At first it was thought that the visual system might be sensitive only to the overall autocorrelation of an image, given by the probability that randomly selected points have the same color.

And in terms of Feynman diagrams the way this happens is by virtual particle-antiparticle pairs of all types and all energy-momenta continually forming and annihilating at all points in the vacuum.

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.

Such circles are no longer flat, but instead are like caps on the sphere—with a circle of radius r containing all points that are geodesic (great circle) distance less than r from its center.

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 in fact even the vacuum Einstein equations for complete universes (with no points left out) have solutions that show curvature.

(This is analogous to what happens for example in classical diffraction theory, where there is an analog of the path integral—with ℏ replaced by inverse frequency—whose stationary points correspond through the so-called eikonal approximation to rays in geometrical optics.)

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.