The result of this is that points in space can always be specified by lists of coordinates—although historically one of the objectives of differential geometry has been to find ways to define properties like curvature so that they do not depend on the choice of such coordinates. … Within say a surface whose points {x 1 , x 2 , … } are obtained by evaluating an expression e as a function of parameters p (so that for example e = {x, y, f[x, y]} , p = {x, y} for a Plot3D surface) the metric turns out to be given by (Transpose[#] . # &) [Outer[D, e, p]] In ordinary Euclidean space a defining feature of geometry is that the shortest path between two points is a straight line. … In ordinary Euclidean space, such paths are straight lines, so that the length of a path between points with lists of coordinates a and b is just the ordinary Euclidean distance Sqrt[(a - b) .

For the points that I make are often sufficiently complex to require quite long explanations.

Note that for a 3D Voronoi diagram with randomly placed points, the average number of faces for each region is 2 + 48 π 2 /35 ≃ 15.5 .)

This is now in practice done by Simplify and other functions in Mathematica using methods of cylindrical algebraic decomposition invented in the 1970s—which work roughly by finding a succession of points of change using Resultant .

And indeed after just thirty steps, the description of the kneading process given above would imply that two points initially only one atom apart would end up nearly a meter apart.

The pattern—in either its square or rounded form—has appeared with remarkably little variation in a huge variety of places all over the world—from Cretan coins, to graffiti at Pompeii, to the floor of the cathedral at Chartres, to carvings in Peru, to logos for aboriginal tribes. … The geometrical pattern was presumably made by first constructing 48 regularly spaced spokes by repeated angle bisection, as in the first picture below, then drawing semicircles centered at the end of each spoke, and finally adding concentric circles through the intersection points.

(Note that given explicit coordinates, one can check whether one is in d or more dimensions by asking for all possible points Det[Table[(x[i] - x[j]) . (x[i] - x[j]), {i, d + 3}, {j, d + 3}]]  0 and this should also work for sufficiently separated points on networks.

But in the abstract there is no reason that these connections should lead to points that can in any way be viewed as nearby in space.

Note that in the main text I have tried to emphasize important points by various kinds of stylistic devices.

And in a case like page 518 —with spacetime always effectively having a fixed finite dimension—points that are a distance t apart tend to have common ancestors only at least t steps back. But in a case like (a) on page 514 —where spacetime has the structure of an exponentially growing tree—points a distance t apart typically have common ancestors just Log[t] steps back.