As well as creating beautiful and unqiue protypes, the DOOC algorithm that produces them has fundamental connections to non-linear dynamics, or "chaos theory" as it is more popularly known. This means that I can use the results and express them on a graph instead of a vector plane, showing just how chaotic, yet ordered, these protypes are:

This graph shows the relative position of the vector to the starting point plotted over previous iterations, for the most mathematically simple cubic protype. An example to see how this relates to chaos is to use this graph to simulate growth in an animal population. When the population starts, there is short term exponential growth (as seen in the curve at the start of the graph), but when the population reaches carrying capacity, it crashes, leaving a small number of organisms alive. Over time, the carrying capacity itself increases, as development of civilisation increases, leading to higher peaks, and more sudden drops, but when the population drops to 0 (i.e. extinction), the whole process repeats and the carrying capacity drops to its starting point.

Clearly, the graph doesn't match any population exactly, but many sections of it are good
analogies of real life situations.

The graph above shows the size of the turn that the vector takes plotted after each iteration has taken place, of EXACTLY the same protype. How can something that looks so simple on a vector plane take such a different shape on a graph? The answer lies in the definition of chaos. The most important thing about chaos is that it is NOT random, like throwing a die. Chaos means, in the words of Edward Lorenz, that:

The present determines the future, but the approximate present does not approximately determine the future

In the context of the DOOC algorithm this means that the position of the vector can be easily deduced from the start, but as soon as the algorithm runs, it is incredibly hard to predict what will happen next. This is because the function (which must be non-linear) will be modulo 360 (as vectors turn by angles), and so an nth term is very hard to find without the starting computations (i.e. before the value of the function goes above 360 for the first time)

Unfortunately the system is not entirely chaotic, because it is entirely hypothetical, and not entirely analogous to
any natural system. Yet.

© Harry Turnbull 2017