First, that if the points are identical the distance is zero, and if they are different, it is a positive number. Second, that the distance between points A and B is the same as between B and A . … If the connections on the underlying network are one-way (as in causal networks) then one no longer necessarily gets the second property, and when a continuum limit exists it can correspond to a (perhaps discontinuous) section through a fiber bundle rather than to a manifold.

Voronoi diagrams The Voronoi diagram for a set of points shows the region around each point in which one is closer to that point than to any other. … Voronoi diagrams for irregularly distributed points have found many applications. … Voronoi diagrams are relevant whenever there is growth in all directions at an identical speed from a collection of seed points.

One definition of a spacelike slice is then a maximal set of points in which no pair are causally related (corresponding to a maximal antichain in a poset). … And an intermediate definition is that a spacelike slice contains points that are not themselves causally related, but which appear in either the past or the future of every other point. Given a spacelike slice in a causal network, it is always possible to construct another such slice by finding all those points whose immediate predecessors are all included either in the original slice or its predecessors.

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. … To a large extent one can.

One approach, illustrated in picture (b), is to use a d -dimensional tree. … The pictures below show how one can build up a kind of memory landscape by successively adding points. … In numbers earlier digits are traditionally considered more important than later ones, and this allows numbers to be arranged in a simple one-dimensional sequence.

Others have leaves with various configurations of sharp points. … But looking at arrays of pictures like the ones on the next page one makes a remarkable discovery: among the patterns that can be generated by simple substitution systems are ones whose outlines look extremely similar to those of a wide variety of types of leaves. … There are patterns with sharp points that look like prickly leaves of various kinds.

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. … And one immediate feature of causal networks that differs from space networks is that their connections go only one way. … To define the spacetime Ricci tensor, one considers—as on page 516 —a sequence of spacelike slices through this

And one can then see that if one follows connections in the network, one is always forced to go progressively down the page, even though one is able to move both backwards and forwards across the page—thus agreeing with our everyday experience of being able to move in more or less any direction in space, but always being forced to move onward in time. … 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.

There are fixed points, limit cycles and so-called strange attractors. … Fixed points correspond to zero-dimensional subsets of the space of possible states, limit cycles to one-dimensional subsets (circles, solenoids, etc.).

Up to 8D, the densest packings of this type are known to be ones obtained by successively adding layers individually optimized in each dimension. And in fact up to 26D (with the exception of 11 through 13) all the densest packings known so far are lattices that work like this. In 8D and 24D these lattices are known to be ones in which each sphere touches the maximal number of others (240 and 196560 respectively).