And by using rules such as s[x___, 1, 0, y___]  {s[x, 0, 1, 0, y], Length[s[x]]} one can keep track of the positions at which substitutions are made. ( StringReplace replaces all occurrences of a given substring, not just the first one, so cannot be used directly as an alternative to having a flat function.)

All however ultimately involve discretization, and can suffer from difficulties that are similar—though often more insidious—to those for finite differences.

This form is inevitable if the underlying pattern effectively has the same structure on all scales.

Yet despite all this, the phenomenon of similarity between forms remained largely a curiosity, discussed mainly in illustrated books with no clear basis in either art or science.

Particularly over the past few decades all sorts of examples of "1/ f noise" have been identified with α ≃ 1 , including flicker noise in resistors, semiconductor devices and vacuum tubes, as well as thunderstorms, earthquake and sunspot activity, heartbeat intervals, road traffic density and some DNA sequences.

In most cells, say in humans, there are two versions of all 23 chromosomes—one from the father and one from the mother.

In supersymmetric models—and string theory—there are typically also all sorts of other types of particles, assumed to have high masses since they have not been observed.

If one computes the product of Exp[  (j 1 + j 2 - j 3 )] for all triangles, then it turns out for example that this quantity is extremized exactly when the whole surface is flat. … In the mid-1990s it was then found that states in 3D loop quantum gravity (see above) could be represented in terms of spin networks—leading for example to quantization of all areas and volumes. … In all cases, however, spin networks and spin foams seem to be viewed just as calculational constructs that must be evaluated and added together to get quantum amplitudes—quite different from my idea of associating an explicit evolution history for the universe with the evolution of a network.

Computer languages are almost always specifically designed once and for all, usually by a single person. … All the computer languages that are in widespread use today are based quite explicitly on context-free grammars.

All of these effects are implied by the standard mathematical formalism of quantum theory. … And in fact I do not consider it at all clear just what is really essential about them, and what is in the end just a consequence of the extreme limits that seem to need to be taken to get explicit versions of them.