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.

pascal triangle

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

Serpinsky 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 Laboratory http://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.