Many years ago, a friend went to grad school to study computer science in the University of Minnesota. I asked him what his research topic was. He said: cellular automata. It sounded interesting as a term, especially "automata". Although he explained it, I didn't understand much. But the term stuck with me along the years.

Recently, I became interested in visualizing math functions [Lorenz Attractor] [Normal Distribution] in Tableau. It's a great way to learn math and learn some advanced features of Tableau. Zen Master Noah Salvaterra's masterful article taught me how to use state iterations in calculating recursive functions. A speech by Stephen Wolfram in which he showed many fun images engendered by cellular automata, picked my curiosity. I want to see if I can render them in Tableau.

After quite a bit of trial and errors, along with self doubts and struggle, I finally made it work! Among the 256 variants, a few of them reproduce the Sierpinski Triangle or similar! Zen Master Joshua Milligan created it in a different approach in Tableau.
The data set
It is very simple. Only 4 rows data are needed.
We only need to define 4 points in the grid. The size along each dimension is defined by a parameter. The other data marks are created through data densification.

The parameters
M: 2M+1 is the width of the grid because we need an odd number.
N: The height of the grid.
Rule: the 256 rules of the cellular automata.

The state in a string
The calculation of each point is a function of the previous row and the two extra wraparound bits. Thus a 2M+3-bit string is used to store those bits. Through iterations, the state string is updated at each iteration. A recursive function is a state machine after all.

The formula is built on previous_value() function which keeps the state string.
The rules
That's it! Click image to access the interactive version. Note that we picked the phone size for the viz. You can view it right from your phone.