It's not quite clear to me yet that this constitutes a proof of nonexistence of even-loopability RFOs, at least for isotropic rules with a large number of statesconfocaloid wrote: ↑September 2nd, 2024, 2:35 amA flipping oscillator with even loopability is impossible, assuming the rules are isotropic.

The impossibility is because otherwise there would be two mirror images of the same reaction at the same time, which will necessarily remain mirror images of each other as they evolve (because the rules are isotropic), and either will never exchange with each other (contradicting the part "flipping") or will collide with each other (contradicting the part "even loopability").

The "will never exchange" part of the above logic isn't necessarily true of isotropic patterns when the line of symmetry is along cell boundaries, right? Two moving patterns traveling on diagonally connected cells in a von Neumann neighborhood will pass through each other very nicely.

Then it's just a question of whether it's possible to find patterns that will loop around after they pass through each other, and turn into their own mirror images

*after switching to the other checkerboard square color*. That's a tall order, for sure, but it seems like it might be possible to come up with a rather complicated engineered example somehow or other -- if we have enough states to work with. (?)

Code: Select all

```
#C diagonalize this isotropic rule (for example) and emulate orthogonally traveling patterns --
#C a mirror-image pair can be arranged to pass directly through each other at 90 degrees.
x = 10, y = 24, rule = ArtemDergachev-vonNeumann3state
2.A$.A.A$4.2A$.A2.2B.AB$.AB.B.3BA$.AB.B.3BA$.A2.2B.AB$4.2A$.A.A$2.A7$
8.A$4.A2B2.A$3.A.5B$2.B3.4B$.A7.A$.2B.2B2.A$A.A$.A!
```