You mention dual systems. In a cellular automata similar to the game of Life, except in an explosive rule where the cells appear to flicker randomly, since the diagonal of a cell is longer than the edges, any chain of cells "moving" diagonally, in going from A to B diagonally, must travel a further distance than going from A to D vertical in the same number of cells, the there is the appearance and the math of diagonal motion travelling faster. What happens in this dual system is that a perfectly straight line shoots almost instantly across the flicker of cells, and a "double circle" or closed path appears somewhere on the screen in the random flicker. Now if the vertical cells could turn on faster so that groupings of cells in the vertical direction now move at the same speed or near-abouts the diagonal, then closed paths still have to occur, but they may not be circles. Since the diagonal is an irrational number, it's uncertain whether you can "square the circle" an ancient problem, and you may have to investigate lunes, or instead analyze what sort of instantaneous motion across all flickering cells, what path it takes instead of a perfectly straight horizontal line that resolves to a double closed circle, it may be a curved line, and the closer you get to the irrational number approximation, perhaps the more highly curved lines that shoot instantly through the flicker, so that it may be an automata that approximates gravity. You can look on an arduino screen that has a slower update on a cheap arduino screen like those small lcd screen in the game of life, and all have a delay, but with the delay you can see how the diagonal increase in speed I mentioned causes the small patterns to make rotations in a circle. I'm talking about varying the update speed of how fast a cell turns on to match the diagonal speed so there is no preferred direction (the diagonals in a square lattice have been criticised for that preferred direction).
It may be that the lunes in the ancient problem of squaring the circle would be the most obvious presence on screen, since you can have exact "squaring of the circle" with certain lunes. If this seems like total nonsense, I did write a post titled "quantum computer" on Conway's Game of Life in the forum -other cellular automata- that shows the pattern, in particular the larger scale placement of the X's, pasted in a hexagonal super pattern in a square underlying grid.
update, There doesn't need to be any change to the rule in the pattern I made in Golly. However, how you present them does need to be changed. You would use the same rules, but as the cells turn on or off, you'd present the product of the computation (where a cell turns on), you would delay everything, but have faster "turn on a cell" if a cell is directly above a cell that is born (not above and to the left, a diagonal), or if a dell is directly to the right (or left) that is born ---- no change in the rule just presenting vertical "toppers" and horizontal "toppers" as they're produced within the sequence of delay, to present those on screen before the diagonal ones even though "NOTHING HAS CHANGED" but the effect is different-----. That's what I learned from the delay in all cells in the arduino screen and also in a lossy form of video compression, they're similar, but I'll have to program a delay just in the -presentation-. I believe that the hexagonal lattice I made in that form (a slightly rotated hexagon), there is a function in Golly, --FLIP PATTERN VERTICAL-- AND --FLIP PATTERN HORIZONTAL--. When you take the original hexagonal pattern, and the paste a flipped one on top in vertical or horizontal, the X's start to line up in rows and columns, a necessary ingredient to have strings flowing through them. I'm going to start working on it, and if you're interested I'll show the results on that Conway forum. The point is to match the diagonal speed and to remove as much circular closed paths as possible.
Edit: If you think about it, having just -in the chain of delay- the vertical cells presented before diagonal born cells, since the diagonals are already "fast", it will make large collections of cells move in a roughly larger circular shape (moving in a rotated square relative to a facing square or a diamond shape, a larger collection of cells moving along the diamond path). Now if you overlay that over another automata (overlay meaning no interaction but you can see both cells as they update) the -another automata- has both vertical and horizontal updating faster than the diagonal within the chain of delay as they are presented, a new presentation of the order of the cells with the same rule. If you think about the -another automata- it's tuned not for rotation since there is no preferred "faster" direction, so you'll have just points or dots or fixed points appear. I'm guessing that the -another automata- are the hidden variables of the large 2d clouds of rotating pixels. Before, with no alteration of anything just the rule B3/S12456 in Golly, upon playing a video in a lossy format, it only produced very small groupings of pixels rotating everywhere, and that's because the X-shaped pattern has a repeating "circular" small oscillator at the center of each X-shape pattern. And instead of just overlaying, you could genetically mutate the two just described -the larger rotation and the fixed points- by overlaying both presentations and having a genetic algorithm mix the two --> in this case the fixed points would get mixed in the larger 2d rotations.
If you just wrap the top and bottom that increases the speed of the vertical, since objects can teleport from top to bottom. That makes a larger rotation with the void area covered by the vertical motion making a continuous area to rotate. Contrast that with a different rotation by wrapping the horizontal. By switching between those two you have a gate (cylinder wrap in golly either vertical or horizontal) and you have a gate ---if you place the X's (minus the diagonals of the X's if you want to have the oscillators interact closer) in a certain pattern, you can switch the gate between vertical and horizontal. By torus wrapping, any fixed points "drop out" since there is no preferred "fast" direction anywhere.