ( s ee le af v eins pat t ern be low right ), as fr actal structur es ar e of ten the most efficient w a y of perf orming a task, fr om a tr ee absorbing sunlig ht to a lung tr ansf erring o xy g en to the blood.