There is a better reason why one-dimensional automata are interesting than the Sierpinski triangle and the Rule-90 (90 is the decimal representation of 01011010) responsible for its creation. This reason is the Rule-110 (01101110). In the 1990's Matthew Cook, a research assistant of Stephen Wolfram proved that Rule-110 is (drumroll, please!) Turing-complete. That's right, folks, these eight extremely simple rules, combined with an infinite one-dimensional strip, form a universal Turing machine, capable of answering any answerable question and simulating the whole Universe, including every one of us with arbitrary precision, given the right input.
Before Rule-110 was found, the simpliest universal Turing machine required not 8, but 28 rules. It is possible, though, that an even simplier machine with 6 rules is possible.
Now behold the beauty, greatness and simplicity of the Rule-110.
Values at t: 111 110 101 100 011 010 001 000
--- --- --- --- --- --- --- ---
Value at t + 1: 0 1 1 0 1 1 1 0
t = 1: *
**
***
** *
*****
** *
*** **
** * ***
******* *
t = 10: ** ***
*** ** *
** * *****
***** ** *
** * *** **
*** **** * ***
** * ** ***** *
******** ** ***
** **** ** *
*** ** * *****
t = 20: ** * *** **** *
***** ** *** * **
** * ***** * ** ***
*** ** ** ******** *
** * ****** ** ***
******* * *** ** *
** * **** * *****
*** ** ** *** ** *
** * *** *** ** * *** **
***** ** *** ****** ** * ***
t = 30: ** * ***** *** ******** *
*** **** *** * ** ***
** * ** * ** *** *** ** *
******** ** ***** * ** * *****
** ****** ******** ** *
*** ** * ** * *** **
** * *** ** *** ** ** * ***
***** ** * ***** * ********** *
** * ***** ** *** ** ***
*** ** ** **** ** * *** ** *
t = 40: ** * ****** ** * ***** ** * *****
I can go on an on, but if you really want to see a lot of generations, like a few billions, get Mathematica and try it yourself with different inputs.
Further watching: a MIT lecture by Stephen Wolfram, available for download at http://mitworld.mit.edu/video/149/ (fast forward to 46th minute for the discussion of Rule-110)