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But for rule 184, an appropriate choice of nested initial conditions yields the highly regular pattern shown below. The pattern produced by rule 184 (shown at left) evolving from a nested initial condition. … With this initial condition, rule 184 exhibits an equal number of black and white stripes, which annihilate in pairs so as to yield a regular nested pattern.
For even though it is intricate, one can see that it actually consists of many nested triangular pieces that all have exactly the same form. … The pattern obtained is intricate, but has a definite nested structure. … Patterns with nested structure of this kind are often called "fractal" or "self-similar".
So if we listen to nested sequences, for example, we have no direct way to tell that they are nested, and indeed all we seem sensitive to are some rather simple features of the spectrum of frequencies that occur. The pictures below show spectra obtained from nested sequences produced by various simple one-dimensional substitution systems. … Frequency spectra of nested sequences generated by one-dimensional neighbor-independent substitution systems.
The nested structure seen in this pattern can then be viewed as a consequence of the fact that rule 184 is able to emulate itself. … And in general it is possible to find quite a few cellular automata that yield nested patterns like rule 184 even from random initial conditions. … Nested structure similar to what we saw in the previous picture is still visible.
[No text on this page] Another example of a cellular automaton that produces a nested pattern even from random initial conditions. … As in rule 184, the nested behavior seen here is most obvious when the density of black and white cells in the initial conditions is equal.
A somewhat related source of nesting relevant in many mathematical systems is the nested pattern formed by the digit sequences of successive numbers, as illustrated on page 117 . … Nested patterns built by the evolution of the rule 90 and rule 150 additive cellular automata starting from a single black cell. Nested patterns obtained by processes in which either two or three branches are formed at regular intervals, and annihilate when any pair of them collide.
So what about other nested patterns? … For a few other nested patterns there exist fairly simple connections with additive cellular automata and similar systems—though usually in more dimensions or with more neighbors. But for most nested patterns there seems to be no obvious way to relate them to ordinary mathematical functions.
And if one does this, one immediately gets all sorts of fairly complicated patterns that are often not just purely nested—as illustrated in the pictures on the next page . … The details of each pattern are different, but in all cases the patterns have a nested overall structure. The presence of this nested structure is an inevitable consequence of the fact that the rule for replacing an element at a particular position does not depend in any way on other elements.
Indeed, if one looks carefully, one can see that every pattern just consists of a collection of identical nested pieces. … The nested structure becomes even clearer if one represents elements not as boxes, but instead as branches on a tree. … In all cases the overall patterns obtained can be seen to have a very regular nested form.
The result, once again, is that one gets an intricate but highly regular nested pattern. … Yet at least in this example, the overall pattern that is ultimately obtained still has a purely nested structure. … The final pattern obtained has an intricate nested structure.