In organisms with a total of just a few thousand cells, the final position and type of every cell seems to be determined directly by the genetic program of the organism; most likely what happens is that each cell division leads to some modification in genetic material, perhaps through rules like those in a multiway system.

For as the picture illustrates, the so-called shift map used in case (d) simply serves to shift all digits one position to the left at each step.

Yet looking at the history of science, one might expect that in the end there would turn out to be nothing special about the rule for our universe—just as there has turned out to be nothing special about our position in the solar system or the galaxy.

But if one identifies time with position down the page, the presence of connections that go up as well as down the page implies that in some sense time does not always progress in the same direction.

Up until a few centuries ago, it was widely believed that the Earth had some kind of fundamentally unique position in space.

But having developed a library of results one is then in a position to pick out substances that might be relevant for a specific technological purpose.

And it is common in spacetime physics to draw "light cones" of the kind shown below to indicate the region that will be reached by a light signal emitted from a particular position in space at a particular time.

Such a circle has area 2 π a 2 (1 - Cos[r/a]) = π r 2 (1 - r 2 /(12 a 2 ) + r 4 /(360a 4 ) - …) In the d -dimensional space corresponding to the surface of a (d + 1) -dimensional sphere of radius a , the volume of a d -dimensional sphere of radius r is similarly given by d s[d] a d Integrate[Sin[ θ ] d - 1 , { θ ,0, r/a}] = s[d] r d (1 - d (d - 1) r 2 /((6 (d + 2))a 2 + (d (5d 2 - 12d + 7))r 4 /((360 (d + 4))a 4 ) …) where Integrate[Sin[x] d - 1 , x] = -Cos[x] Hypergeometric2F1[1/2, (2 - d)/2, 3/2, Cos[x] 2 ] In an arbitrary d -dimensional space the volume of a sphere can depend on position, but in general it is given by s[d] r d (1 - RicciScalar r 2 /(6(d + 2)) + …) where the Ricci scalar curvature is evaluated at the position of the sphere.

Computer communication Most protocols for exchanging data between computers have in the end traditionally had rather simple structures—with different pieces of information usually being placed at fixed positions, or at least being arranged in predefined sequences—or sometimes being given in name-value pairs.

With a grid of cells set up in advance, each step in this type of Eden model can be achieved with AStep[a_] := ReplacePart[a, 1, (# 〚 Random[ Integer, {1, Length[#]}] 〛 &)[Position[(1 - a)Sign[ ListConvolve[{{0, 1, 0}, {1, 0, 1}, {0, 1, 0}}, a, {2, 2}]], 1]]] This implementation can readily be extended to generalized aggregation models (see below ).