[Webinar] SpiraTest - Setting New Standards in Quality Assurance
Pheade 2011
1. Shapes and
Forms in Nature,
from the perspective of
Complexity Science
Dr. Florin Munteanu
2. Summary:
• 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
3. Breakthroughs of XX-th century
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
4. Complexity science
To start with a definition:
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
5. Some islands of the complexity theory
• An important knowledge
database;
• new scientific domains;
• a candidate for a new
paradigm = Complexity
6. Dissipative systems
Ilya Prigogine's non-equilibrium thermodynamics
Order, Predictability Between Order and disorder a Disorder, Randomness,
Analitical approach realm of Complexity Statistical approach
Disipative systems, Patterns
Bifurcations,
•sensitivity to initial conditions
INFORMATION
•self-organize criticality
•context dependency
•synchronizations of chaotic
oscillators
……
•new models
•new tools
www.complexity.ro •new techniques
www.complexity.ro
7. A profound bifurcaton
1) The studies of the flux of energy and matter
Interactons, force, impusl, inteactions, gradient,
thermodynamics, Dynamical systems,
2) The studies of the generaton,
propagation and decoding messages
form a streamimg of data (embedded in • Concept • Perception
a energo-material flux). • Constrains • Representaton
• Control • Knowledge
• Regulation • Meaning
• Data, codess, • Wisdom,
ACTIV patterns,
signal Comunication • Procedures
Semantic pocessor
PASIV • Instructions
meaning reacton
12. Self – similar, rough, never identical …
Smooth, ever identical, not
Foto Chuon Szen Ong
related with environment..
13. Natural vs. Artifact Logos?
external
streaming of
information- the
human mind
Project
Semantic
filed ?
Innovation
Internal streaming of information; Cognitive process Discovery,
self referential in growth o David Hall
www.complexity.ro
14. Complicated…
…or complex?
or COMPLEX
Photo by Dinu Lazar
15. Complicated to Euclid…
Mankind has simplified Nature in order to copy, control, and dominate it.
“Structure” vs. reproducibility
16. …but not to Mandelbrot!
Benoit B. Mandelbrot
http://www.math.yale.edu/users/mandelbrot/
The need for a new language
19. Formal recursion: IFS
xi T xf T n T T ... T
yi yf n
N N
cos sin a 0 b1
T R A B i
b
i 1 i 1 sin cos 0 a 2 i
xi
N=3
F limT
n ai =1/2
T nF F
yi
n bi =0, 1/4, 1/2
Fixed point of T
20. 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 ...
21. 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.
22.
23. 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 …
24. bringing the philosophy into engineering in a natural and useful way =
contemplate and understand Natural things
25. 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
27. Ag – the center of convergence of the recursive process.
“Fragmentation" of a golden rectangle
B4
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. ...
30. Golden section - a prototype of thinking!
Can be extrapolated in 3D? …but in nD?
1,1,2,3,5,… xn,, x n+1,…
xn+2=xn-1+x n
x n+1/xn -> 1.618034…
A0 A s2 a/b=x
A0 / As1 = A1/A s2
A1
b A s1
x2-x-1=0; x= (1+√5)/2 = Φ
a
31. 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 …
32. … 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
[2]
It has the value
its decimal expansion begins with 1.324717957244746025960908854....
The plastic number is also sometimes called the silver number
33. 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
36. 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
38. 1988-89 3D
Leafs from the same tree archetype ?
45 o
60 o
The social revolution of 1989… a result of a
profound change in… Logos? Something
more profound is happening?
39. 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:
40. 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
41. 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
n-1
equation x(x-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:
42. c) The harmonic mean of the sides of a golden volume, generated in n-
dimensional space:
it can verify the expresion:
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
43. 2004 Summer school in Gura Humorului
We build a golden Volume, talking about natural engineering
