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This yields a chord such as
Play[Evaluate[Apply[Plus, Flatten[Map[Sin[1000 # t] &, N[2 1/12 ]^Position[list, 1]]]]], {t, 0, 0.2}]
A sequence of such chords can sometimes provide a useful representation of cellular automaton evolution.
One can scan a quadrant of an infinite grid using the σ function on page 1127 , or one can scan a whole grid by for example going in a square spiral that at step t reaches position
(1/2(-1) # ({1, -1}(Abs[# 2 - t] - #) + # 2 - t- Mod[#, 2]) &)[ Round[ √ t ]]
The angles are particularly accurate in, for example, flower heads—where it is likely the positions of elements are adjusted by mechanical forces after they are originally generated.
Sqrt[1 - 4 x]/2 yields a sequence with 1's at positions 2 m , as essentially obtained from the substitution system {2 {2, 1}, 1 {1, 0}, 0 {0, 0}} . … EllipticTheta[3, π , x]/2 gives a sequence with 1's at positions m 2 .
(In base 2 this number has 1's essentially at positions Fibonacci[n] ; as discussed on page 914 , the number is transcendental.)
Since numbers can be factored uniquely into products of powers of primes, a number can be specified by a list in which 1's appear at the positions of the appropriate Prime[m] n (which can be sorted by size) and 0's appear elsewhere, as shown below.
In the pictures below, the n th point has position ( √ n {Sin[#], Cos[#]} &)[2 π n GoldenRatio] , and in such pictures regular spirals or parastichies emanating from the center are seen whenever points whose numbers differ by Fibonacci[m] are joined.
In rule 154R, each diagonal stripe is followed by at least one 0; otherwise, the positions of the stripes appear to be quite random, with a density around 0.44.
that a digit which appears at a particular position in their result can depend on digits that were originally far away from it.
(These correspond to positions of a particle bouncing around in an idealized box, as discussed on pages 971 and 1022 .)