This procedure is based on the standard repeated squaring method of finding 2 n by starting from 2, and then successively squaring the numbers one gets, multiplying by 2 if the corresponding base 2 digit in n is 1.

For while simple infinite quantities like 1/0 or the total number of integers can readily be summarized in finite ways—often just by using symbols like ∞ and ℵ 0 —the same is not in general true of all infinite processes.

Two-operator logic [axioms] If one allows two operators then one can get standard logic if one of these operators is forced to be Not and the other is forced to be And , Or or Implies —or in fact any of operators 1, 2, 4, 7, 8, 11, 13, 14 from page 806 .

Reversal-addition systems The operation that is performed here is n  n + FromDigits[Reverse[IntegerDigits[n, 2]], 2] After a few steps, the digit sequence obtained is typically reversal symmetric (a generalized palindrome) except for the interchange of 0 and 1, and for the presence of localized structures.

It was discovered in antiquity that Euclid's algorithm starting with {x, 1} terminates only when x is rational.

History [of 2D cellular automata] As indicated on pages 876 – 878 , 2D cellular automata were historically studied more extensively than 1D ones—though rarely with simple initial conditions.

Extensive work has been done since the early 1900s on so-called elliptic curve equations such as x 2  a y 3 + b whose corresponding algebraic surface has a single hole (genus 1). … Over the history of number theory the sophistication of equations for which proofs of no solutions can be given has gradually increased—though even now it is state of the art to show say that x  y  1 is the only solution to x 2  3 y 4 - 2 . … Writing equations in the form p[x 1 , x 2 , …, x n ]  0 the distribution of values of p will in general be complicated (see page 1161 ), but as a first approximation one can try taking it to be purely random.

Another approach was to look at actual possible transformations between partitionings, and this led from the late 1950s to various studies of so-called shift-commuting block maps (or sliding-block codes)—which turn out to be exactly 1D cellular automata (see page 878 ). … In the 1950s and 1960s—quite independent of symbolic dynamics—there was a certain amount of work done in connection with ideas about self-reproduction (see page 876 ) on the question of what configurations one could arrange to produce in 1D and 2D cellular automata.

In the first case shown, starting for example at position 4 the dot then visits positions 5, 0, 1, 2 and so on, at each step going from one node in the network to the next. … The picture below shows the network obtained from a class 1 cellular automaton (rule 254) with 4 cells and thus 16 possible states.

Particle masses The measured masses of known elementary particles in units of GeV (roughly equal to the proton mass) are: photon: 0, electron: 0.000510998902; muon: 0.1056583569; τ lepton: 1.77705; W : 80.4; Z : 91.19. … Then there is also a direct mass: gluons 0; u : ~0.005; d ~0.01; s : ~0.2; c : 1.3; b : 4.4; t : 176 GeV.