Shapes and

Forms in Nature, from the perspective of

Complexity Science

Dr. Florin Munteanu

• 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:

The science of XX-th century bring up the importance of the

Man in the metabolism of the socio-economical system (GAIA

vision)

- The science of Complex Systems that define a

nonlinear approach on Reality (fractals, chaos theory, dissipative

systems, constructal theory etc.)

- Computational science (or scientific computing) that

create 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 of

the Reality

- The science of Mind – a new approach on the study of

the Consciousness and Free Will

Breakthroughs of XX-th century

Latin word complexus, signifies "entwined", "twisted together". This may

be interpreted in the following way: in order to have a complex you need two or

more 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

Some islands of the complexity theory

• An important knowledge

database;

• new scientific domains;

• a candidate for a new

paradigm = Complexity

www.complexity.ro

Order, Predictability

Analitical approach Disorder, Randomness,

Statistical approach

Between Order and disorder a

realm of Complexity

Disipative systems, Patterns

Bifurcations,

INFORMATION •sensitivity to initial conditions

•self-organize criticality

•context dependency

•synchronizations of chaotic

oscillators

……

•new models

•new tools

•new techniques

Dissipative systems

Ilya Prigogine's non-equilibrium thermodynamics

www.complexity.ro

• Concept

• Constrains

• Control

• Regulation

• Data, codess, patterns,

• Procedures

• Instructions

A profound bifurcaton

1) The studies of the flux of energy and matter

2) The studies of the generaton,

propagation and decoding messages

form a streamimg of data (embedded in

a energo-material flux). • Perception

• Representaton

• Knowledge

• Meaning

• Wisdom,

Semantic pocessor Comunication

PASIV

ACTIV

reacton

signal

meaning

Interactons, force, impusl, inteactions, gradient,

thermodynamics, Dynamical systems,

www.complexity.ro

Shapes and Forms in Nature

www.complexity.ro

Pattern

FORM +

DYNAMICS

www.complexity.ro

www.complexity.ro

Behavior

FORM + DYNAMICS + “history”

www.complexity.ro

Foto David Hall

Notice the differences

between Natural and

Artifact…

Foto Chuon Szen Ong

Self – similar, rough, never identical …

Smooth, ever identical, not

related with environment..

Internal streaming of information;

self referential in growth o David Hall

Project

Cognitive process Innovation

Natural vs. Artifact

Semantic

filed ?

www.complexity.ro

Logos?

external streaming of information- the human mind

Discovery,

Complicated…

…or complex?

Photo by Dinu Lazar

or COMPLEX

Complicated to Euclid…

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

“Structure” vs. reproducibility

…but not to Mandelbrot!

Benoit B. Mandelbrot

http://www.math.yale.edu/users/mandelbrot/ The need for a new language

The recursion surprise

The recursion surprise

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

ii i i

baT R A B

ba

N=3

ai =1/2

bi =0, 1/4, 1/2 F lim in

ni

xT

y

nT F FFixed point of T

Recursively generated structures

If such geometric objects are materialized,

what kind of new physical properties may

have? How it shows the field distribution

around them if through them pass an electric

current, or are exposed to mechanical 1/f

noise, or ...

Fractal antennas

• Fractal antenna theory uses a fractal geometry that is a natural extension of Euclidian geometry.

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

From Golden Section to

Golden Volume - the need for a new approach in the design of Artifact

- From efficiency to … harmony

- Beyond functionality, esthetics, ergonomics …

bringing the philosophy into engineering in a natural and useful way =

contemplate and understand Natural things

Qualitative and Quantitative

Morphogenesis - qualitative growth - structuring

process

Development - quantitative growth, which preserve

the form, but increase the weight (homothetic growth)

A1 < A2 < A3…

B1 < B2 < B3 …

t1 < t2 < t3 … A1/B2 = A2/B2=..ct

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

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 collinear

A1, Ag, A3 are collinear

The two lines are cut at right angles

Etc. ...

esthetics and the theory of art - Matila Ghyka

A0

Golden section - a prototype of thinking!

Can be extrapolated in 3D? …but in nD?

1,1,2,3,5,… xn,, x n+1,…

x n+1/xn -> 1.618034…

x2-x-1=0; x= (1+√5)/2 = Φ

A A

1

s1

A s2

a

b

a/b=x

A0 / As1 = A1/A s2

xn+2=xn-1+x n

1,1,1,2,2,3,4,5,7,… an,, a n+1,…

a n+1/an -> ?

x3-x-1=0;

What we discover was an

original geometrical 3D

interpretation of …

… The plastic number

In mathematics, the plastic number ρ (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

The name plastic number (het plastische getal in Dutch) was given to this number in 1928 by Dom Hans van der Laan. Unlike the names of the golden ratio and silver ratio, the word plastic was not intended to refer to a specific substance, but rather in its adjectival sense, meaning something that can be given a three-dimensional shape

An other approach to the same

problem

The Golden

Volume

The name plastic number (het plastische getal in Dutch) was given to this number in 1928 by Dom Hans van der Laan. Unlike the names of the golden ratio and silver ratio, the word plastic was not intended to refer to a specific substance, but rather in its adjectival sense, meaning something that can be given a three-dimensional shape

2D alfa = 45 0

3D alfa = 60 0

1988-89 3D

archetype ?

45 o

60 o

The social revolution of 1989… a result of a

profound change in… Logos? Something

more profound is happening?

Leafs from the same tree

Let V1 (n), an initial n-dimensional

parallelepiped; the sides L1> L2> L3 >...> Ln,

which can be viewed as a set of type:

we cut from V1 (n), a parallelipipedic

volume:

whose projection in 0x1xn plan is a square. The remaining volume,denoted

by C 1 (n) is defined as:

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 = space

dimension

n=2 x2=x+1 x(x-1)-1=0 ɸ= 1.618034… Golden 2D number; Golden section

n=3 x3=x+1 x(x-1)2-1=0 ξ = 1.324717…. Golden 3D number; Golden Volume

n=4 x4=x+1 x(x-1)3-1=0 Ψ4=1.220744…

n=5 x5=x+1 x(x-1)4-1=0 Ψ5=1.1673043..

n xn=x+1 x(x-1)n-1-1 =0

XP is a positive real root of the equation x(x-1)

n-1-1 =0

Returning to the general case of n-dimensional, can be

calculated:

a) The arithmetic mean of the sides of the body:

b) The geometric mean of the sides:

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 its

sides, raised to the power n:

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

2004 Summer school in Gura Humorului

We build a golden Volume, talking about natural engineering

Philosophers, Engineers, Artists…

students and professors together …

Thanks for your attention

Florin Munteanu

florin@complexity.ro http://implexus.org/profile/FlorinMunteanu