Mathematical Functions

The last section showed that individual numbers obtained by applying various simple mathematical functions can have features that are quite complex. But what about the functions themselves?

The pictures below show curves obtained by plotting standard mathematical functions. All of these curves have fairly simple, essentially repetitive forms. And indeed it turns out that almost all the standard mathematical functions that are defined, for example, in Mathematica, yield similarly simple curves.

But if one looks at combinations of these standard functions, it is fairly easy to get more complicated results. The pictures on the next page show what happens, for example, if one adds together various sine functions. In the first picture, the curve one gets has a fairly simple repetitive structure. In the second picture, the curve is more complicated, but still has an overall repetitive structure. But in the third and fourth pictures, there is no such repetitive structure, and indeed the curves look in many respects random.

Plots of some standard mathematical functions. The top row shows three trigonometric functions. The bottom row shows three so-called special functions that are commonly encountered in mathematical physics and other areas of traditional science. In all cases the curves shown have fairly simple repetitive forms.

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From Stephen Wolfram: A New Kind of Science [citation]