Random Explorations in Automata Theory Gary J. Shannon Created: Mar. 14, 2003 Last updated: Mar. 14, 2003

Spin State Automata: Some Experiments

Presented here are the results of some preliminary experiments with spin state automata. The same rule is used in all of these experiments in order to observe any differences in the texture of space as indicated by the propagation of a simple square wave of states through that space.  The rule is that an empty cell with any number of neighbors in state 1 (HEAD) will go to state 1, any cell already in state 1 will go to state 2 (TAIL) and any cell in state 2 will return to the empty state.

In an ordinary Moore neighborhood CA this rule results in a square of HEAD cells shown in red, that propagates outward as a growing square while being followed by an inner square in the TAIL (magneta) state as shown in figure 1.

Fig. 1. The any-neighbor rule.

Using the spin state rule that implements the Margolus neighborhood and again starting with a single cell at state 1 the CA evolves into a continuous set of concentric squares rather than a single outwardly moving sqaure. The corner of the Margolus block that held the original 1 state leaves its mark on the configuration in the form of a notched corner.

Fig. 2. The same rule in the
Margolus neighborhood.

Using the same Margolus-type spin states but setting the spin rate to +90 degrees per tick for all cells results in a set of concentric dotted bands.

Fig. 3. 90-degree spin rate.

A 45-degree spin rate results in a brief splash lasting 8 ticks.  At its largest extent it is no more than a few living cells.

Fig. 4. Tiny splash.

Even much large configurations of initial states die out within 16 ticks.

Using a spin rate of +135 degrees with the 3-neighbor Margolus spin window results in a growing square who's inside is filled with a random-looking pattern that gives the appearance of swirling clockwise in some areas and counter-clockwise in others.

Fig 5. Swirling interior with
135-degree spin rate.

If the view window of the cell is set to see only two neighbors positioned 180 degrees apart the area fills with the checkered pattern shown in figure 6. If the spin rate is 90 degrees the outer perimeter of the area is square, but if the spin rate is 45 degrees then the outer perimeter becomes octagonal even though the interior pattern remains the same.

Fig 6. The 180-degree view window.