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Note (d) for Growth of Plants and Animals

(In modern times this idea is associated for example with the notion that just a few measurements are sufficient to specify the fitting of clothes.)

Note (a) for Uniqueness and Branching in Time

The basic idea of so-called critical pair completion procedures has arisen several times—notably in the Gröbner basis approach of Bruno Buchberger from 1965 to finding canonical forms for systems of polynomials.

Note (f) for Substitution Systems

(e) For large t the number of elements increases like λ t with λ = ( √ 13 + 1)/2 ; there are always λ times as many white elements as black ones.

Note (a) for Undecidability and Intractability

Among such computations are Plus , Times , Divide , Det and LinearSolve for integers, as well as determining outcomes in additive cellular automata (see page 609 ).

Note (d) for The Notion of Numbers

But already in Babylonian times, practical problems of geometry began to require square roots.

Preface

My brief times in college (Oxford) and graduate school (Caltech) enhanced my enthusiasm and confidence in science, and allowed me rapidly to begin life as a professional scientist.

Note (g) for More Cellular Automata

The pattern obtained after t steps is then given by NestList[f[RotateRight[#], #]&, init, t] The pictures below show results with f being Times , and cells having values (a) {1, -1} , (b) the unit complex numbers {1,  , -1, -  } , (c) the unit quaternions.

Note (g) for Biological Pigmentation Patterns

Following Turing's work, the fact that simple reaction-diffusion equations can yield spatially inhomogeneous patterns has been rediscovered—with varying degrees of independence—many times.

Note (a) for Human Thinking

Particularly in early investigations of neural networks, it was common to consider systems more like very simple cellular automata, in which the s[i] corresponded not to states of successive layers of neurons, but rather to states of the same set of neurons at successive times.

Note (b) for Quantum Phenomena

(Versions of this were noticed by Richard Feynman in the 1940s in connection with his development of path integrals, and were pointed out again several times in the 1980s and 1990s.)