# Author Archives: tc

## ECA matrices

More art? Rule 22 Rule 37 …and more.

## factor pushing

Factor pushing is an idea I came up with when trying to establish an easy proof of the existence of a prime in $(n,n^2)$. I haven’t managed that yet, but it’s still on my bucket list. To do it, you take some set of numbers, write them out in a row, and then generate new […]

## h vs. v

1. horizontal union 2. vertical union

## On the persistence of zeros

I think I have a proof of the aperiodicity of the center column in Wolfram’s Rule 30 open question. Outline Instead of assigning ones and zeros, we’ll be using boolean algebra to represent cell values. This allows us to ratchet up the complexity of the system in a more understandable way. We let $k$ be […]

## Proof summary for aperiodicity of Rule 30

This is rule 30. The center column is highlighted, as we are interested in determining whether or not it’s possible for it to ever degenerate into a cyclic pattern. Today, I have a pretty good outline for how to show that it cannot. Rule 30 as boolean algebra We can always substitute variables in place […]

## Variable displacement

When $b=1$, the center column must be periodic. This means that all cells not dependent on $b$ must be periodic regardless of anything else. When $b=0$, then $a$ controls whether the left or center column behaves as the typical center column. If $a=0$, then the center is the center and must be periodic; if $a=1$, […]