But in fact even the vacuum Einstein equations for complete universes (with no points left out) have solutions that show curvature. If one assumes that space is both homogeneous and isotropic then it turns out that only ordinary flat Minkowski space is allowed. … One example is parity violation; another is the presence of long-range forces other than gravity.

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 ).

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.

In the 1870s Maxwell also suggested that mechanical instability and amplification of infinitely small changes at occasional critical points might explain apparent free will (see page 1135 ).

But as mathematical methods developed, they seemed to apply mainly to physical systems, and not for example to biological ones. … By the late 1800s advances in chemistry had established that biological systems were made of the same basic components as physical ones. … And indeed many of the points I had made about the promise of the field were being enthusiastically repeated in popular accounts—and there were starting to be quite a number of new institutions devoted to the field.

So in particular one sees class 4 behavior. … But as soon as one goes to slightly more complicated rules—though still very simple—one can find examples. … But as soon as one looks at a one-dimensional slice—as on page 249 —what one sees is immediately strikingly similar to what we have seen in many one-dimensional class 4 cellular automata.

In general, if one wants to find a piece of data that has a certain property—either exact or approximate—then one way to do this is just to scan all the pieces of data that one has stored, and test each of them in turn. … So what can one then do? … It turns out that one does not.

For at any point the system will always in effect be able to do more things that one did not expect. … But if one had a system that was complete and consistent then it is easy to come up with such a procedure: one just runs the system until either one reaches the string one is looking for or one reaches its negation. For the completeness of the system guarantees that one must always reach one or the other, while its consistency implies that reaching one allows one to conclude that one will never reach the other.

One might have thought that it would be purely a matter of history. … And indeed if one starts from the beginning of the list one finds that most of the theorems can readily be derived from simpler ones earlier in the list. … So what happens if one applies the same criterion in other settings?

To say that one is not moving means that one imagines one is in a sense sampling the same region of space throughout time. But if one is moving—say at a fixed speed—then this means that one imagines that the region of space one is sampling systematically shifts with time, as illustrated schematically in the simple pictures below. … And in fact one can just view different states of motion as corresponding to different such choices: in each case one matches up space so as to treat the point one is at as being the same throughout time.