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

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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."

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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."

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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."

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Stewart (1995). "Nature's Numbers: Discovering Order and Pattern in the Universe." Phoenix. p. 1, 1, 3, 16, 21, 142, 168, respectively.

"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 ..."

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

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

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

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

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

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

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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."

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Pickover, C.A. (Ed.) (2005). "The 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."

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Graham, B. (2006). "The Intelligent Investor." Revised edition, Updated with new commentary by Jason Zweig. Collins, p. 220.

"Humans are pattern-seeking animals."

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

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

Pickover, C.A. (2015). "The Mathematics Devotional." Sterling, New York, p. 1.

"Some of the individuals who are quoted have likened mathematicians to artists who seek patterns and who have aesthetic perspectives          on their subject matter." 

My Fragments - From My Personal Point of View

For the complete list, please see Tamari.

MF1:  Fractals relate to attractors like cobblestones in roads.   Added 1 January 2000.

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

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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: "A picture is worth a thousand words" (like a Mobius strip).

A metaphor is worth a thousand pictures.

An allegory is worth a thousand metaphors.

A ? is worth a thousand allegories.

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

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

     Wislawa Szymborska (2005) "Colon" "The nightmare of the poet" (trans. by Deena Land, 15 July 2010).

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

Patterns

God is an economical architect. He visited our earth, left a number of “master plans” that would continue operating on their own based on unknown principles (including a feedback mechanism), and then departed for subsequent tasks. Now we must decipher them using the “tools” at our disposal – tools created in His image, according to His principles and His plans. These tools are: our intellect, our imagination and our skills, as well as our predisposition for language, metaphor and mathematics. Following is an attempt to compile, and as far as possible to classify, a collection of the patterns around us, which the author believes will help us understand, when the time comes, the master plans that He bequeathed to us, and that depict the “laws” in effect (like scaffolding that is removed after a building is constructed). For an example of this kind of attempt, see Speculation 1.

Patterns in Nature:








Dendrites:




Rivers and Deltas:




Cracks: Glass, Mud, Metal, Rocks and Wood respectively:







Drops, Splashes and Bubbles:




Currents, Whirlpools, Swirls, Vortices and Symmetry:





Short Movies of Water Patterns: Video: Ben Tamari, Music: Mordechai Cohen




Clouds:




Multifractals:





Electricity:




Diffractions:




Interferences:




Vibrations and Cymatics:




Self-Organization, Emergence and Complexity:




Astronomy:





Cellular Automata, Nano-technology:
         



Sea Creatures, Diatoms and Microscopic Fungi:




Shells:




Trees:




Leaves:




Plants:




Animals:




The Human Body and...:






Architecture:





Arts:




Graphics:




Religion and Symbols:




2D Computer-Generated Fractals, figured with FractInt software.


 


3D Computer-Generated Fractals and more:




Cobwebs:




Near-Mathematical Creatures in Nature:
Mandelbrot Set:



Mobius Strip:                   Sierpinski Fractal:                



Natural Fractals:



Natural Attractors:



Attractors in the Brain, Music, Economics and Earthquakes:




3D Computer-Generated Attractors:

Chua, Ikeda, Lorenz, Pickover, Rossler and Tamari attractors respectively, figured with  Eco2 software.



Humor in Attractors and Fractals: