Elementary Linear Cellular automata example

Lets consider linear cellular automata with Wolfram rule 90

Elementary cellular automata is essentially string of two symbols, lets call them 0 and 1, equipped with rules (and starting configuration, of course) 

Rule takes symbol in the position, as well as left and right neighbors. Ie – when calculating symbol of the cell, we should consider 3 cells, cell to the left, cell itself and cell to the right.

Ie – suppose we have set of rules(*):
<em>111 - 0</em>
<em>110 - 1</em>
<em>101 - 0</em>
<em>100 - 1</em>
<em>011 - 1</em>
<em>010 - 0</em>
<em>001 - 1</em>
<em>000 - 0</em>

Or graphically:


So if we have 10011 for starting configuration, calculation goes like this(from top to bottom):

1 0 0 1 1
(0) 1 1 1 (1)
(1) 1 0 0 (1)

(italics) means that we dont know which symbol is to the left and right to the starting configuration, for simplicity I consider here absent symbols to be 0s (ie …00000100110000…)

Now, what is 90, which is called Wolfram rule.

It is obvious, that  3 cells define state, and since we have two symbols, all possible combinations are , and since we have two possible symbols for each combination – rule of CA can be expressed as (0,2⁸=256) number.

90 in binary would be (*), which is enough to describe elementary ca behavior.