To model the pouring of sand into a pile one can consider a series of cycles, in which at each cycle one first adds 4 to the value of the center cell, then repeatedly applies the rule until a new fixed configuration FixedPoint[SandStep, s] is obtained. … The pictures at the top of the next page show some of the final fixed configurations, together with the number of steps needed to reach them. … With a total initial s value of m , the number of steps before a fixed point is reached seems to increase roughly like m 2 .

But ultimately the whole point of causal networks is that their connections represent all possible ways that effects propagate. … But whatever these values are, a crucial point is that their ratio must be a fixed speed, and we can identify this with the speed of light.

Lengths of [number] representations (a) n , (b) Floor[Log[2, n] + 1] , (c) Tr[FixedPointList[Max[0, Ceiling[Log[2, #]]] &, n + 2]] - n - 3 , (d) 2 Ceiling[Log[3, 2n + 1]] , (e) Floor[Log[GoldenRatio, √ 5 (n + 1/2)]] .

that no fixed point is reached, and instead there is exponential growth in total size—with apparently rather random internal behavior.

In addition: • GoldenRatio is the solution to x  1 + 1/x or x 2  x + 1 • The right-hand rectangle in is similar to the whole rectangle when the aspect ratio is GoldenRatio • Cos[ π /5]  Cos[36 ° ]  GoldenRatio/2 • The ratio of the length of the diagonal to the length of a side in a regular pentagon is GoldenRatio • The corners of an icosahedron are at coordinates Flatten[Array[NestList[RotateRight, {0, (-1) #1 GoldenRatio, (-1) #2 }, 3]&, {2, 2}], 2] • 1 + FixedPoint[N[1/(1 + #), k] &, 1] approximates GoldenRatio to k digits, as does FixedPoint[N[Sqrt[1 + #],k]&, 1] • A successive angle difference of GoldenRatio radians yields points maximally separated around a circle (see page 1006 ).

Quite what gravitational effects such zero-point energy might have has never been clear.) … What happens in essence is that the modes of fields in different background spacetime structures differ to the point where zero-point excitations seem like actual particle excitations to a detector or observer calibrated to fields in ordinary fixed flat infinite spacetime. … But presumably this is just a reflection of the idealization involved in looking at quantum fields in a fixed background classical spacetime.

Look at the array of Julia sets and ask for each c whether the Julia set includes the point z = 0 . … The pictures below show a generalization of this idea, in which gray level indicates the minimum distance Abs[z - z 0 ] of any point z in the Julia set from a fixed point z 0 .

But since one can always just start by proving the new axioms, the change can only be by a fixed amount. … A crucial point, however, is that for theorems of a given length there is always a definite upper limit on the length of proof needed.

Indeed, in all the systems that we have discussed so far there is in effect always a fixed underlying geometrical structure which remains unchanged throughout the evolution of the system. … But the crucial point is that these positions have no fundamental significance: they are introduced solely for the purpose of visual representation.

And the point is that causal invariance then implies that the same underlying rules can be used to update the network in all such cases. … Most likely the intrinsic properties of a particle—like its electric charge—will be associated with some sort of core that corresponds to a definite network structure involving a roughly fixed number of nodes.