From the Middle Ages many Persian gardens (such as those of the Taj Mahal from around 1650) have had fairly regular nested structures obtained by a few successive fourfold subdivisions. … Ever since Babylonian times arithmetic has been done by repeatedly applying simple rules to digits in numbers. … But beyond forms like syllogisms little seems to have been studied in the way of generating identifiable patterns from them.

Universal Diophantine equation The equation is built up from ones whose solutions are set up to be integers that satisfy particular relations. … Given an integer a for which IntegerDigits[a, 2] gives the cell values for a cellular automaton, a single step of evolution according say to rule 30 is given by BitXor[a, 2 BitOr[a, 2a]] where (see page 871 ) BitXor[x, y]  BitOr[x, y] - BitAnd[x, y] and a is assumed to be padded with 0's at each end. … Note that it is potentially somewhat easier to construct Diophantine equations to emulate register machines—or arithmetic systems from page 673 —than to emulate cellular automata, but exactly the same basic methods can be used.

And by about 200 BC the development of gears had made it possible to create devices (such as the Antikythera device from perhaps around 90 BC) in which the positions of wheels would correspond to positions of astronomical objects. … But much more widely known were the machines built in the 1640s by Blaise Pascal for doing addition on numbers with five or so digits and in the 1670s by Gottfried Leibniz for doing multiplication, division and square roots. … And from the late 1800s until about 1970 mechanical calculators were in very widespread use.

In general, the list for a particular rule can be obtained with the function ElementaryRule[num_Integer] := IntegerDigits[num, 2, 8] Given a rule together with a list representing the state a of a cellular automaton at a particular step, the following simple function gives the state at the next step: CAStep[rule_List, a_List] := rule 〚 8 - (RotateLeft[a] + 2 (a + 2 RotateRight[a])) 〛 A list of states corresponding to evolution for t steps can then be obtained with CAEvolveList[rule_, init_List, t_Integer] := NestList[CAStep[rule, #]&, init, t] Graphics of this evolution can be generated using CAGraphics[history_List] := Graphics[ Raster[1 - Reverse[history]], AspectRatio  Automatic] And having set up the definitions above, the Mathematica input Show[CAGraphics[CAEvolveList[ ElementaryRule[30], CenterList[103], 50]]] will generate the image: The description just given should be adequate for most cellular automaton simulations. … But since the C program explicitly updates values sequentially from left to right, the left-hand neighbor of a particular cell will already have been given its new value when one tries to updates the cell itself.

For certainly in the standard formalism, quantum probability amplitudes are taken to be continuous quantities in which an arbitrary number of digits can be specified. … But any sequence of bits one extracts must be deduced from a corresponding sequence of measurements. … So what this suggests is that there may in the end be no clear way to tell whether randomness is coming from an underlying quantum process that is being measured, or from the actual process of measurement.