H6X9H


Irrational numbers

The procedure of computing directions and drawing a figure can be generalised to (positive) irrational numbers. However, the problem is then that these figures have an infinite size. So, a figure can only be drawn partly after computing a finite number of directions.
The formula is similar to the rational numbers: N = K * D + E , where N, D and E are positive irrational numbers, K is a positive integer, and 0 < E < D . In principle D < N ; bigger values of D can be transformed just as for the rational numbers:

The rules for reduction are similar as for the rational numbers.
A figure can be reduced to remove the "noise" of the higher generations. Because of the irrational numbers a reduction process newer ends. A figure based on irrational numbers can be approximated for gaining insight, as follows. Reduce the figure for a few generations, and store the values of K and s . Replace the (reduced) values of N and D at the lowest generation by resembling integers. And then build up the higher generations again using the stored values of K and s. For example the pair (π, 1) can be approximated by (3230883, 1028422) .

Because of the usage of irrational numbers we can divide both sides of the characteristic equation by the same value without any further consequences. An advantage for doing such a division is to scale the numbers to convenient values. The equation can always be scaled such that D becomes 1 . In that situation, we define the numbers as being normalised.