Stephen Wolfram's Rule 30 pattern clarification

Question

The problem

I was going through the methods by which Wolfram's Mathematica generates random numbers and found that it uses the Cellular Automata Rule 30. The underlying explanation of it goes as follows:

The evolution of an elementary cellular automaton can completely be described by a table specifying the state a given cell will have in the next generation based on the value of the cell to its left, the value the cell itself, and the value of the cell to its right.

Since there are 2 x 2 x 2 = 2^3 = 8 possible binary states for the three cells neighboring a given cell, there are a total of 2^8 = 256 elementary cellular automata, each of which can be indexed with an 8-bit binary number (Wolfram 1983, 2002). For example, the table giving the evolution of rule 30 (in binary, rule 30 is written as 30 = 00011110) is illustrated down below. In this diagram, the possible values of the three neighboring cells are shown in the top row of each panel, and the resulting value the central cell takes in the next generation is shown below in the center.

The Question

How does the above diagram relate to the individual patterns in the upper part of the table? I know that the diagram is composed of them but confused as to how it forms sequentially. I don't see any row/column distinguishing feature that could perhaps tell me that those layers have been built using the upper rows in the given table. In short, I want to see the tetris sort of phenomenon behind this

Answer 1

The pattern forms sequentially, row-by-row. We start with the top row: a single black cell.

In the next row, every cell looks at the three neighbour cells in the row above and compares that pattern to the rule. So, the left cell sees --X, the middle cell sees -X-, and the right cell sees X--; according to the rules, all three of those produce black cells (the bottom cell of each rule).

In the next row, the middle five cells see --X, -XX, XXX, XX- and X--, so they become XX--X after transforming with the rule.

Note any cell that sees three white cells above it will also be white, so we don't have to worry about the cells outside the 45-degree pyramid. Also note that, because --X -> X and X-- -> X, the 45-degree pyramid always has one black cell on each side in each row.

Answer 2

The main diagram is not composed of the patterns in the upper portion, it is algorithmically generated by them.

The main diagram is evolved one row at a time. Starting with a single black cell in the centre of row 1.

The upper part of the table are the rules used to produce each new row of the lower part of the diagram.

The colour ( black/white ) of each cell on the new row is determined by the 3 cells above the new cell. The rules determine the colour. For instance, the first rule in your example says that if the 3 cells above are all black, then the cell in the new row will be white.

Every diagram uses the same 8 patterns to determine the colour of the next row's cells.

The output from those 8 rules forms a binary word, as shown in the diagram, this is the Rule. In this case the output is 00011110, which is 30 in binary. Hence the name Rule 30.

The underlying interest in these rules is that highly complex patterns can be generated from such simple principles, sorry, no Tetris here :)