Cellular Automata in Excel

[Lepton]

I was stuck in a meeting today with a new notebook, so I decided to see if I could create the cellular automata that are introduced in Wolfram's "A New Kind of Science". I used functions in each cell to display either a 1 or a 0 (if you actually do this, set up your spreadsheet so that 0s aren't displayed) depending on the values of the three cells directly and diagonally above. The simplest example would be:

[ralphmerridew]

With just addition and subtraction, mod 2, there are only 8 rules you can create. I'm not sure whether adding abs improves that any.

It's fairly easy to create any binary rule with addition and multiplication, mod 2, though.

[ChickenMarengo]

Can't you create 16 rules, as you can also optionally add 1 to invert a rule?

Allowing multiplication, I get:

Rule 30: =mod(A1+B1+C1+B1*C1,2)

Rule 110: =mod(A1*B1*C1+B1*C1+B1+C1,2)

To get these, start with rules that will only match one pattern from the previous generation. e.g. A*(B+1)*(C+1) mod 2 will only give you a 1 when the previous generation is 100.

Add up these rules for all the patterns that you want to give a 1, then multiply out and simplify by removing pairs of duplicate terms.

[lostdummy]

Well, it seems like it is possible to do it even without multiplication, if what is given in example is allowed.

In fact, if ABS could be used, then I can make it even without "mod 2" .

I will show example with defining "functions" to shorten up writing, but in excel cells those functions can be expanded to be entire formula instead:

sgn(n)= abs(n+0.5)-abs(n-0.5)
as(n)= abs( sgn(n) )
is0(n)= 1-as(n)

sumN= A1+A1+A1+A1+B1+B1+C1

cell= is0(sumN-1) + is0(sumN-2) + is0(sumN-3) + is0(sumN-4)

This will put in "cell" value needed for Rule 30. One expanded cell in real excel sheet at position I2 would look like: