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 (they may share links)
- eulerian cycles, using each link only once
- hamiltonian cycles, using each node 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. Having neither, the component is isolated; if every node is connected to every other, however remotely, the component is connected.
Seven regions
| 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. It contains only a single strand because more would interact on a cylinder of circumference 6, but wider cylinders could have them. |  |
| 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 one which makes barberpoles. Those terminate the phoenix with preblocks to create finite patterns, but left alone or with trailing dominoes, the result is a fuse. |  |
| F | A genuinely two dimensional shifting agar, as it resists being split up into wicks. Truncated from below with a row of dominoes it makes a cleanly burning two-dimensional fuse. However, it includes an appendage to the cycle E according to which a vertical domino can occasionally replace and upspace rows of the phoenix E, possibly creating a random texture. |  |
| Z | All roads lead to Rome |  |
Analysis
Layout of a graph is important for its comprehension; the one shown is slightly cramped for compactness, but seven prime or nearly prime cycles can be discerned.
Of these, Z represents the vacuum with only incoming links. That means that there are no freetanding patterns because such a pattern would have to connect to the vacuum at both ends. But all the others do connect to the vacuum, leaving fuses which will have trailing sparks if the connection is not direct.
Region A is the first of three clusters connecting directly to Z but not with each other. If the wick terminates abruptly it becomes a fuse, the progenitor of many makers, which could appear in diagrams for wider cylinders or different periodicities.
In the second cluster, the 2-cycle E and the 3-cycle F form an interesting combination. E alone generates strands whose stable truncations include tubs and the barges but those, being finite, are not shift invariant. F alone, omitting the backlink between 011 and 202 is still another phoenix.
The third and remaining cluster has two poles, namely the absorbing self-linked nodes Z and B, with cross connections C and D. They are respectively phoenices of displaced dominoes, horizontal for C and vertical for D. At the bottom, either can be a fuse, with such sparks as there may be. At the top, they may be connected as tagalongs to a diagonal double strand acting as a wickstretcher. Such a double fuse is rather uncommon.