Apeirogon

The apeirogon (Schläfli symbol {∞}) is the limiting case as the number of sides in a regular polygon tends to infinity. It can be considered a tiling of one-dimensional space.

One-dimensional cellular automata are considered as running in an apeirogonal universe.

Coordinates, directions and displacements

Cartesian coordinate system

The lattice points (and equivalently the constituent line segments) of an apeirogon correspond directly one-to-one with the set of integers. An object's position can simply be described using one integer, and the magnitude of displacement also only requires one integer.

Notating velocity

For periodic moving objects such as spaceships, puffers and replicators, we need to also know the period of the object alongside the distance it travels in a period cycle. This forms a simple string:

nc/p

where:

• n is the object's displacement
• p is the object's period

c is not a variable, and instead is used to represent the object's velocity in relation to the speed of light.

Since in one-dimensional space there are only two possible directions, the direction of an object can be communicated using one of two words, often left and right.

Neighbourhoods on the apeirogon

In most cases, the neighbourhood is simply the two cells to each side of a given cell. Whether the cell itself is also counted as part of the neighbourhood varies.

For more complex rules, we can also consider cells further than one cell from the center as being part of the neighbourhood.

Symmetries

Static

For any given pattern, there are three ways in which it can be symmetric:

• asymmetry
• mirror symmetry, centered on a cell
• mirror symmetry, centered on a vertex

Kinetic

In addition to these 3 static symmetries, periodic patterns can also conform to one of 5 possible distinct kinetic symmetries. 3 of these duplicate the static symmetries above, whereas the other 2 arise for periodic patterns which have a mod equal to half of their period.

These two "bonus" symmetries are patterns which are asymmetric, but halfway through their evolution, become their mirror image; this can happen centered either on a cell or on a vertex.

Topologies

It is possible to either enforce boundaries at defined "edges" of the grid, or connect one side of the grid to the other; the latter can be seen as running a cellular automaton on a p-gon, with p being the number of faces.

Software support

One-dimensional rules are most often run as two-dimensional rules - successive generations are translated downwards, and the axis perpendicular to the universe contains prior copies of the pattern as to display evolutionary history.