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The prototype of the modern cellular automata can be regarded as Pascal's triangle, known from the tenth century. Here's what Wikipedia says: “The rows of Pascal's triangle are conventionally enumerated starting with row n = 0 at the top. The entries in each row are numbered from the left beginning with k = 0 and are usually staggered relative to the numbers in the adjacent rows. A simple construction of the triangle proceeds in the following manner. On row 0, write only the number 1. Then, to construct the elements of following rows, add the number above and to the left with the number above and to the right to find the new value. If either the number to the right or left is not present, substitute a zero in its place. For example, the first number in the first row is 0 + 1 = 1, whereas the numbers 1 and 3 in the third row are added to produce the number 4 in the fourth row.

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If we replace the odd numbers with the points, and even make transparent, then we obtain the Sierpinski triangle:

The cellular automaton

Again we turn to Wikipedia : “Von Neumann cellular automata are the original expression of cellular automata, the development of which were prompted by suggestions made to John von Neumann by his close friend and fellow mathematician Stanisław Ulam. Their original purpose was to provide insight into the logical requirements for machine self-replication and were used in von Neumann's universal constructor.”

Live Pixels algorithm on an infinite field multiplies any initial location of active pixels, that is, solves the problem of von Neumann. Identification of a cellular automaton from Wikipedia: “A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model studied in computability theory, mathematics, physics, complexity science, theoretical biology and microstructure modeling. It consists of a regular grid of cells, each in one of a finite number of states, such as "On" and "Off"”

Rather extensive analysis of cellular automata made in the program Cellular Automata Laboratoryhttp://www.fourmilab.ch/cellab/cellab.zip In the program implemented 37 types of machines, which are described on the page: http://www.fourmilab.ch/cellab /manual/rules.html

Examples of the program do not use an algorithm that repeats the Live Pixels, the study of the descriptions of algorithms shows that the Live Pixels are a special case of the Edward Fredkin’s algorithm Fredmem. Also close to the Live Pixels is an algorithm Fractal, who is also a special case of the algorithm Fredmem. Edward Fredkin indicates that its algorithm is an example of self-replication, the initial pattern in a certain number of iterations is transformed into several patterns like initial. This picture we see in the Live Pixels on an infinite field. Even more complete collection of cellular automata can be found at: http://www.mirekw.com/ca/rullex_wlif.html

Algorithms "Evolution" and then "Live Pixels" were developed and implemented in software in 2012. Till now many aspects of the algorithms are not still studied. The reader can take part in the analysis of "Live Pixels" and "Evolution" and search for interesting structures using the free program NeoNeuro Live Pixels.

Download NeoNeuro Live Pixels - Evolution

Appearance:

The program shows a field with “pixels” and the description of each algorithm. Simple and easy to use editor allows you to create new designs and analyze their work and save to a file that can be put online for public access.

Left panel

Allows you to select a scene for each of the three algorithms:

• Live Pixels
• Evolution
• Game of Life

Central panel

Here is the field of cellular algorithms.  Move can be made by mouse click. "Cell Type" in the right pane allows you to change the functionality of mouse click. The description of a particular algorithm is at the bottom.

Right panel

Move: makes one move. It is equivalent to clicking the mouse when the “Cell Type” is “Move”.

Back:  takes one move back.

First: returns the field to the starting position.

Clear:  clears the field.

Run!:  automatically changes the scene. Button becomes "Faster" after click.

Stop:  stops automatic scene changing.

Cell type: choose the functional of mouse click on the matrix. Moving the mouse while holding down the left button on a matrix allows you to "draw" the desired pattern more quickly.

At the bottom of right-hand column are situated the matrix core configuration options.

The program interface is implemented in English, Russian and German languages, users have the ability to add new translations.

The algorithm simulates the development of species - from the simplest microbes to giant dinosaurs and intelligent people. This algorithm was designed before "Live Pixels", and it is independent cellular automata.

Rules

Element is a set of pixels that lie side by side. The elements are colored according to the number of pixels in them.

1. At each move algorithm adds an element of one pixel in a random place on the field.
2. If there is a collision of elements with the same number of pixels, they form a new element, one pixel greater. In nature it is the appearance of species with larger body with there are enough species with smaller body.
3. If there is a  collision of elements of different sizes, there is only the largest stays, the smaller disappears. In nature it is the struggle for survival, food chain.
4. The existence of elements is limited by time, which is directly proportional to the number of pixels in the element.In nature the lifetime of the organisms is limited and there is also a dependence of the lifetime from the average size of species.

"Evolution" can be regarded as the development of individual living beings, though more correct to regard it as the development of species. Every element is a certain kind of organisms.

For the convenience the elements are drawn with the color regarding their size. Single pixel size elements are shown in green, two pixels size are dark red, the three are light red, the four are blue. The most common are the elements of one or two pixels. The figure shows that the element of the four pixels is only one. The major elements live a long, long time and then disappear - like dinosaurs.

Sea

Life came out of the ocean. The image gray bars at the bottom simulate the rugged seabed. It is interesting to see how the size of the animal species dependent on the environment - the more space, the greater the size of the largest living beings. At the same time small animals are everywhere.

It is interesting to see a clear dependence between the size of the species and the size of their habitats. Though to the bottom of the cleft can "climb" any "beast", mainly there are elements of one or two pixels.