User:H. V. McIntosh/Northeast single shift agar
Being that an agar is a shift periodic pattern, taking them them from an associated de Bruijn diagram is the surest way to meet this goal, although only the smallest shifts and periods produce sufficiently small diagrams to be usable.
The diagram at the left is a second stage diagram for partial vertical neighborhoods two cells high (from splitting a Moore neighborhood horizontally) and six cells long (that being the circumference of the cylinder chosen to contain the period). To identify the nodes, the cells are grouped into three squares whose bits are to be read as hexadecimal numbers in the bigendian style. Nodes connect downwards, although this shft runs to the northeast. Strings of nodes linking only to each other in sequence can be consolidated, to make the diagram more compact and legible.
Links show which nodes can overlap each other, but for the sake of economy only symmetry classes of nodes are shown, and notations such as 2L (rotate two cells left), rot (any rotation) and so on let the links span classes.
A good way to read a diagram is to start with a node and add the bottoms (omit the top line) of successive nodes to the pattern represented by a walk. Multiple links mean alternative walks, generating multiple patterns.
Details of interest include:
- prime cycles, containing no subcycle
- eulerian cycles, using their links only once
- hamiltonian cycles, using their nodes only once
- clusters of nodes, connected more to each other than to others.
- extreme clustering: a cluster has only outgoing links, incoming links, neither, or both.
- if they have neither, the component is isolated.
- if every node is connected to every other, however remotely, the component is connected.
| region | commentary | |
|---|---|---|
| A | Region A contains an approximate self loop, except for the notation 2L. The pattern generated is a diagonal wick; if terminated by an excursion to Z, a diagonal fuse. | | |
| B | Diagonal pairs can merge into one or the other of two diagonal phoenices, but only in one direction to preserve the northeast shift. They cannot be mixed more than once and if broken, dissipate as fuses. | | |
| C | Stepwise diagonal phoenices, oscillators of period 2, nevertheless appear to shift between generations as dominoes appear and disappear. They can also break off by connecting to Z, possibly trailing sparks. | | |
| D | The dominoes in diagonal phoenices are aligned either horizontally or vertically and therefore occupy separate regions of the de Bruijn diagram and cannot mix; their shared link to Z is always one-way. | | |
| E | A cylinder of circumference 6 is wide enough to contain another kind of phoenix, the ones which make barberpoles. | | |
| F | A genuinely two dimensional shifting agar, even though it is also a phoenix, as it resists being split up into wicks. | | |
| Z | All roads lead to Rome | | |
