How to interpret Turing Machine illustrations on p79 of Stephen Wolfram's “A New Kind of Science” book?

Question

I am reading Stephen Wolfram's "A New Kind of Science".

At present, I cannot understand how the cellular automata illustrations on p79 are created.

In the patterns, the active cell, representing the head, appears to change orientation between up and -45 degrees. However, none of the rules seem to include an active cell with an orientation other than up or down. How does the active cell orientation of -45 degrees come about in the patterns?

Am I missing something obvious (I am a beginner in this area)?

Answer 1

You have a simple rule. Just a mapping from 3 binary digits to 1 binary digit. For example:

111 - 0
110 - 0
101 - 0
100 - 1
011 - 1
010 - 1
001 - 1
000 - 0

Then you have some sequence of digits during time t0. For example 00111010. To find what will happen at time t1, you need to use this mapping. So 001 will be 1, then 011 will be 1, 111 = 0, then ... and 010 = 1. This way you will receive the sequence of the same length for the second generation (t1). And you move on further and further again till you will see repetition.

So on that pictures your X axis is this sequence (empty square 0, gray square 1). On your Y axis is how this sequence evolve under specific rule. In my example it was a rule 30 (because 00011110 = 30 in binary)

You can read high level overview here. Also these rules are simple, they can give rise to complex behavior.

P.S. paper was published in Nature (high level science journal) and considered revolutionized because it showed that complicated structures and motifs like dots on the leopard's skin or the image on the shell can arise from really simple rules.

Answer 2

I think that it is an inconsistency between the printing of the rules and the diagrams.

If the downward (-90 degrees) arrows in the rules are replaced with arrows pointing to the bottom right (-45 degrees) then the rules seem to make sense.