Gustavo6046 wrote:I will try to find a diagonal spaceship witb small synthesis and a small bounding box (maybe a bit bigger than a glider),
The smallest known c/4 diagonal spaceship, other than a glider, is the 25-bit Crab, which (coincidentally) is the only other c/4 diagonal spaceship that has a synthesis, and that synthesis is very recent. It is highly unlikely that there are any that are smaller (although Josh Ball's discovery of the Loafer has shown that the entire object space has not been fully explored). Still, likely the only way to find one (or prove definitely that there are none) is to do an exhaustive search of all patterns of suitably small population and/or bounding box.
As for finding a synthesis for such a spaceship, if one indeed exists, is also a very difficult problem. Until very recently, there was only one exotic spaceship (i.e. excluding the four basic ones, plus flotillae of multiple *WSS) that had a known synthesis, and that synthesis was very complicated. There has been recent work at synthesizing some of the more exotic spaceships, but again, these syntheses are mostly extremely complicated, and not something that should be even attempted by novices. (The one notable exception was the Loafer.)
Gustavo6046 wrote:but rather faster, at least c/3 or c/2.
It is trivially easy to prove that no spaceship in Life can move orthogonally faster than c/2. It is also easy (although not quite so trivial) to prove that no spaceship in Life can move diagonally faster than c/4. You should try playing with Life some more at a microscopic level (i.e. running small patterns one step at a time and observing what they do and how), and this should become fairly obvious to you.
Gustavo6046 wrote:This would ease the creation of a low-period (and rather small-sized) Geminoid.
I don't see how this could possibly work (i.e. you could make a Geminoid, but not a smaller or faster one). Gliders are very easy to synthesize, which makes logic circuitry that uses them fairly small and easy to make. Logic circuitry that uses other kinds of spaceships would require much larger factories, and a Geminoid's instruction tape would have to be much larger, to create and destroy such much more complicated structures, not to mention that the spacing between bits of information on the tape would have to be larger, if one uses larger spaceships.
gmc_nxtman wrote:It has been proven impossible for any diagonal spacecraft to move faster than c/4. This is because no orthogonal spaceship can travel faster than c/2, and, technically, the two are equivalent.
Actually, the two are not equivalent, even though this just happens to work in Life. There are many rules that permit diagonal velocities of c/3 (which is a provable maximum). In general, if one looks at all Moore-neighborhood cellular automata that don't allow infinite growth at velocity c (i.e. no births on 0, or 1, or 2 adjacent orthogonal cells), it's easy to show this.
Consider a super-rule that is superset of all such possible rules (e.g. B345678/S012345678 plus also birth on any 2 cells that aren't orthogonally adjacent). Any pattern run under this super-rule will necessarily contain all living cells that would have been alive under any of the other sub-rules. Conversely, any cell not alive under the super-rule cannot be alive under any of the other sub-rules.
If you start with a small solid rectangle, it expands into an octagon, whose north, south, east, and west points (actually, edges that are the same length as those of the original rectangle, and don't increase in size) expand at c/2 orthogonally, and whose northwest, northeast, southwest, and southeast points expand at c/3 diagonally, with lines of slope (1,2) along the edges connecting these corners. It is thus impossible for any pattern in any Moore-neighborhood CA (other than the light-speed growth ones I mentioned above) to expand faster than this.
Life, and a few other rules like it, are exceptional, in that the diagonal speed has a hard limit of c/4 rather than c/3.
gmc_nxtman wrote:Here are the only known diagonal spaceship speeds, and the smallest spaceship of each speed.
This is technically correct, although it is possible to construct Geminoid spaceships of any desired velocity and angle (including diagonal) below a certain (very small) limit.
-- I don't even have the synthesis, imagine the really slow rake; I still haven't found such... (a px spaceship where x>>54)