The VirtualDynamics~ Org    Creative Computing Centre

VirtualDynamicsSoft:   Science & Engineering Virtual Labs

Javier Montenegro Joo                    The VirtualDynamics~ Org    



Wonder wavelets

Download - Edu Virtual Labs

 Art with Ellipses 


Recursive-Computing Curves

Computer-made scenes

Rainbow Odyssey


Iterated Function Systems, IFS 


VirtualDynamics Art

www VirtualDynamicsSoft com

Algorithmic Art Exhibition Centre

Javier Montenegro Joo




For your eyes only: The Beauty of Algorithmic Masterpieces


The images exposed in this document have all been created by means of Standard Procedures known as "Algorithms" in the language

of Mathematics, these amazing and most of the times, appealing images, are evidence of the power of Mathematics to produce beauty.


Originals of the images exposed here are considerably much larger and much better quality.
Javier Montenegro Joo, author of all the images exhibited in this document reserves for himself absolutely all the rights.

"Virtual Dynamics" is the name of an Algorithm created by JMJ in 1988 to Computer Simulate the Stochastic Aggregation of Diffusing Particles (Thesis, San Marcos University).

VirtualDynamics is a Registered Trademark of the Palatinus Development Foundation. All Rights are Reserved.

Author's email:    

EduVirtualLabs by VirtualDynamicsSoft


Algorithmic Art Exhibition Centre

Algorithmic Art



         Images shown here are in compressed formats so that they  are easily and quickly loaded onto the web page.

Originals are much larger in size (Usually 60 cm x 60 cm) and they are in non-compressed format hence their quality is optimal.



The Algorithmic Art (AA) is just one of the several manifestations of the Digital Art or Computer Art, which owes its existence to that of the computers. Among the products of AA are the beautiful and exceptional images developed by computation protocols (algorithms) executed on mathematical expressions like equations, formulas, etc.

The famous and captivating images of Fractal Objects created for Benoit Mandelbrot are works of AA, because they are the result of certain iterative processes on mathematical equations, specifically in the so called Complex Numbers Space.

Computers are indispensable in order to create AA works, for they can operate millions of times on mathematical expressions, introducing a very small variation each time, and generating the corresponding graphic representation, a work that if made by hand would be highly tedious, extremely slow and prone to errors.

From the point of view of computers an image is a tridimensional matrix with components (x,y,z), where x and y indicate the position, the (x,y) coordinates of a dot (pixel) in the image, and z represents the color in that position. In grey-leveled images z goes from 0 to 255, indicating an intensity of grey color. In color images, the value of z has three components (the RGB values) representing the combination of different intensities of Red, Green and Blue to be placed in the point of coordinates (x,y).

Algorithmic art images are not necessarily the result of mathematical computations on equations. Some outstanding and interesting images result from logic protocols, consisting of logic transformations like translations, rotations, scale changes, etc.

Some very well known and commonly used techniques to create AA works are Fractal Modeling, Iterated Function Systems, Cellular Automata Tessellations, Polar Inversion and Numerical Computation. Obviously there must be AA techniques known by only a few, which are consequently of very limited use.

Some AA Works show the temporal evolution of certain silhouettes, shapes and colors. Even though the final product is in itself a work of art, each step towards the final product generates also an astonishing image, this is the case of objects generated with the Cellular Automata technique.

JMJ, author of the images shown in this web page, had many times the opportunity of appreciating the hypnotic effect produced in the auditorium, by a display in slow motion of one of his creations, executed with the technique of Cellular automata.

Since the dawn of the computer era JMJ has created some extraordinary images, which cannot be reproduced again mainly because they were devised when the computer operating system was the DOS, and this is not available any longer.

JMJ has devised at least three different AA techniques, which have allowed him to create astonishing images, not only because of its beauty, but also due to their intricate complexity and visual effects.


The two images above are Fractal Objects from a field of Mathematics known as Complex Dynamical Systems. These were developed by JMJ using a technique created by Benoit Mandelbrot, the mathematician father of Fractals, and consisting in the iteration of a Complex Space function, several thousands of times before escaping to infinity.

It can be seen that the Fractal Objects are replicated very many times with changes of scale, position and orientation, this is the objects have self similarity. The different colors are associated to the different number of  function iterations needed to reach a certain pre-defined value.

Left:  image created with the   Rainbow Odyssey  algorithm, based on

Strange Attractors and  devised by JMJ.



Left: An image developed with the Angular Evolutions algorithm,               created by JMJ

The art is not only the final image (shown here). The slow-motion          evolutions visualized on the computer screen while the image                develops itself are breathtaking.



Original images are much better quality but they are too heavy.


   Return to Index    


Download my article ( The Breathtaking algorithmic art ) about Algorithmic Art:




Algorithmic Masterpieces


Wonder Wavelets

Algorithmic Art

The following images have been created with an algorithm developed by JMJ, this algorithm allows to create interlocking silhouettes, tessellations, and intricate patterns.  

       Return to Index  

Intuitively-easy-to-use EduVirtualLabs by VirtualDynamicsSoft


Educational  Virtual Labs created by VirtualDynamicsSoft

EduVirtualLabs on  Physics,  Mathematics and  Digital Image Processing,for use from high school to university.


 <<< Download Demos:   Click link  >>>

Download  Physics Virtual Lab (PVL)  Demo

Download_Math Virtual Lab (Visual Math) Demo

Download_Imagery  (Digital Image Processing)  demo

Download_Math & Physics based Computer Games Demo

   Return to Index

Computer-made scenes


Computer-made scenes


Collection of Cubes

The image of a single object may be uninteresting, however, there is no doubt that many times it is the distribution of some objects which result alluring and interesting. Some distribution of objects are horrible while others are quite appealing.

JMJ made a computer program to sketch cubes and then he allowed the computer to place the cubes not only at random but also with random orientations, this means that the distribution of cubes shown in this frame (which includes cubes location and their orientations) might never be reproduced again.

A cube has the interesting ambiguity feature that the same observer sees it concave in a moment, and convex the next moment

   Return to Index  

Rainbow Odyssey Image Gallery

Rainbow Odyssey  Image Gallery



These images have been created by means of the Rainbow Odyssey  Algorithm, based on Strange Attractors and developed by JMJ.


Original Images are much larger (up to 70 cm x 70 cm), and much better quality. These images -like all other material in this document- can not be reproduced, they are under copyright by the author.





         Return to Index  

Art with Ellipses

Carefully combining the equations of several ellipses JMJ obtained this image. Watching the slow motion development of this object is a very interesting experience. Here only the final product is shown.



Gradual overlapping of color ellipses. Some years back when computers were not as quick as they are today, watching the development of this object was literally a breath taking experience. Nowadays with ultra fast computers, one only sees the final result (shown here) and the experience is not as astonishing as it was in the past.



In an effort to understand the (artificial) world around us, we study Geometry at school, but that Geometry does not allow us to understand the Nature, for it seems to be that Nature has its own geometry and not precisely the one we are familiar with.

Benoit Mandelbrot discovered the Geometry of Nature, based on Fractals, which brings us closer to the real geometry of natural things around us.

After getting to know about Fractals, people's vision of trees, clouds, leaves, feathers and other natural things, is not the same any longer. It is like dividing people's life in before and after Fractals.

As an example, consider a tree. Before Fractals, we see that in a tree all branches look just the same, we would say that a tree occupies a volume, a 3D volume. After Fractals, a tree is a non regular structure having statistical self-similarity, it seems to occupy a 3D space, but actually it occupies neither a 2D nor a 3D space, it occupies a space somewhere between 2D and 3D.

The image displays an object developed by JMJ using a technique known as Iterated Function Systems (IFS,  created by Michael F. Barnsley), it can be seen that the object is a Fractal, because the whole object looks like any part of it (self similarity), it is composed of replicas of itself, which appear in different sizes and locations and with diverse orientations. Obviously the whole object does not occupy a 2D space, it occupies a space whose dimension is rather a fraction between 1 and 2.

IFS not always generates fractal objects.


   Return to Index  

Recursive-computing Curves

Recursive-computing based images

A computer program is recursive when its functioning invokes to itself.

As an example of a recursive process, consider the Factorial n! of an integer number n.
The factorial of n is given by  n! = n  (n-1)!
and this means that the factorial of n is the factorial of the number n-1 multiplied by the number n.  This short story repeats for the Factorial of  n-1, and thus goes on successively until the Factorial of 1 that, by definition, is 1.

In the case of a recursive computer program, the program appeals to the execution of the same program.




The graphic patterns shown here consist of the superposition of several curves.

Long before the invention of computers the mathematicians Hilbert and Sierpinski created the patterns carrying their names.


The computer programs to make these patterns would be extremely cumbersome if they were not recursive.  


   Return to Index  

Complex Dynamical Systems

Each one of the images below is the result of a computational algorithm dealing with a Complex Variable Equation which is iterated several hundreds of times. The color distribution in these images is associated to the necessary number of times for the results of an iteration to escape to infinity. Each image corresponds to a different equation which deals with six parameters (18-digits-long each).  These are images of Fractals, they have self similarity.
Original Images are much larger and much better quality, here they have been reduced.


   Return to Index  


Iterated     Function Systems ( IFS )

This self-similar ferns are JMJ's variations of the Michael F. Barnsley's Fern developed with the Iterated Function System (IFS)  algorithm.

Notice that these objects are composed of replicas of themselves, placed in different sizes and locations and with diverse orientations.

Left: Wheat Spike created by JMJ





   Return to Index  

Cellular Automata


Discrete Dynamical Systems

Regular objects developed by means of  Cellular Automata.

Cellular Automata (CA) were originally developed by Konrad Zuse, Stanislaw Ulam and Jhon von Neumann, at the time of the first computers.  During the 1970ís and 1980ís  Stephen Wolfram, carried out extraordinary work on CA.  JMJ owes his knowledge of CA to the works of Stephen Wolfram.

A CA is a Discrete Dynamical System represented by an array of cells whose states are updated in parallel. The CA evolution is due to the iteration of a simple deterministic rule. Notwithstanding the simple evolution rules of a Cellular Automaton, this is capable of supporting complex emerging behavior.

Although CA were originally applied to physics, fluid mechanics and biology, they have also been applied to social sciences and even to studies of immune system related topics.  Seeing the accompanying images it is obvious that CA can be used to generate interesting graphical objects, not only in 2D but also in 3D.

In graphical applications CA generates objects starting from a single cell or from just a few cells usually originally placed in the center of the object.

The time development of this cell or cells is governed by a growth rule, which after a number of generations develops an object.

Thus the objects shown here are not final objects, they are just the objects generated after some evolution time. Usually objects generated by means of Cellular Automata are regular in shape, but this may be avoided with an algorithm.

 The objects shown here are a fraction of those developed by JMJ when the Operating System of computers was the DOS, hence great many of those images have been lost.


   Return to Index