Just visited your site ...
https://www.cut-the-knot.com/ctk/Sierpinski.shtml

You mention various ways to create the Sierpinski triangle. One of them in particular caught my attention...

8.1-dimensional automata. Consider an infinite string of cells that might be of two kinds (say, 0 and 1). At discrete times, cells transition depending on their kind and the kind of their immediate right neighbors. The like kinds produce 0. Two unlike kinds result in 1 . Depict consecutive states of a string in rows growing downwards.

However, the reference doesn't match any of the ones listed..

1. I.Aharoni, From Tower of Hanoi to Pascal's Triangle, Alefefes, #4, 1996 (in Hebrew)
   
2. M.Barnsley, Fractals Everywhere, Academic
   Press, 1988
   
3. G.A.Edgar, Measure, Topology, and Fractal Geometry, Springer, 1990
   
4. M.Gardner, Mathematical Carnival, Vintage Books, 1965-1977
   
5. P.Hilton, D.Holton, J.Pdersen, Mathematical Reflections, Springer, 1997
   
6. B.Mandelbrot, Fractal Geometry of Nature, W.H. Freeman & Co, 1983
   
7. C.A.Pickover, Computers, Patterns, Chaos,
   and Beauty, St. Martin's Press, 1990
   
8. I.Stewart, Another Fine Math You've Got Me Into, W.H. Freeman & Co, 1992
   
9. D.Wells, You Are a Mathematician, John Wiley & Sons, 1995
   

I've done something similar and was wondering if someone else had done it before. That one seems to be the closest match. If you can tell me which reference it came from, I'd really appreciate it.

Thanks!

Pete Gerken

All the best,
Alexander Bogomolny