Working with cell labels, combining both the neighborhood invoked and the state of the cell, got me thinking about "bits".
Bits are traditionally noted as 0 and 1, though any two lables, say "a" and "q" or "red" and "green", do the same thing. 0 and 1 works well especially when it comes to developing formulas and calculations.
Consider bits in the one dimension of classical geometry where points and lines live. There is nothing other than the point or combinations of points (lines). In one dimension, there is only active. Passive implies that there is an environment surrounding the active and that introduces the idea of a second dimension. So in one dimension, there is only 1 (or "a" or "red").
The thought is that active passive bits (0 and 1, etc.) can only come into being in two dimensions. The classic geometry the second dimension comes into being when two lines intersect creatng a plane. This also creates an evironment between the active lines filled by passive space.
I got to thinking about the active bit. Without context, there's no way to know whether it describes a one- or two dimenstional bit. Seems to me that two dimensional bits should properly desigate both their state and the alternative state, like 01 and 10. Such a notation clearly differentiates between the active state in both one and two dimensions.
I'm not sure that "dimension" is really the right term to use. One issue that comes up is that elementary cellular automata are called one dimensional. Even though most elementary automata are presented as two-dimensional patterns with each row of the automata displayed below its predecessor, the formal definition is for each row to replace the predecessor, leaving just one row. In truth, this isn't really one dimensional because the space between active lines is formally defined by passive spaces. Such is one of the problems of talking about dimensions, so I decided to think about these as first and second order bits.
This all began when I started thinking about the state of a grid cell prior to its state being determined. It reminded me of that idea from quantum mechanics: super-position. The grid cell can be either active or passive, depending on the rule and seed conditions specified. The fact is that cellular automata are not quantum because once the rule and seed conditions are specified, the state of future cells in a grid have been already determined. Still, the idea of super-position remains intriguing.
It can make sense with by thinking of a third order bit. In the third order, there are three bit states. Bits are defined as "0 not 1", "1 not 0", and "0 or 1" where the state of "0 or 1" is not pre-determined. In this construct, the super-position idea becomes a formally defined third state.
The third order begs that there be a fourth order, where the fourth bit is defined as "0 and 1". This gets interesting because the addition of two states must create something that is neither 0 or 1, but is both. In classic dimesional terminology, this fourth bit is called "time".
This is not to say that time is defined at "0 and 1". Instead, it suggests to me that there may be other states of matter which have not been discovered or identified. It also suggests that time is an entity separate from matter/space. If this is the case, perhaps we should be thinking about time as a multi-order entity. Is our current understanding of time akin to being stuck in a one dimensional universe?
Along this line, there have been recent articles suggesting that time is an emergent property of particle entanglement: