For while rule 30 does generate such a pattern, it also does a lot else that appears irrelevant to this purpose. … But while this cellular automaton seems to have little extraneous going on, it operates in a slow and sequential way, and its underlying If the purpose is to generate a uniformly expanding pattern it seems more plausible that the top cellular automaton should have been the one created for this purpose.

And almost all the arguments in the book—while often not conceptually simple—require no specialized scientific or other knowledge to follow. … And while I hope that all the effort I have put into presentation in this book will make it easier for others, I do not expect it to be a quick process.

In case (a), the fraction of black elements fluctuates around 1/2; in (b) it approaches 3/4; in (d) it fluctuates around near 0.3548, while in (e) and (f) it does not appear to stabilize.

Properties [of logical primitives] Page 813 lists theorems satisfied by each function. {0, 1, 6, 7, 8, 9, 14, 15} are commutative (orderless) so that a ∘ b ＝ b ∘ a , while {0, 6, 8, 9, 10, 12, 14, 15} are associative (flat), so that a ∘ (b ∘ c) ＝ (a ∘ b) ∘ c .

The digits that lie directly below and to the left of the original 1 at the top of the pattern correspond to the whole number part of each successive number (e.g. 3 in 3.375), while the digits that lie to the right correspond to the fractional part (e.g. 0.375 in 3.375).

mathematical fact that the area of a two-dimensional circle is π r 2 , while the volume of a three-dimensional sphere is 4/3 π r 3 , the volume of a four-dimensional hypersphere is 1/2 π 2 r 4 , and so on.

But while these models have often proved extremely useful in practice, I have usually considered them intellectually quite unsatisfactory.

But after a while I became convinced that they were—and I realized that I had seen a sign of a quite remarkable and unexpected phenomenon: that even from very simple programs behavior of great complexity could emerge. … But while I did find that ideas from areas like chaos theory and fractal geometry helped in explaining some specific features, nothing even close to the phenomenon as a whole seemed to have ever been studied before.

A few percent exhibit repetitive behavior, while only one in several million exhibit more complex behavior.

Theological implications [of ultimate models of physics] Some may view an ultimate model of the universe as "leaving no room for a god", while others may view it as a direct reflection of the existence of a god.