This site is permanently under construction. It is work in progress

My interests in Generative Art combined with natural phenomena were very much influenced by my academic teachers Alfred Pfeiffer, the founding director of the Institute for Control Engineering of the Technical University Chemnitz, then Institute of Technology Karl-Marx-Stadt, who presented a very acclaimed lecture on Philosophical Issues in Cybernetics, and his successor Manfred Peschel, who founded and edited together with Werner Ebeling the monograph series "Mathematical Ecology and Evolution" of the Akademie-Verlag, Berlin, and initiated the yearly "Wartburg Workshops" at Wartburg Castle, Eisenach -- an Unesco World Heritage site.

From the Preface of the Exhibition Catalogue **Art Machines Machine Art** (Kehrer Verlag, Heidelberg 2007) we have

"The most intense and original exploration of the machine as an autonomous creative apparatus is manifested in the foundational work of the artist
Jean Tinguely"

Max Hollein

Director

Schirn Kunsthalle Frankfurt

Guido Magnaguagno

Director

Museum Tinguely Basel

An example is shown in the following painting (produced by my grandson Paolo at the Museum Tinguely Basel in cooperation with a Méta-Matic Drawing Machine of Jean Tinguely during the exhibition tinguely@tinguely on March 30, 2013)

The following video shows the fountain in front of the Museum Tinguely Basel in motion. The fountain was created by Jean Tinguely.

We have to state that the computer is also a machine though a very versatile one. Powerful hardware and advanced system architectures combined with sophisticated and complex software systems make it more and more intelligent. Will computers eventually take control ?

A possible description of Generative Art is given here !

We will look at Generative Art from a developmental and evolutionary perspective !

"Organic form itself is found, mathematically speaking, to be
a function of time....

We might call the form of an organism
an event in space-time, and not merely a
configuration in space."

( from d'Arcy Thompson.
On growth and form. University Press, Cambridge, 1952.)

An L-System or Lindenmayer system is a parallel rewriting system and a type of formal grammar which was invented by Aristid Lindenmayer for the developmental modelling of organisms like plants. A beautiful and very convincing book by Prusinkiewicz and Lindenmayer is titled The Algorithmic Beauty of Plants (Printed book by Springer-Verlag, New York 1990).

Plant generated using an L-system in 3D. (Image by Solkoll (public domain))

The growth of artificial neural nets during learning is an ongoing subject of research especially with respect to generalization capabilities and engineering effort.

The work of Benoit Mandelbrot is -- beside the mathematical ancestors like Henri Poincarè et. al. -- closely related to fractal geometry. He discovered the Mandelbrot set of never-ending fractal shapes. His autobiography, The Fractalist: Memoir of a Scientific Maverick, was published in 2012.

With the event of powerful computer systems and computer graphics Heinz-Otto Peitgen and Peter Richter, whom I met during a Wartburg-Workshop, made the rather abstract subject more accessible to a broader audience by the very striking book The Beauty of Fractals (Springer-Verlag, Heidelberg 1986).

Flying Balls (privately owned) from the Fractal Worlds Gallery. I created this image by using ChaosPro and PovRay (The Persistence of Vision Raytracer) in 2005. Both software packages are very powerful.

The title of Jaques Monod's influential and much debated book Chance and Necessity:
An Essay on the Natural Philosophy of Modern Biology, New York, Alfred A. Knopf, 1971,
was according to the introduction inspired by a line attributed to Democritus,

"Everything existing in the universe is the fruit of chance and necessity".

When we, Irene and myself, visited Ulla and Heinz Mühlenbein -- at that time he was very much involved at the GMD, Sankt Augustin, in Evolutionary Computation research, especially gleaned from livestock breeding (advanced by Douglas Scott Falconer and his influential book Introduction to Quantitative Genetics) -- we made a sightseeing trip to nearby Cologne visiting also the famous Cologne Cathedral -- an Unesco World Heritage site. I was very much impressed by the stained glass window designed by Gerhard Richter.

Stained glass window by Gerhard Richter in the Cologne Cathedral, 20 metres (66 ft) tall

(© 2015 by Heinz Mühlenbein (private communication))

In contrast to the patches of the window I did not use a fixed color palette of 72 colors but 24-bit true color when creating the images of the Chance & Necessity Gallery.The creation of those images does not follow pure chance but the images are the result of an evolutionary optimization path depending on the to be optimized function and on a more or less random initial state. What can be seen is an individual of a population at a certain time (generation) and a video of the whole optimization process for a special optimization function.

Measured individual n° 22 of a population at generation 200 in the course of an evolutionary optimization (minimization) process.

Here, the to be optimized function is the famous sphere function
$f\left(x\right)=\sum _{i=1}^{n}{{x}_{i}}^{2}$
. The function has its minimum at the origin. It was used by
Ronald A. Fisher to explain the nature of adaptation in his fundamental book
The Genetical Theory of Natural Selection and by
Motoo Kimura who's work culminated in his book
The Neutral Theory of Molecular Evolution.
Ingo Rechenberg used the function to compute progress rates for Evolution Strategies
(German: *Evolutionsstrategien*),
which he invented together with Hans-Paul Schwefel.
It was also used by Kenneth DeJong from
George Mason University.
For the creation of the images I programmed a soft sensor using GNU Octave to transform the linear vector
$x=[{x}_{1}\xb7\xb7\xb7{x}_{n}]$
to a true-color 2D-array as shown. Since the optimal (minimal) value of the sphere function
is given by the origin the corresponding image should have the RGB-color Black.

The following video shows the whole evolutionary optimization process scanned every 200 generations and depicting the first individual of the population.

The music comes from a live performance of John Michel (Wikimedia Commons). I created the video using the MAGIX Video de Luxe (© MAGIX GmbH) software.

See also the tutorial Introduction to Quantitative Genetics with Applications to Evolutionary Computation.

Evolutionary Game Theory (replicator dynamics) is a special field of game theory.
It was invented by John Maynard Smith and George R. Price for the explanation of different types of animal behavior and further analyzed and extended by
Karl Sigmund, Joseph Hofbauer et. al. I met Karl Sigmund
in 1994 at a conference in Jerusalem
where we discussed replicator systems during the breaks. In my book Evolution and Optimization: An
Introduction to Solving Complex Problems by Replicator Networks a so called diversified replicator dynamics is defined which takes into account several replicator dynamics' in different but connected habitats.
Examples are shown for its use for solving optimization problems on graphs.
Results with respect to Generative Art will be shown in future galleries. So, stay connected and visit often.

© 1994- Hans-Michael Voigt

& Generative Art