But it is nevertheless possible to find two-dimensional cellular automata that yield less regular shapes.

The next page shows the kinds of persistent structures that can be generated in rule 110 from blocks less than 40 cells wide.

And one can then see that if one follows connections in the network, one is always forced to go progressively down the page, even though one is able to move both backwards and forwards across the page—thus agreeing with our everyday experience of being able to move in more or less any direction in space, but always being forced to move onward in time.

But as soon as one looks at a picture of how rule 110 actually behaves, the idea that it could be universal starts to seem much less absurd.

But in a field like set theory this is less true.

So given, say, an ordinary piece of rock in which there is all sorts of complicated electron motion this may in a fundamental sense be doing no less than some system of the future constructed with nanotechnology to implement operations of human thinking.

Note that the total number of integers less than n which can be expressed as a sum of three squares increases roughly like 5n/6 , with fluctuations related to IntegerDigits[n, 4] . … The total number of ways that integers less than n can be expressed as a sum of d squares is equal to the number of integer lattice points that lie inside a sphere of radius Sqrt[n] in d -dimensional space.

For neighbor-independent rules, the growth for large t must follow an exponential or an integer power less than the number of possible colors.

Digits of pi The digits of π shown here can be obtained in less than a second from Mathematica on a typical current computer using N[ π , 7000] .

And as faster digital electronics makes it easier to implement these it seems likely that it will become less and less common to have simple carriers with definite frequencies.