And second, even if one did see such phenomena, it was almost impossible to tell whether in fact they were genuine consequences of the underlying models or were just the result of approximations made in implementing the models on a computer.

And after a while I realized that if there was going to be any dramatic progress made, I was the one who was going to have to make it.

color will occur at any specific step, one still knows for example that black and white will on average always occur equally often.

One might have thought that with all their successes over the past few centuries the existing sciences would long ago have managed to address the issue of complexity.

From the tradition of the existing sciences one might expect that its answer would depend on all sorts of details, and be quite different for different types of physical, biological and other systems.

But the fact remains that despite all the various methods of mathematical and other analysis that have been developed, our visual system still represents one of the most powerful and reliable tools we have.

By looking at picture (d), one can begin to see how it might be possible for a cyclic tag system to be emulated by rule 110: the basic

One general result, however, is that all so-called Sturmian sequences Round[(n + 1) a + b] - Round[n a + b] with a an irrational number must yield discrete spectra. … (b) (Fibonacci-related sequence) This sequence is a Sturmian one.

But if one wants to find out what happens after an arbitrarily large number of steps, one needs to use the μ operator, yielding a general recursive function.

Closely related to my discussion of the absence of t 2 terms in volume growth for 4D spacetime cones is the statement that if one sets up a small 3D ball of comoving test particles then the volume it defines must have zero first and second derivatives with time. … And indeed the so-called initial value formulations constructed in the 1960s allow one to start with a 3D metric and various extrinsic curvatures defined for a 3D spacelike hypersurface, and then work out how these change on successive hypersurfaces. … One can potentially view the causal networks that I discuss in the main text as providing another approach to setting up an initial value formulation of the Einstein equations.