Table Of ContentA Concise Introduction to
Hypercomplex
Fractals
A Concise Introduction to
Hypercomplex
Fractals
Andrzej Katunin
Boca Raton London New York
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Contents
Preface vii
Chapter 1(cid:4) Introduction to Fractals on a Complex Plane 1
1.1 BIRTH OF COMPLEX FRACTALS – JULIA AND
FATOU SETS 3
1.1.1 Preliminaries 3
1.1.2 Coloring FractalSetson aComplex Plane 6
1.1.3 Julia and Fatou Sets within the Polynomials of
Various Orders 12
1.1.4 Other Variations of Julia and Fatou Sets on a
Complex Plane 16
1.2 FRACTAL REVOLUTION – THE MANDELBROT SET 19
1.2.1 History andConceptof theMandelbrot Set 19
1.2.2 Geometrical Structure and Atoms of the
Mandelbrot Set 20
1.2.3 Mandelbrot Set within the Polynomials of
Various OrdersandOtherVariations 25
1.3 SEARCHING FOR THE HOLY GRAIL – 3-D
MANDELBROT SET 29
1.4 WHAT’S NEXT? FURTHER EXTENSIONS OF THE
MANDELBROT SET 33
Chapter 2(cid:4) Fractal Beasts in Terms of Normed Division
and Clifford Algebras 37
v
vi (cid:4) Contents
2.1 QUATERNIONIC FRACTAL SETS 38
2.1.1 History andPreliminaries ofQuaternionCalculus 38
2.1.2 QuaternionicJulia andMandelbrotSets 39
2.2 OCTONIONIC FRACTALSETS 49
2.2.1 History andPreliminaries ofOctonions 49
2.2.2 Octonionic Julia andMandelbrotSets 51
2.3 IS THERE ANYTHING ALIVE IN HIGHER-
DIMENSIONAL HYPERCOMPLEX SPACES? 55
2.3.1 Preliminaries toClifford Algebras 56
2.3.2 Cliffordean HypercomplexFractalSets 57
2.4 THE PLACES (AND HYPERSPACES) WHERE
FRACTALS CANNOT LIVE 58
Chapter 3(cid:4) Tensor Product Fractal Mutants 63
3.1 BICOMPLEX, TRICOMPLEX, ... MULTICOMPLEX
FRACTAL SETS 63
3.1.1 Julia and Mandelbrot Sets in Bicomplex Vector
Space 63
3.1.2 Julia and Mandelbrot Sets in Tricomplex Vector
Space 70
3.1.3 Julia and Mandelbrot Sets in Multicomplex
Vector Space 72
3.2 TENSOR PRODUCT HYPERCOMPLEX AND
MULTIHYPERCOMPLEXFRACTAL SETS 73
3.2.1 Julia and Mandelbrot Sets in Biquaternionic
Vector Space 73
3.2.2 BeyondBiquaternionicJulia andMandelbrotSets 78
Bibliography 81
Index 93
Preface
Since I started working with fractals, my family, friends, colleagues, and
studentswerepuzzledhearingabouthypercomplexfractals.Thisismainly
duetothespecificnomenclatureusedbothinthefractaltheoryaswellas
inthehypercomplexnumbertheory,withalloftheirattractors,repellers,
octonions, zero divisors, and nilpotents. These words naturally scare
them! And at the same time, they love colorful images generated by
my computersimulations. Theseimages, however, are just nice pictures
to them.I amconvinced that manynon scientists, whohad contact with
complex and hypercomplex fractals have the same feelings. Therefore, I
decidedtowriteabookwhichwouldpresentcomplexandhypercomplex
fractals in a concise and comprehensible manner omitting mathematical
formalism as muchas possible. This ideahas germinatedin mymind for
afewyears,andfinallyIcanplacethefruitofafewyears’workintoyour
hands.
This book concisely presents the complete story on complex and
hypercomplex fractals, starting from the very first steps in complex
dynamics and resulting complex fractal sets, through the generalizations
of Julia and Mandelbrot sets on a complex plane and the Holy Grail of
fractalgeometry–a3 DMandelbrotset,andendingwithhypercomplex,
multicomplex, and multihypercomplex fractal sets that are still under
consideration of scientists. I tried to write this book in a simple way in
order tomake it understandableto most people whose math knowledge
covers the fundamentals of complex numbers only. Moreover, the book
is full of illustrations of generated fractals and stories about great
mathematicians,numberspaces,andrelatedfractals.Inmostcases,only
informationrequiredforproperunderstandingofanatureofagivenvector
space or a construction of a given fractal set is provided; nevertheless, a
moreadvancedreadermaytreatthisbookasafundamentalcompendium
on hypercomplex fractals, with references to purely scientific issues like
dynamicsandstability ofhypercomplexsystems.
The preparation of this book would not be possible without the
reviewers:ProfessorWojciechChojnackifromtheUniversityofAdelaide,
vii
viii (cid:4) Preface
Australia,whoisanoutstandingspecialistincomputerscience,computer
graphics,andmathematicsrelatedtothesedisciplines, andDr.Krzysztof
GdawiecfromtheUniversityofSilesia,Poland,whoistheeminentfractal
researcher working on developing new types of fractal sets. They both
contributed many corrections and additions to make this book even
better, and I am very grateful to them for their valuable comments and
discussion.
I would like to thank my dear fiance´e, Angelika, who supported me
duringmywritingofthisbook,readitsdraftversion,andprovidedagreat
feedback.Iamverygratefultoherforthis help.
Finally,IwouldliketothankCRCeditorialstaff,especiallyRickAdams,
Jessica Vega, and Robin Lloyd Starkes, for their great and professional
supportduringtheentiretyofthepublishingprocess.
AndrzejKatunin
January,2017
1
CHAPTER
Introduction to Fractals
on a Complex Plane
Fractal.Thismysteriouswordpenetratesmodernpop cultureandsociety.
Butwhatdoesit mean?Whatis the definition ofa fractal?
Actually, considering a great variety of types, shapes, and properties
ofobjects thatarecalled fractals, itis really hardtoformulateauniversal
definition. In 1975, the word “fractal” was introduced by Benoit B.
Mandelbrot, the father of a fractal geometry, and popularized in his
famous book, The Fractal Geometry of Nature [79]. It comes from the
Latin fractus, which means “broken” or “fractured.” This name explains
itsnature,whichisusuallycharacterizedbyaverycomplexshapeand,in
mostcases,fractionaldimension.Beyondthisproperty,fractalshavefew
otherdifferencesfromothergeometricalobjects.Thenextpropertyisthe
self similarity of fractals, which means that they looks exactly the same
nomatterhowbigthemagnificationofthefractalsare.Roughlyspeaking,
fractals are constructed from smaller copies of themselves. And the last
thing that characterizes a fractal (which results from already presented
properties) – it cannot be represented by a closed form expression, but
byarecurrentdependency.
However, fractals and hierarchical structure of objects were known
a long time before the Mandelbrot. The best examples are the Indian
temples built in the Middle Ages1 (see example in Figure 1.1). Many
proofs of self similarity of Hindu temples were given by numerous
1StrikingexamplesofIndiantemplesthatuseself similarityintheirconstructionsare
the Kandariya Mahadeva Temple built in 1030 in Khajuraho, the towers of Meenakshi
AmmanTemplebuiltin1623–1655inMadurai,theShvetaVarahaswamyTemplebuiltin
1673–1704inMysore,andmanyothers.
1
