Rule 30

Rule 30 ( English Rule 30) is a one-dimensional cellular automaton, which was discovered in 1983 by Stephen Wolfram. The rule specifies how the state of a particular cell in a one-dimensional lattice changed (black or white). If many versions usually worn with each other, creating a typical rule for the two-dimensional pattern ( see figure below). According to Wolfram's classification scheme is part of the cellular automaton "class 3", which means that it is aperiodic, chaotic behavior.

Description

The rule 30 is of particular interest, because it causes complex, seemingly random structures which have clear local regularities. Exactly these structures, the shells of Weber cone, a Meeresschneckenart on. With the rule, also a random number generator for Mathematica was developed and proposed a stream cipher to encrypt information. However, it was shown that other cellular automata for encryption are more suitable.

Definition

The primarily studied by Stephen Wolfram elementary cellular automata consist of an infinitely long, one-dimensional grid of cells. These cells can state 0 ( white) or 1 (black) accept. Initially, the configuration of the cells is determined, such as a single black cell. In each subsequent time step, a rule is applied to each cell, determines whether the cell in the next step is black or white. Here, the respective next state depends on the cell itself and of its left and right neighboring cell. A rule must therefore define possible cell combinations, such as 010 ( the cell is black, left and right neighbors are white ):

Each of the eight possibilities (000 to 111) can be assigned to an arbitrary state 0 or 1. Overall, there is therefore elementary cellular automata. Their designation performed according to the established pattern of tungsten, by reading the states written eight side by side as a binary number, e.g., 00011110, the corresponding decimal number giving the name of the elementary point (in this case 30).

Your reflection, complement, and complementary mirror image are the rules 86 (01010110), 135 (10000111) and 149 ( 10010101 ).

The following diagram shows the sequential execution of a rule, where at the beginning just a single cell are colored black and all the other white. The vertical axis is the time, and each horizontal line indicates the state of the cells at a given time step.

Properties of the sample

Two main structures are visible: The frequent occurrence of white triangles and regular striped pattern on the left side. The number of black cells at a particular time describes the sequence

And is about the same.

Chaos

The Rule 30 seems chaotic, not only for aesthetic reasons, but it also fulfills the mathematical conditions of chaos:

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