Others have leaves with various configurations of sharp points. … There are patterns with sharp points that look like prickly leaves of various kinds.

A simple way to define the distance between two points is to say that it is the length of the shortest path between them. And in ordinary space, this is normally calculated by subtracting the numerical coordinates of the positions of the points.

But one way to do it is simply to pick two points in the network, then to say that paths in the network are going in the same direction if they are segments of the same shortest path between those points.

Concatenation sequences, as well as generalizations formed by concatenating values of polynomials at successive integer points, are also known to yield numbers that are transcendental.

But with rules (a) and (b) only a limited number of points in space can ever be reached. The other rules shown do not, however, suffer from this problem: in all of them progressively more points are reached in space as time goes on.

It is thus important to read these notes in parallel with the sections of the main text to which they refer, since some necessary points may be made only in the main text.

Moire patterns The pictures below show moire patterns formed by superimposing grids of points at different angles. … The second row of pictures illustrates what happens if points closer than distance 1/ √ 2 are joined.

Of these, {2, 4, 12, 40, 144, 544, 2128, 8544, …} are themselves fixed points. Of combinator expressions up to size 6 all evolve to fixed points, in at most {1, 1, 2, 3, 4, 7} steps respectively (compare case (a)); the largest fixed points have sizes {1, 2, 3, 4, 6, 10} (compare case (b)). At size 7, all but 2 of the 16,896 possible combinator expressions evolve to fixed points, in at most 12 steps (case (c)).

The first six give basic properties of betweenness of points and congruence of line segments. … The axioms given can prove most of the results in an elementary geometry textbook—indeed all results that are about geometrical figures such as triangles and circles specified by a fixed finite number of points, but which do not involve concepts like area.

In general the density for an arrangement of white squares with offsets v is given in s dimensions by (no simple closed formula seems to exist except for the 1 × 1 case) Product[With[{p = Prime[n]}, 1 - Length[Union[Mod[v, p]]]/p s ], {n, ∞ }] White squares correspond to lattice points that are directly visible from the origin at the top left of the picture, so that lines to them do not pass through any other integer points.