The rates at which shell material is secreted at different points around the opening are presumably determined by details of the anatomy of the animal. … The rule on the left shows the amount of material added at each stage at different points around the opening; the line from the center indicates the progressive lateral displacement of the opening.

But vastly more common in practice is instability only at specific critical points—say bifurcation points—combined with either intrinsic randomness generation or randomness from the environment.

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.

In many cases this maximal rectangle overlaps those found at subsequent points.

Definitions of distance Any measure of distance—whether in ordinary continuous space or elsewhere—takes a pair of points and yields a number. … 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 .

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.).

[Memory analogs with] numerical data In situations where pieces of data can be thought of as points in space similarity can often be defined in terms of spatial distance. … Another approach, illustrated in picture (c), is to set up a continuous function with minima at the existing points, and then to search for the closest minimum. … The pictures below show how one can build up a kind of memory landscape by successively adding points.

General topology [and axioms] The axioms given define properties of open sets of points in spaces—and in effect allow issues like connectivity and continuity to be discussed in terms of set theory without introducing any explicit distance function.

In each case the result of 10 steps of evolution is shown, and the pictures are scaled so that all points above the bottom of the original stem can be included.