For patterns (a) through (f) on page 188 , k = 3 and form is given respectively by (a) {1, 1} , (b) {0 | 2, 0 | 2} , (c) {0 | 2, 0 | 2} | {1, 1} , (d) {i_, j_} /; j > i , (e) {0, 2} | {1, 1} | {2, 0} , (f) {0, 2} | {1, 1} .

But through the development since the late 1920s of Ramsey theory it has become clear that this is not the case.

And in fact up to 26D (with the exception of 11 through 13) all the densest packings known so far are lattices that work like this.

In plants, cells typically expand—normally through intake of water—only for a limited period, after which the cellulose in their walls crystallizes to make them quite rigid.

If the connections on the underlying network are one-way (as in causal networks) then one no longer necessarily gets the second property, and when a continuum limit exists it can correspond to a (perhaps discontinuous) section through a fiber bundle rather than to a manifold.

The mathematical framework developed for relativity theory in the early 1900s (see page 1042 ) treated space and time very symmetrically, leading popular accounts of the theory to emphasize a kind of fundamental equivalence between them and to try to make this seem inevitable through rather confusing thought experiments on such topics as idealized trains travelling near the speed of light.

But as I have discussed above, the only randomness that can actually come out of such a system is randomness that was explicitly put in through the details of its initial conditions.

But there are two important effects which tend to limit this randomness, or indeed any randomness that is obtained through the mechanism of interaction with the environment.

So what this means is that the randomness we observe in fluid flow cannot simply be a reflection of randomness that is inserted through the details of initial conditions.

And as an example of a simple approach to modelling this, one can consider having a collection of discrete eddies that occur at discrete positions in the fluid, and interact through simple cellular automaton rules.