But once again, while these strategies may in some cases lead to greater efficiency, they do not usually lead to qualitative differences.

And while I have gone to considerable effort to ensure that its main elements are correct, ultimate objective confirmation is usually impossible. … And while with sufficient effort it is usually possible to give fairly simple explanations for fundamental ideas in science, the same may not be true of their history.

The element at position n in the first sequence discussed above can however be obtained in about Log[n] steps using ((IntegerDigits[#3 + Quotient[#1, #2], 2] 〚 Mod[#1, #2] + 1 〛 &)[n - (# - 2)2 # - 1 - 2, #, 2 # - 1 ]&)[NestWhile[# + 1&, 0, (# - 1)2 # + 1 < n &]] where the result of the NestWhile can be expressed as Ceiling[1 + ProductLog[1/2(n - 1)Log[2]]/Log[2]] Following work by Maxim Rytin in the late 1990s about k n+1 digits of a concatenation sequence can be found fairly efficiently from k/(k - 1) 2 - (k - 1) Sum[k (k s - 1) ((1 + s - s k)/(k - 1)) (1/((k - 1) (k s - 1) 2 ) - k/((k - 1) (k s + 1 - 1) 2 ) + 1/(k s + 1 - 1)), {s, n}] Concatenation sequences can also be generated by joining together digits from other representations of numbers; the picture below shows results for the Gray code representation from page 901 .

And while there are various triangulation methods that for example avoid triangles with small angles, no standard method yields networks analogous to the ones I consider in which all triangle edges are effectively the same length. … But while explicit coordinates and lengths are not usually discussed, it is still imagined that one knows more information than in the networks I consider: not only how vertices are connected by edges, but also how edges are arranged around faces, faces around volumes, and so on. And while in 2D and 3D it is possible to set up such an approximation to any manifold in this way, it turns out that at least in 5D and above it is not.

In each case, the column on the right shows the sequence of base 2 digits in the number, while the box on the left shows the remainder at each of the steps in the computation.

The plots in the second picture indicate that for a while the sizes of numbers obtained by the evolution of the system in these two cases are indistinguishable.

And in the pictures below one connection is therefore always shown going above the line of nodes, while the other is always shown going below.

In the first case, the connections from the new node are exactly the same as the connections from the existing node, while in the second case, the "above" and "below" connections are reversed.

And while it is indeed true that for almost every rule the specific pattern produced is at least somewhat different, when one looks at all the rules together, one sees something quite remarkable: that even though each pattern is different in detail, the number of fundamentally different types of patterns is very limited.

On subsequent steps, rule 255 allows only sequences containing just black cells, while rule 4 allows sequences that contain both black and white cells, but requires that every black cell be surrounded by white cells.