Using Signature Sequences in Weaving Drafts

An interesting class of fractal sequences consists of signature sequences for irrational numbers. The signature sequence of the irrational number x is obtained by putting the numbers i + j × x i , j = 1, 2, 3, … in increasing order. Then the values of i for these numbers form the signature sequence for x, which is denoted by S(x). Here's the signature sequence for φ, the golden mean: Both upper trimming and lower trimming of a signature sequence leave the sequence unchanged. Signature sequences have a characteristic appearance, but they vary considerably in detail depending of the value of x. Signature sequences start with a run 1, 2, …, n+1, where n = x ⎣ ⎦ , the integer part of x. The larger the value of x, the more quickly terms in the sequence get larger. Most signature sequences display runs, either upward or downward or both — which one is usually a matter of visual interpretation. At some point, most signature sequences become interleaved runs. This sometimes gives the illusion of curves. Although signature sequences are defined only for irrational numbers, the algorithm works just as well for rational numbers. Although signature sequences for rational numbers are not fractal sequences, they are as close as you could determine manually. The structure of a signature sequence depends on the magnitude of x. Furthermore, there are irrational numbers arbitrarily close to any rational number. There is no difference in the initial terms of signature sequences for numbers that are close together. For example, S(3.0) and S(π) do not differ until their 117th terms. It's also worth noting that there really is no way, in general, to perform exact computations for irrational numbers. Computers approximate real numbers, and hence irrational numbers, using floating-point arithmetic. A floating-point number representing an irrational number is just a (very good) rational approximation to the irrational number. For example, the standard 64-bit floating-point encoding for π is