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2D [discrete] transitions [in cellular automata]
The simplest symmetrical rules (such as 4-neighbor totalistic code 56) which make the new color of a cell be the same as the majority of the cells in its neighborhood do not exhibit the discrete transition phenomenon, but instead lead to fixed regions of black and white.
Note that with my definition of dimension for networks, the fact that a network is planar does not necessarily mean that it has be two-dimensional—and for example the networks on page 509 are not.
The first obvious but crucial thing to do is to explain and interpret what is already in the book. … Like any serious intellectual pursuit, doing well the new kind of science in this book is not easy. … Not that it is easy to do this.
Why does such nesting occur?
In a class 2 system with random initial conditions, a similar thing happens: since different parts of the system do not communicate with each other, they all behave like separate patterns of limited size.
But with all the first million initial conditions, only one other structure is produced, and this structure is again one that does not move.
So how does this work?
How complicated do the rules need to be in order to get universality?
So between 2 states and 2 colors and 2 states and 5 colors, where does the threshold for universality in Turing machines lie?
In most cases, many strings are never produced—so that there are many possible statements that simply do not follow from the axioms given.