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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.
"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: Director@VirtualDynamics.Org jmj@VirtualDynamicsSoft.com 
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 noncompressed 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 greyleveled 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 predefined 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 slowmotion evolutions visualized on the computer screen while the image develops itself are breathtaking.

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

Download my article ( The Breathtaking
algorithmic art ) about Algorithmic Art:
https://www.researchgate.net/publication/288827387_The_Breathtaking_Algorithmic_Art

Wonder Wavelets 
The following images have been created with the algorithm Wonder Wavelets developed by JMJ, this algorithm allows to create flowerresembling objects, interlocking silhouettes, tessellations, and intricate patterns. 





















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EduVirtualLabs: 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 >>> 
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Computermade 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 
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.

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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. 
Recursivecomputing based images


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. 

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 (18digitslong each). These are
images of Fractals, they have self similarity. 



This selfsimilar 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 



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 1970’s and 1980’s 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.

author: Javier Montenegro Joo 

Warning: Non professional physicists might not understand this.  
In highly degeneratestates physicists do it ..... 

With a cat (Schrodinger's), with twins (Einstein's) ...  
With Strange Attractors, Random Walkers, Annihilators, ...  
With Models in Super Positions and with Chaotic Motion.  
With Rigid Bodies moving Back and Forth with Friction.  
Spontaneously with Random Walkers on Inclined Planes.  
With Random Motions with The Nearest Neighbors  
With Freely Falling Bodies and Transverse Oscillations  
Slowly changing face and Temperature with Annihilators  
With Chains (Markov's), Vacuum pumps, Vibrators and Strings  
With SelfExcited Motions in Closed Isolated Environments  
Approaching the Critical Point with Noisy Vibrating Bodies.  
Swinging back and forth.  
With Intermittency, with Noise, With Distortions and Aberrations  
Dissipating Energy.  