Patterns, Fractals and  Attractors


Published 2000,  Update 07-January-2014


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Patterns Quotes

Norbert Wiener (1954) "The Human Use of Human Beings", Da Capo Press. p. 21, 95, 96 respectively.

"Messages are themselves a form of pattern and organization."

"Organism is opposed to chaos, to disintegration, to death, as message is to noise."

"A pattern is a message, and may be transmitted as a message."

"We are but whirlpools in a river of ever-flowing water."

"We are not stuff that abides, but patterns that perpetuate themselves."

"It is the pattern maintained by this homeostasis, which is the touchstone of our personal identity."


Carl G. Jung (1964) "Man and his Symbols", Aldus Books-Jupiter Books. p. 308.

"Our conscious representations are sometimes ordered (or arranged in a pattern) before they have became conscious to us."


Ni Hua-Ching (1983) "The Book of Changes and the Unchanging Truth", Seven Star Communications, Santa Monica. p. 23.

ying yang "All Movement, all change, is a pattern of the interaction of yin and yang."

Rucker R. (1987) "MIND TOOLS: The Five Levels of Mathematical Reality", Houghton Mifflin Company, Boston. p. 3.

"Mathematics is the study of pure pattern, and everything in the cosmos is a kind of pattern."


Rolf I. P.  (1989) "Rolfing",  Healing Art Press. p. 15.

"As in all matter organized into biological units, there is a pattern, an order, in human bodies."

Gleick J. and Porter E. (1990) "Nature's Chaos", Little, Brown and Company. p. 35.

"The human mind is a powerful pattern-recognition machine, more powerful than any computer yet built.
Sometimes it is too powerful; it detects patterns where they do not really exist.

Hardy G.H. (1992) "Mathematician's Apology", Canto ed., Cambridge University Press.  p. 84.

"A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas."

Casti L. J. (1992) "Reality Rules: I Picturing the World in Mathematics - The Fundamentals", John Willey. p. 229.

"We can think of the evolution of a cellular automaton as a pattern-recognition process, in which all initial configurations in the basin of attraction of a particular attractor are thought of as instances of some pattern with the attractor being the "archetype" of this pattern. Thus, the evolution of the different state trajectories toward this attractor constitutes recognition of the pattern."


Stewart (1995) "NATUR'S NUMBERS: Discovering Order and Pattern in the Universe", Phoenix. p. 1, 1, 3, 16, 21, 142, 168, respectably.

"We live in a universe of patterns."

"Nature's patterns are not just there to be admired, they are vital clues to the rules that govern natural processes."

"Patterns possess utility as well as beauty."

"Each of nature's patterns is a puzzle, nearly always a deep one."

"Mathematics is the science of patterns, and nature exploits just about every pattern that there is."

"Even when the consequence of natural laws seem to be patternless, the laws are still there and so are the patterns."

"Nature's patterns are "emergent phenomena, They emerged from an ocean of complexity ..."

Capra F. (1996) "The Web of Life", Anchor Books, p. 80.

"Understanding of life begins with the understanding of patterns."

Devlin K. (1998) "The Language Of Mathematics: Making the Invisible Visible", Freeman / Owl books. p. 3.

"...mathematics is the science of patterns."

Sprott. J. C. (2003) "Chaos and Time-Series Analysis", Oxford UP. p. 272.

"Fractals are to chaos what geometry is to algebra."

Mandelbrot  B. B. and Hudson R. L. (2004) "The (mis) Behavior of Markets", Basic Books. p. 21.

Patterns are the fool's gold of financial markets.

Pickover, C. A.  ed. (2005) "PATTERN BOOK: Fractals, Art and Nature", World Scientific. p. 403

Yuzo Nakano: "I did my research as an artist, not as a scientist: ... the reality of nature is the shape of patterns formed by the continual oscillation and transformation of energy."

Graham B. (2006) "The Intelligent Investor", Revised Edition, Updated with New Commentary by Jason Zweig, Collins. p. 220.

"Humans are pattern-seeking animals."

Wolfram S. (2009) "Cellular Automata as Models of Complexity" (1984).

"Natural systems from snowflakes to mollusc shells show a great diversity of complex patterns. The origins of such complexity can be investigated through mathematical models termed `cellular automata'."






My Fragments - From My Personal Point of View, The Complete list see Tamari page:


MF1:  Fractals relate to attractors as cobblestones in roads.   added: 1. January 2000.


MF3:  Fractalization of the Language  I      added 21 April 2002.

A word is worth a thousand ...?... .

A picture is worth a thousand words , as the old saying goes.

But sometimes: One good sentence is worth a thousand pictures.
for example the sentence: "A picture is worth a thousand words" (following Mobius strip). (added: 5-April-2007).

A metaphor is worth a thousand pictures.

An allegory is worth a thousand metaphors.

A .....?..... is worth a thousand allegories.



MF10:  Fractalization of the Language  II    added 15 November 2009.

Between any two 'points' in a set of: points, numbers, words, sentences, paragraphs, thoughts, etc.,
we can put another 'point',and if the set is not big enough, we can add another 'point'.


Wislawa Szymborska (2005) "COLON" "The nightmare of the poet" (translate Deena Land, 15-July-201)

"An allocated number of words. Never a needless one
Meaning - no poetry
And no philosophy and no religion.


MF9:  Fractals and chaos is like sewing and cooking.   added 28 March 2009.






Patterns in Nature:
Siberian Traps/Wadi HadramawtLake ErepecuNamibiaUtah Manantah Reservoir, Mali Jeff AluGreenland2012
Siberian Trap               Wadi Hadramaw            Lake Erepecu               Namibia                        Utah             Manantah Reservoir, Mali  Jeff Alu    Greenland 2012

volcanofiji IslandsThe Zagros Mountains - IranThe poleNature Fractal Christchurch, New Zealand
Volcano                           Fiji Islands                                      The Zagros Mountains, Iran           The pole                         Héctor Garrido       New Zealand  

Sand PatternsPele FingerprintsAlluvial FanIn Southern IranHimalaya
Zeph Van Allen              Springs                 Macaskill-Sand Patterns  Hanson-Pele Fingerprints    Alluvial Fan In Iran             Himalaya               Bill Burton - Puma        

NapoleoniteBeijing OncolitesPatternSand FractalDolomitPattern
Denovich-Napoleonite        Beijing Oncolites - Shanan  

Diffusion Limited Aggregation (DLA)

Rivers and Deltas:
Delta MesopotamiaDelta GangesDelta Delta the Selenga RiverDelta Nasser LakeDelta Greenlanddelta MissuriYukon delta
Mesopotamia   Ganges              Aerial Snaking Delta    Selenga River           Nasser Lake               Greenland dela                Missouri          Yukon delta - NASA

Cracked: glass, mud, metal, rocks and wood respectively:
Harold Edgerton Crack GlassCrack Glass Crack Glass Crack Glass Crack GlassCrack Glass
crack mudcrack mudcrack mudcrack mudcrack mudcrack snow
crack metalcrack metalcrack lavacrack rockscrack rockscrack asphalt
crack woodcrack woodcrack woodcrack woodcrack woodcrack woodcrack wood

Drops, Splashes and Bubbles:
Harold EdgertonRaffaele De MatolaIrene MullerHarold EdgertonsplashBubbleBubble
Harold Edgerton  

Flow, Whirlpools, Swirls, Vortices and Symmetry:
water patternsflawwater flawwater flawwater flawwater flawmargarinClark LittleflamsmokesmokefireVorticeVortice Ying Yangwhirlpool
NASA                                                                                                                                                                                Alexander Lillevik                  Symmetry

Short Movies of Water Patterns:  video: Ben Tamari, music: Mordechai Cohen
SwirlWater LightWater FlawWater Gold lineswater ringWater StarsWater Flaw
Double Swirl                         Water Moon Light           Water Flaw    Water Gold lines             Water Patterns                   Water stars                 Water Interference

CloudsCloudsCloudsCloudsCloudsCloudsAsh CloudAsh CloudAtom BombSuper cell Thunder storm
NASA                              NASA Mount C leveland           J. Warren                                           Atom bomb                            
Sarychev Peak Volcano

Multi Fractals:
Multi FractalsMulti Fractals
Multi FractalsMulti Fractals
Multi fractalMulti FractalsMulti Fractals
Multi FractalsMult iFractals  James Van Gundy
Multi FractalsmultifractalMulti Fractals/Multi Fractal

Lightning        Kirlian "lightning storm"    Static Electric Ball       Lichtenberg figure           Electric spark             Electric spark             Positive discharge

Diffractions and Interferences:
InterferencesInterferencesInterferencesQuasicrystals InterferencesinterferenceInterferences

Vibrations:                                                                                    Cymatics:
VibrationsVibrationsAtoms in an enzyme MoleculeVibrationsSound VibrationChladni Vibrationssound waves
SubAtomic Particles   Atom                                           Berenice Abbott      Peter Cadogan           Cymatics                                          Chladni                        Sound waves

Self Organization, Emergence and Complexity:
Self OrganizationSelf OrganizationSelf OrganizationSelf OrganizationSelf OrganizationSelf Organization Jerry MatchettSelf OrganizationSelf OrganizationSelf Organization
Space: The finel frontier               Gardhaia, Algeria       Robert B. Haas                                         Jerry Matchett

Astronomyspiral starsun eclypsesun
Saturn                                   spiral star                                      Sun

Microscopic fungus, Diffusion, Cellular Automata, Nano Technology:

Sordaria Fimicola AndersoncellCellular Automata Jonathan McCabeCellular Automata Jonathan McCabenano technologynano technology
Sordaria fimicola       Cell                               Jonathan McCabe                                        Nanotechnology 1          Nanotechnology 2

Sea Creatures:
diatomssea creaturesea creaturesea creature
Diatoms                                      Jellyfish 1                          Jellyfish 2                             Tina Carvalho    William White

sea creature
Emiliania Huxley                  Radiolarian                                                Maurice Burton


Quiver treeTree    baobabtreeTree  Dracaenatreetreestrees

Antonio Franchi          


skinskinsnake skingirapha skinzebra skinmetalagama tailfractalmuthbirdfractalpeacocksnail

Human Body:
brain Andre AasenbrainCyst Of Kidney
he Brain                             The Brain Scan          The Heart, McNally   The Eye, McClenaghan   The Ear, Nilsson   The Kidney, Kage      Cyst

Fractal 5D-2L   Comroe            Blood in The Brain                   Blood in The Head                             The Nerves in Brain             DNA molecule            Vitamin

seashell-houseMonastery Of Saint Anthony EgyptAnabela Sequeira LaosAustralia Sydney Opera HouseBelgrade Sreten PantelicIndiaMyanmarMoroccofractal
The Nautilus             Egept                                   Laos           Australia, Sydney Opera House   Belgrade     India       Myanmar           Morocco             France

Muqarnas Dome BaghdadIran VilaMIQELfractalchandelier fractal
India                        Muqarnas Dome, Baghdad   Iran Vila                      Italy, floor mosaic  Roman Vila             MIQEL                                                  Chandelier

Mandelbroth Esher LikeHokusai fractal LikeHiroshige fractal likepersian art fractal likealvador Dali  visage of warAfrican fractalPoul Klee Fugue in Red
Mandelbroth set    Hokusai                                                                Hiroshige          Iran art           Salvador Dali - visage of war African fractal   Klee-Fugue in Red

EscherHyperbolic Tiling William ShawPeitgen and Saupe Yuzo NakanoFelipe Arte Mandalaeltic circle fractalTom ShanonJos Leys fractal
Escher                        William T. Shaw          Peitgen and Saupe       Yuzo Nakano          Felipe Arte        Celtic circle fractal       Tom Shannon            Jos Leys

Yin Yang FractalMandalaMandalaMandalaHamsa Mira Krispel Ethiopian Cross
Yin and Yang             Mandala:                                                                                                                             Hamsa Mira Krispel
N EthiopianCross

Fractals in Computer 2 Dimensions (figured with FractInt).

KAM, Kolmogorov-Arnold-Moser     Dynamic                            Barnsley                                    Circle                                       Complex Newton

Mandelbrot set                      Lyapunov                                    Popcorn                                    Spider                                          Unit

Fractals in Computer 3 Dimensions:

Walrus uses 3D. hyperbolic geometry to display graphs under a fisheye-like distortion, it is being developed by Young Hyun at CAIDA.
Based on research by Tamara Munzner - figured with Walru s - Graph Visualization Tool .

Cobweb Robert Donovancubweb Moonjazzcubweb KieranCobwebcubwebcubweb

Almost Mathematical Creatures in Nature:
Mandelbrot Set
Almost Mandelbrot, Dendi Caldera EthiopiaAlmost Mandelbrot Paul CheslAlmost Mandelbrot Set John MumawAlmost Mandelbrot SetAlmost Mandelbrot Se Campello Boris Ice ThingAlmost Mandelbrot Set - Antarctica IceAlmost Mandelbrot Set Ben TamariAlmost Mandelbrot Set Ben Tamari
Dendi Caldera, Ethiopia   Paul Chesley, Yellowstone   North pole of Saturn         John  Mumaw       Campello Boris - Ice Thing
    Antartica's Moving Ice  

Mobius Strip                                 Sierpinski fractal                
Almost Mobius StripMount Sierpinski
Drew Patrick                            Martin Richard                             

Natural Fractals:
similaritySimilarityLightninng FractalFungsNaturalFractalNaturalFractal
Pickover                                              Designin Nature

Natural Attractors
Cats Eye NebulaCats Eye NebulaCats Eye Nebulamoon's gravity
An X Class Flare Region on the Sun  A  Mysterious Hexagonal Cloud System on Saturn   The Cat's Eye Nebula
TRACE Project NASA                                                                                                                                                            Russell E. Cooley IV

Attractors in Brain, Music, Economics and Earthquake:
graphics of Beethoven 9 1 movement
Tamari Attractor.gifA sand earthquake tracing pendulum
Walter J. Freeman                        
   Rolf Wallin              Beethoven - Stephen Malinowski      Tamari attractor    A sand earthquake tracing pendulum

Attractors in Computer 3 Dimensions: figured with software
Chua attractorIkeda AttractorLorenz AttractorPickover AttractorRossler attractorTamari Attractor
Chua attractor                    
 Ikeda Attractor                Lorenz Attractor         Pickover Attractor        Rossler Attractor          Tamari Attractor

Humor in Chaos, Fractals and Attractors:
Humor - Mini Fractal
Norway economicsHumor - fractalHumorHumor-BabushkaHumor - elevator fractalhomur- fractalFractal Hands.gif






Speculation 1, 19-October-2013 On Patternness (y) as function of Aggregation [ y = f(x) ].

The three Law's regimes (RSD) - The Random laws (gases-like), the Statistical (chaotic) laws (liquids-like) and the Deterministic laws (solids-like).

pattern aggregation determinism



Deterministic Laws: Lagrangian and Lorentz transformations

Statistical Laws: Weather - Lorenz Attractor and Economics - Tamari Attractor

Random Laws: Cellular Automata - The rule 110 of Stephan Wolfram and Games of life






Books and Articles:

John A. Adam (2003) "Mathematics In Nature: modeling patterns in the natural world", Princeton UP.

Donald M. Anderson (1961) "Elements of Design", Holt, Rinehart and Winston.

Michael F. Barnsley (1993) "Fractals EveryWhere", 2ed. Academic Press.

Michael F. Barnsley (2006) "SuperFractals: Patterns of Nature", Cambridge UP.

John Briggs (1992) "Fractals: The Patterns of Chaos", Simon and Schuster.

Peter Cadogan (1985) "From Quark to Quasar", Cambridge UP.

John L. Casti (1992) "Reality Rules: I Picturing the World in Mathematics - The Fundamentals", John Willey.

John L. Casti (1992) "Reality Rules: II Picturing the World in Mathematics - The Frontier", John Willey.

Keith Devlin (1998) "The Language Of Mathematics: Making the Invisible Visible", Freeman / Owl books.

Field and Golubitsky (1992) "Symmetry In Chaos: A Search for Pattern in Mathematics, Art and Nature", Oxford UP.

Priya Hemenway (2005) "Divine Proportion : F (Phi) In Art, Nature, and Science", Sterling Pub. NY.

Janus H. (1957) "Nature As Architect",  Frederick Ungar Pub. NY.

andelbrot B.B. (1983) "The Fractal Geometry of Nature", Freeman.

Mandelbrot B. B. (2004) "Fractals and Chaos", Springer.

Novak M. M. (Ed.), (2004) "Thinking in Patterns Fractals and related Phenomena in Nature", World Scientific.

Rucker R. (1987) "MIND TOOLS: The Five Levels of Mathematical Reality", Houghton Mifflin Company, Boston.

Peitgen and Saupe, (Ed.), (1988) "The Science of Fractal Images", Springer-Verlag.

Peitgen, Jurgens and Saupe (1992) "Chaos and Fractals, New Frontiers Of Science", Springer-Verlag.

Pickover C.A. (Ed.), (2005) "PATTERN BOOK: Fractals, Art and Nature", World Scientific.

Pickover C. A. (2009) "THE MATH BOOK: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics", Sterling.

Sprott J. C. (1993) "Strange Attractors: Creating Patterns in Chaos", M T Books, NY.

Stewart I (1995) "NATUR'S NUMBERS: Discovering Order and Pattern in the Universe", Phoenix.

Wille L. T. (Ed.), (2004) "New Directions in Statistical Physics: Econophysics, Bioinformatics, and Pattern Recognition", Springer.

Wolfram S. (2002) "A New Kind of  Science", Wolfram Media, Inc..


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