And unlike for example electric charge the observed masses of elementary particles never seem to be in simple ratios—so that for example the muon is about 206.7683 times the mass of the electron, while the tau lepton is about 16.819 times the mass of the muon.

[History of] truth tables The method of finding results in logic by enumerating all possible combinations of truth values seems to have been rediscovered many times since antiquity.

(Different entities would nevertheless still often need liquidity at different times.)

In 3D the arrangement one gets is typically not very regular—although as noted at various times since the 1600s individual objects often have pentagonal faces suggestive of dodecahedra. … But in 1993 Denis Weaire and Robert Phelan discovered a layered repetitive arrangement of 12- and 14-faced polyhedra (average 13.5) that yields 0.003 times less total area.

And after n steps the positions of all tips generated are given simply by Nest[Flatten[Outer[Times, 1 + #, b]] &, {0}, n]

And indeed the question of whether the halting times for a system grow only like a power of input size is in general undecidable. And if one tries to prove a result about halting times using, say, standard axioms of arithmetic or set theory, one may find that the result is independent of those axioms.

The number of cells that are not white on row t in this case is given by Apply[Times, 1 + IntegerDigits[t, k]] . … Mod[Binomial[t, n], k] is given for prime k by With[{d = Ceiling[Log[k, Max[t, n] + 1]]}, Mod[Apply[Times, Apply[Binomial, Transpose[ {IntegerDigits[t, k, d] , IntegerDigits[n, k, d] }], {1}]], k]] The patterns obtained for any k are nested.

But in fact, as the pictures below show—and as we have seen many times in this book—it is also possible for randomness to arise intrinsically just through the application of simple underlying rules.

The functions And , Xor and Not are equivalent to Times , Plus and 1 - # & for variables modulo 2, and in this case algebraic functions like PolynomialReduce can be used for minimization.

In recent times, models based on wavelets have also been used.