Shapes and Forms 2011

8/3/2019 Shapes and Forms 2011

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Shapes andForms in Nature, from the perspective ofComplexity Science

Dr. Florin Munteanu


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About Complexity Science

Shapes and Forms (natural and artifact) The essential role of the recursive process The golden number

The golden Volume The n-D generalization of the Golden cut Conclusions for a natural technology

Summary:


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The science of XX-th century bring up the importance of theMan in the metabolism of the socio-economical system (GAIAvision)

- The science of Complex Systems that define anonlinear approach on Reality ( fractals, chaos theory, dissipative systems, constructal theory etc.)

- Computational science (or scientific computing ) thatcreate extraordinary tools for the modeling and simulations of the

dynamic and evolutions of real objects- Quantum physics, Cognitive Science focus on the

importance of the Observer and his intentionality on the evolution ofthe Reality

- The science of Mind a new approach on the study ofthe Consciousness and Free Will

Breakthroughs of XX-th century


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Latin word complexus , signifies "entwined", "twisted together". This maybe interpreted in the following way: in order to have a complex you need two ormore components, which are joined in such a way that it is difficult to separate

them. Similarly, the Oxford Dictionary defines something as "complex" if it is"made of (usually several) closely connected parts".

Escher

To start with a definition:Complexity science


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Some islands of the complexity theory

An important knowledgedatabase;

new scientific domains; a candidate for a new

paradigm = Complexity


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www.complexity.ro

Order, PredictabilityAnalitical approach

Disorder, Randomness,Statistical approach

Between Order and disorder arealm of Complexity

Disipative systems, PatternsBifurcations,INFORMATION sensitivity to initial conditions

self-organize criticalitycontext dependency

synchronizations of chaoticoscillators new modelsnew tools

new techniques

Dissipative systems Ilya Prigogine's non-equilibrium thermodynamics

www.complexity.ro


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Concept Constrains Control Regulation Data, codess,

patterns, Procedures Instructions

A profound bifurcaton1) The studies of the flux of energy and matter

2) The studies of the generaton,propagation and decoding messagesform a streamimg of data (embedded ina energo-material flux).

Perception Representaton Knowledge Meaning Wisdom,

Semantic pocessorComunication

PASIV

ACTIV

reacton

signal

meaning

Interactons, force, impusl, inteactions, gradient,thermodynamics, Dynamical systems,


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www.complexity.ro

Shapes and Forms in Nature


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www.complexity.ro

Pattern

FORM +

DYNAMICS

www.complexity.ro


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www.complexity.ro

Behavior

FORM + DYNAMICS +

history www.complexity.ro


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Foto David Hall

Notice the differencesbetween Natural andArtifact


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Foto Chuon Szen Ong

Self similar, rough, never identical

Smooth, ever identical, notrelated with environment..


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Internal streaming of information;self referential in growth o David Hall

Project

Cognitive processInnovation

Natural vs. Artifact

Semanticfiled ?

www.complexity.ro

Logos?

external streaming of

information- thehuman mind

Discovery,


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Complicated

or complex?

Photo by Dinu Lazar

or COMPLEX


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Complicated to Euclid

Mankind has simplified Nature in order to copy, control, and dominate it.Structure vs. reproducibility


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but not to Mandelbrot!

Benoit B. Mandelbrothttp://www.math.yale.edu/users/mandelbrot/

The need for a new language


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The recursion surprise


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The recursion surprise


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Formal recursion: IFS

i

i

x

y

f

f

x

y

T ...

n

n

T T T T

1

1 1 2

cos sin 0

sin cos 0

N N

i i i i

b a T R A B

b a

N =3a i =1/2b i =0, 1/4, 1/2

F lim i n n

i

x T

y

n T F F Fixed point of T


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Recursively generated structures

If such geometric objects are materialized,what kind of new physical properties mayhave? How it shows the field distributionaround them if through them pass an electriccurrent, or are exposed to mechanical 1/fnoise, or ...


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Fractal antennas

Fractal antenna theory uses a fractalgeometry that is a natural extensionof Euclidian geometry.

The geometry of the fractal antenna isa candidate for a multiband solution and also as a small (physical size)antenna . We expect that a self-similarantenna (which contains many copiesof itself at several scales) operate in asimilar way at several wavelengths.


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From Golden Section toGolden Volume- the need for a new approach in the design of Artifact- From efficiency to harmony - Beyond functionality, esthetics, ergonomics


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bringing the philosophy into engineering in a natural and useful way =contemplate and understand Natural things


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Qualitative and Quantitative

Morphogenesis - qualitative growth - structuringprocess

Development - quantitative growth , which preservethe form, but increase the weight (homothetic growth)

A1 < A2 < A3 B1 < B2 < B3

t1 < t2 < t3 A1/B2 = A2/B2=..ct


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1,1,2,3,5,8,13,21,34,55,89,..


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the center of convergence of the recursive process.Fragmentation" of a golden rectangle

B4

Ag

If A0 B0 B1 B2 is a golden rectangle,then:A0, Ag, A2 are collinearA1, Ag, A3 are collinearThe two lines are cut at right anglesEtc. ...


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esthetics and the theory of art - Matila Ghyka
http://en.wikipedia.org/wiki/Matila_Ghykahttp://en.wikipedia.org/wiki/Matila_Ghykahttp://en.wikipedia.org/wiki/Matila_Ghykahttp://en.wikipedia.org/wiki/Matila_Ghyka


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A0

Golden section - a prototype of thinking!Can be extrapolated in 3D? but in nD?

1,1,2,3,5,xn,,

xn+1,

x n+1 /x n -> 1.618034

x2-x-1=0; x= (1+ 5)/2 =

A A 1

s1

As2

a

b

a/b=xA0 / As1 = A1 /A s2

xn+2 =xn-1 +x n


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1,1,1, 2,2, 3,4 ,5, 7, an,, a n+1,

a n+1 /a n -> ?

x3-x-1=0;

What we discover was anoriginal geometrical 3Dinterpretation of


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The plastic number

In mathematics , the plastic number (also known as the plastic constant ) is a mathematical

constant which is the unique real solution of the cubic equation

It has the value [2]

its decimal expansion begins with 1.324717957244746025960908854....

The plastic number is also sometimes called the silver number


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The nameplastic number (het plastische getal inDutch) was given to this number in 1928 by DomHans van der Laan. Unlike the names of thegolden ratioandsilver ratio, the word plastic was notintended to refer to a specific substance, but rather in its adjectival sense, meaning somebe given a three-dimensional shape


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The GoldenVolume


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The nameplastic number (het plastische getal inDutch) was given to this number in 1928 by DomHans van der Laan. Unlike the names of thegolden ratioandsilver ratio, the word plastic was notintended to refer to a specific substance, but rather in its adjectival sense, meaning somebe given a three-dimensional shape


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2D alfa = 45 0 3D alfa = 60 0


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1988-89 3Darchetype ?

45 o

60 o

The social revolution of 1989 a result of aprofound change in Logos? Something

more profound is happening?

Leafs from the same tree


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Let V1 (n), an initial n-dimensionalparallelepiped; the sides L1> L2> L3 >...> Ln,which can be viewed as a set of type:

we cut from V1 (n), a parallelipipedicvolume:

whose projection in 0x 1xn plan is a square. The remaining volume,denotedby C 1 (n) is defined as:


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In these conditions,

Quantitative relationship between the parts resulting from this iterative algorithm(analogous from our adopted model in the transition from the 2D in 3D) is:

and lead to determine the n- dimensional "golden number in a similar manner to that defined in the transition from 2D to 3D

n = spacedimension


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n=2 x2=x+1 x(x-1)-1=0 = 1.618034 Golden 2D number; Golden sectionn=3 x3=x+1 x(x-1)2-1=0 = 1.324717. Golden 3D number; Golden Volumen=4 x4=x+1 x(x-1)3-1=0 4=1.220744 n=5 x5=x+1 x(x-1)4-1=0 5=1.1673043..

n xn=x+1 x(x-1)n-1-1 =0 XP is a positive real root of theequation x(x-1)n-1-1 =0

Returning to the general case of n-dimensional, can becalculated:a) The arithmetic mean of the sides of the body:

b) The geometric mean of the sides:


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it can verify the expresion:

c) The harmonic mean of the sides of a golden volume, generated in n-dimensional space:

The Volume of the n-dimensional golden body is equal to the Mg of itssides, raised to the power n:

http://www.csc.matco.ro/conferinta.html
http://www.csc.matco.ro/conferinta.htmlhttp://www.csc.matco.ro/conferinta.html


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2004 Summer school in Gura HumoruluiWe build a golden Volume, talking about natural engineering


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Philosophers, Engineers, Artistsstudents and professors together


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Thanks for your attention

Florin [email protected] http://implexus.org/profile/FlorinMunteanu
mailto:[email protected]://implexus.org/profile/FlorinMunteanuhttp://implexus.org/profile/FlorinMunteanumailto:[email protected]

Shapes and Forms 2011

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