(Page 0148) making the generated pattern nested. But when α is not a square root the pattern can be more complicated. And if more than two sine functions are involved there ...

(Page 0269) In a sense what is happening here is that even though rule 126 usually shows class 3 behavior, it is possible to find special initial conditions that make it ...

(Page 0914d) Notes for: Systems Based on Numbers | Mathematical Constants Continued fractions The first n terms in the continued fraction representation for a number x can be ...

(Page 0057) rules 0 and 128 all the cells become white, while in rule 255 all of them become black. There are also rules such as 7 and 127 in which all cells alternate ...

(Page 0956a) Notes for: Starting from Randomness | Special Initial Conditions General associative [cellular automaton] rules With a cellular automaton rule in which the new ...

(Page 1087c) Notes for: Processes of Perception and Analysis | Cryptography and Cryptanalysis [Cryptographic] properties of rule 30 Rule 30 can be written in the form Xor[p, ...

(Page 1091k) Notes for: Processes of Perception and Analysis | Traditional Mathematics and Mathematical Formulas Standard mathematical functions There are an infinite number ...

(Page 0141) It turns out that square roots are certainly not alone in having apparently random digit sequences. As an example, the table on the next page gives the digit ...

(Page 0977c) Notes for: Mechanisms in Programs and Nature | The Phenomenon of Continuity Random walks In one dimension, a random walk with t steps of length 1 starting at ...

(Page 1188b) Notes for: The Principle of Computational Equivalence | Intelligence in the Universe [History of] SETI First claims of extraterrestrial radio signals were made ...