By 1934 these were known to be equivalent, and in 1935 Alonzo Church suggested that either of them could be used to do any mathematical calculation which could effectively be done.

But especially with the advent of the Christian religion the notion that humans can at some level make free choices—particularly about whether to do good or not—emerged as a foundational idea.

In addition: • GoldenRatio is the solution to x  1 + 1/x or x 2  x + 1 • The right-hand rectangle in is similar to the whole rectangle when the aspect ratio is GoldenRatio • Cos[ π /5]  Cos[36 ° ]  GoldenRatio/2 • The ratio of the length of the diagonal to the length of a side in a regular pentagon is GoldenRatio • The corners of an icosahedron are at coordinates Flatten[Array[NestList[RotateRight, {0, (-1) #1 GoldenRatio, (-1) #2 }, 3]&, {2, 2}], 2] • 1 + FixedPoint[N[1/(1 + #), k] &, 1] approximates GoldenRatio to k digits, as does FixedPoint[N[Sqrt[1 + #],k]&, 1] • A successive angle difference of GoldenRatio radians yields points maximally separated around a circle (see page 1006 ).

In the pictures in the main text, the black region is connected wherever it does not protrude into the shaded region, which corresponds to disconnected patterns, in the pictures above.

And indeed a crucial point for my discussion in the main text is that in formulating general relativity one actually does not appear to need all the structure of a simplicial complex.

What practical computers always basically do is to repeat millions of times a second a simple cycle, in which the processor fetches an instruction from memory, then executes the instruction.

And what this would do is to take just a tiny region and make it large enough to correspond to everything we can now see in the universe.

Operations on sequences of digits had been used since antiquity in doing arithmetic. … (Notions in physics like the Ising model do not appear to have had a direct influence.) … Apparently motivated in part by questions in mathematical logic, and in part by work on "simulation games" by Ulam and others, John Conway in 1968 began doing experiments (mostly by hand, but later on a PDP-7 computer) with a variety of different 2D cellular automaton rules, and by 1970 had come up with a simple set of rules he called "The Game of Life", that exhibit a range of complex behavior (see page 249 ).

And adding more variables does not seem to help.

The first concentrated on doing fairly controlled experiments on humans or animals and looking at responses to specific stimuli.