The Mathematical Definition of the Timewave by Peter Meyer Written 1999-08-30, slightly revised 2006-03-31

The current version of HTML does not support mathematical signs (one has to use little GIFs for the infinity sign, the capital sigma, etc., which I don't have at present and can't be bothered looking for) so in this part I shall describe the mathematical definition rather than present it. Full details are given in my The Mathematics of Timewave Zero, which appeared in the 1994 edition of Terence and Dennis McKenna's The Invisible Landscape. In passing I'll note that this mathematical appendix replaced the mathematical appendices by Royce Kelley and Leon Taylor in the 1975 edition of that book. It would probably have been better to include in the 1994 version the original appendices as well as my mathematical reformulation, since the original appendices contain material concerned with the derivation of the 384 data points from the King Wen Sequence, whereas my mathematical reformulation takes the 384 numbers as primitive and simply fractalizes them in a mathematically rigorous way. This reformulation was not based upon the 1975 mathematics, and goes beyond it in a manner which results in a truly fractal mathematical object. My mathematical reformulation, and the associated lemmas and theorems, can be understood by any student of mathematics familiar with infinite series.

The timewave is defined by means of a formula (this is not the formula found by Matthew Watkins) which takes as input a non-negative real number x and gives as output a non-negative real number y. The formula is interpreted as follows: Firstly a point in historical time is chosen as, in the jargon of Timewave Zero, the "zero point". The original zero point was 6 a.m. (La Chorrera, Colombia, time) on 2012-12-21; another may be used if so desired. Then, given that x is the number (not necessarily a whole number) of days prior to the zero point, y is the value of the timewave (or the degree of Novelty) at that point in time x days prior to the zero point.

The formula is the sum of a doubly-infinite series, i.e., it is of the form:

y(x) = ... + t(-3) + t(-2) + t(-1) + t(0) + t(1) + t(2) + t(3) + ...

where t(i) is the ith term in the series, and has the form:

v(x/64^i) * (64^i)

(where 64^i is 64 — the so-called "wave factor" — raised to the ith power). When x is an integer, v(x) is simply the xth number in the set of 384 numbers (for x = 0, 1, ...; after 383 we use x modulus 384). When x is not an integer, v(x) is a value between v(k) and v(k+1), where k is an integer and k < x < k+1 (the value v(x) is obtained simply by linear interpolation between v(k) and v(k+1)).

Now it turns out that, because of a property possessed by the set of numbers, in practice the doubly-infinite series is actually singly-infinite. The property is that the first two numbers are zero (actually the first three are zero in the usual four number sets). This means that the function v() is 0 for all real numbers from 0 to 1. Thus, for any value x, when evaluating the terms in the equation for y(x) which proceed toward the right, i.e.

v(x/64)*64,  v(x/64^2)*(64^2),  v(x/64^3)*(64^3),

and so on, we will eventually reach some i for which x/64^i is less than 1, so that v(x/64^i) is 0, so v(x/64^i)*(64^i) is 0. The same is true of all terms to the right of this one (i.e. for larger i's), so on the right there is always a limit to the non-zero terms.

Now to consider the terms which proceed toward the left, i.e.

v(x/(64^-1))*(64^-1),  v(x/(64^-2))*(64^-2),  v(x/(64^-3))*(64^-3),

and so on. The function v() is a saw-tooth graph which is obtained by reproducing the saw-tooth graph for the 384 numbers, in the range 0 through 383, all along the x axis. Among the 384 numbers there is, of course, one which is a maximum, namely, 79 (in the Kelley set; the other three number sets contain less than 79 different numbers). Thus the function v() never exceeds 79. Therefore the sum of the terms proceeding toward the left is less than the sum of the terms 79*(64^-1), 79*(64^-2), 79*(64^-3), and so on, or in other words the sum is less than

79 * ( 1/64 + 1/(64^2) + 1/(64^3) + ... )

This sum within the parentheses converges to 1/63, so the sum of the terms proceeding to the left converges to less than 79/63.

Thus the expression for y(x) is always well-defined and can be calculated in a finite number of steps to any given degree of precision.

Any function v(), from the non-negative real numbers to the non-negative real numbers, can be fractalized in this way provided the function satisfies two conditions:

(i) There is a finite non-zero real number n such that v(x) is 0 for all 0  <=  x  <=  n, and
(ii) For all x >= 0 the value v(x) is in the range 0 through m for some positive real number m.

Thus any set of about five or more numbers, beginning with two zeros and containing non-zero numbers, can be fractalized into a timewave-like fractal wave by the method described above.

A way to tinker with the timewave is to change the number 64 (in the construction above) to some other number. The Fractal Time software allows you to do this by changing the value of the wave factor (which is 64 by default but can be set to any integer in the range 24 through 68). Changing this number produces a gradual distortion (a contraction or an expansion) of the original timewave.

Fractal Time Software