User blog:Captain Bendy Ball/Captain bendy ball's Life function

Ok so Im sure you all know about Conways game of life.

If you dont please read this.

We can make a function out of this ofc by saying L(n) denotes the # of gens in the longest finite terminating game of life with n starting cells. But these rules that are the norm are only one in some other variations like this. (see pic to the right because wiki wraps text around it



So the possiblities are really endless as we see. but how do we get so many?

I will be going through just that.

Neighborhood
Every cellular automaton has a global and local neighborhood. The global neighborhood is the cell space as a whole which is usually infintite. The local neighborhood is the area around an individual cell used to determine the next gen state of course.

this area is an 8 square ring in norm game of life but can be any shape as long as it has 4 fold symmetry so theres no confusion about what cells are taken into consideration when determining next gen states.

now my idea is this: take the cells in one octant of four axes superposed on this cell, and now ennumerate them in a snake pattern. So any binary number corresponds to whether or not a cell in the local neighborhood will be consdiered a square to check when doing next gen calcs. Remember since this is only one octant this is done 8x, so there is no cell confusion.

next, just split it into 2x the number of cells around every cell for higher cell spaces which change global geometry so for hexagons it would be 6 axes and dudekatants. Anyways its best explained in pictures (coming SOON

The point is make a list of all possible local neighborhoods for every geometry (since pentagons around pentagons is only possilbe in hyperbolic space ofc

States
There are multiple states for each cell but the exact names "dead" and "alive" do not matter. That is just english. What really matters is this. you see. We can list the rules for next gens based on states like listing neighborhoods and this makes the second part of our eventual function.

Captain bendy ball's Life function
L(a,b,c,d) is the max gen length of an a-gon tiled global neighborhood with the loval neighborhood ennumeration not exceeding b with the ruleset ennumeration not exceeding c using d initial cells.

If we want all possible rulesets to be taken for max effectiveness just set c to a really high number like grahams numbers which is bigger than most possible rulesets. It takes a bit of thinking to though because the bigger a is the bigger b and c are. Well really only c but b might be involved idk.

Captain bendy ball's Life function Number
Captain bendy ball's Life function Number is L(googol,googolduplex,googolduplex,googolplexian).