Here is a thinner (but slightly wider) period 3 edge spark spaceship than one which I reported on a long while ago:

The top edge is closer to the row having the edge spark, and so this spaceship might fit into places that the previous edge spark spaceship wouldn't fit. So that you can compare them, that previous ship was:

I still have hopes to find an even thinner version of this edge spark.

BCNU, -dbell-

Playing with my c/3 puffer technology, I just found a reasonably small period 45 backwards rake which uses my period 45 puffer and four period 3 edge spark spaceships:

The amazing thing about this rake was that when I discovered it, it emitted two backwards gliders on nearly the same path. So the rearmost c/3 spaceship in the rake is only there to delete the extra glider! I could not find any perturbation that would create just one glider without leaving junk. (I can leave a loaf or a boat along with the glider, for example.)

Using two of these backwards rakes in a kickback reaction, we immediately get a period 45 forwards rake:

There is bound to be a smaller way to create a p45 forwards rake. For example, running a backwards rake's gliders into the debris from another p45 puffer should create a clean forwards glider somehow.

Now that we have period 45 rakes, we might almost have the technology to create puffers with an infinite number of odd periods. I know Dean knew of a period doubling method for puffers. Can anyone think of a period tripling method for rakes? Because then you could make an infinite class of puffers having periods 45 * 3^N, which are all odd.

BCNU, -dbell-

I should have held off on my previous announcement a little while, for here is an even better c/3 edge spark spaceship:

This one is thinner, and even shorter than the original edge spark spaceship. The cells at the bottom near the edge spark are slightly different, but don't appear to detract from its usefulness.

Also, about my previous message concerning odd period spaceships. I remember now that period doubling puffers had to make use of temporary spaceships in order to remember a bit of information. For c/2 puffers, this is easy enough. But for c/3 puffers, creating temporary c/3 spaceships is a bit beyond our capabilities! And so period doubling or tripling reactions for c/3 puffers might be hard. However, we might do even better by continuing to use temporary c/2 spaceships in these c/3 puffers. As in my period 90 based series of c/3 spaceships with arbitrarily large periods, we could make an arithmetic series of periods using period 45 c/3 puffers. I don't know for sure but I hope that these periods would also be odd. The only problem then is to make standard spaceships out of three p45 glider streams. Bit they are close together and impact the new spaceships before they can escape. This can probably be solved by using one of the 3-fold, 5-fold, or 7-fold period multiplying puffers I already found, and then cleaning up the debris with many p45 rakes. Then we would have c/3 rakes of period 135, 225, or 315. (But HUGE).

Can anyone make spaceships that can escape from the onslaught of the p45 glider streams?

BCNU, -dbell-

Yea, I know you are all bored by this already, but here is another result. This is a few columns shorter than the previous c/3 edge spark spaceship:

The search is still going...

BCNU, -dbell-

Using my new period 3 c/3 edge spark spaceship, I can make smaller versions of my period 45 forwards and backwards rakes.

The new period 45 c/3 backwards rake:

The new period 45 c/3 forwards rake:

I like how the arms of those two back rakes nestle together closely. You're not usually so lucky.

My search finished for a smaller edge spark spaceship. So that's all on that topic for now.

BCNU, -dbell-

Here is a new period 45 c/3 forwards rake which works by running the gliders from a p45 backwards rake into some p45 puffer output:

This has a smaller population and is thinner than the p45 forwards rake which used a kickback reaction, but is longer. In addition, the gliders have a longer distance to travel before they are "out of the shadow" of the puffer.

I also found an alternative period 90 backwards rake than the original one I found a while ago. This one works by running the gliders from two p45 backwards rakes together:

This rake is much shorter than the old p90 rake, but is wider.

BCNU, -dbell-

Well, it's ugly and huge, but here it is. This is a period 225 c/3 forwards rake composed of an old period 225 dirty puffer with the addition of three p45 backwards rakes and a few period 3 spaceships to clean up the huge mess:

Note that 225 is 45 * 5. There should be an analogous rake with period 135 = 45 * 3, but I never saved a dirty puffer of this period, and haven't yet tried making a new one.

I thought I did pretty good cleaning up all that debris with only 3 rakes! Of course, there are bound to be better solutions.

BCNU, -dbell-

This describes a method of creating spaceships and puffers which have an arbitrarily large odd period. This is based on c/3 period 45 rakes, a c/3 period 45 spaceship, and some LWSS's and boats. I think this can make all periods which are multiples of 45.

Basically, this works similarly to my previous arbitrarily large period spaceships which were based on c/3 period 90 rakes. (But all of those spaceships had even periods.) The idea is to use c/3 rakes to create a set of LWSS's which run forwards into a c/3 spaceship. The reactions produce still lifes which the rakes catch up to and thereby trigger more LWSS's.

Here is the first example of the series of spaceships with odd periods. It is a period 495 c/3 spaceship:

In this spaceship, there are two LWSS's (or boats) traveling in the loop at any time. Removing one of them produces the "base" spaceship having the period 990. This base period is the important number used to calculate and construct the next spaceship in the series.

To create the next spaceship in the series, you need to do the following painful steps. Delete all LWSS's and boats except for the single LWSS directly behind the top period 45 c/3 spaceship. Move that front c/3 spaceship and it's LWSS upwards by 30 cells. Doing this will increase the period of the base spaceship by 270 generations. In this example, the new period will become 990+270 = 1260. Then divide out all factors of two from the period. Here, this becomes 1260/4 = 315. This period will become the period of the next spaceship in the series, and the number of LWSS's and boats in that ship will be the quotient, in this case 4. Now to actually create the new spaceship, remember the position of the original LWSS, and run the new base spaceship the number of generations indicated for the next odd period spaceship (e.g., 315). Insert another LWSS behind the front c/3 spaceship in the same position as the original LWSS. Continue running and inserting until the loop is fully populated. You may also have to run a few more cycles in order to make the "tail" of blinkers and loafs stabilize. Then you have the new odd period spaceship.

Here is the period 315 spaceship after this construction:

Here are the first few periods in the series:

BASE SPACESHIP ODD PERIOD SPACESHIP QUOTIENT -------------- -------------------- -------- 990 495 45 * 11 2 1260 315 45 * 7 4 1530 765 45 * 17 2 1800 225 45 * 5 8 2070 1035 45 * 23 2 2340 585 45 * 13 4 2610 1305 45 * 29 2 2880 45 45 * 1 64 3150 1575 45 * 35 2 3420 855 45 * 19 4 3690 1845 45 * 41 2 3960 495 45 * 11 8 4230 2115 45 * 47 2As you can see, the periods jump around wildly, depending on the factorization of the base spaceship's period of 90 * (11 + 3N). I haven't investigated the pattern any, but I think this method will give all odd multiples of 45, but I don't know how to predict which ship will give a desired period.

I am sorry this is so complicated. I tried to make this all work with a single LWSS in the loop, but it won't do that. If anyone knows of a good rule to insert the LWSS's and boats without having to actually run the thing, please let me know. Or if someone can easily determine which ship in the series has a specified period let me know that too.

Leaving one factor of 2 in the final spaceship lets you also make spaceships which have the periods which are even multiples of 45 (ie, multiples of 90). But spaceships with those periods were known previously.

Finally, these ships can trivially be made into puffers by removing the trailing c/3 spaceship. It is there only to clean up occasional gliders.

BCNU, -dbell-

Mostly to David Bell

Nice work!

As you can see, the periods jump around wildly, depending on the factorization of the base spaceship's period of 90 * (11 + 3N). haven't investigated the pattern any, but I think this method will give all odd multiples of 45, but I don't know how to predict which ship will give a desired period.Actually, you'll get all odd multiples of 45 except the multiples of 135. You'll get period 45M iff M is the largest odd factor of 11+3N for some N>=0. Note that 11+3N is never divisible by 3, so its largest odd factor is never divisible by 3. On the other hand, if M is >= 11, odd, and not divisible by 3, then either M or 2M will be congruent to 2 (mod 3), so it will have the form 11+3N. Hence M will be the largest odd factor of 11+3N, so you'll get a period 45M spaceship. And your table already covered the cases M = 1, 5, and 7.

Dean Hickerson dean@ucdmath.ucdavis.edu

I found another version of my second type of period 3 edge spark spaceship, which is thinner by two rows than the previous one. Unfortunately, it is also much wider than the old ship. :-( I am still looking for ships which are better in both ways. The long arm here certainly looks like it should be able to be made shorter, but it is proving difficult:

Being thinner, it might be able to reach some c/3 debris and perturb it where the old fatter version couldn't fit.

BCNU, -dbell-

Here is a new c/3 period 15 spaceship that I have been trying to find for over a year, unsuccessfully until now:

This ship has both arms going in the same direction, maximizing access to the engine's output. The only new reaction that this p15 ship allows is to perturb a backwards glider into a boat (which is the reason I have looked for this ship for so long):

With this new reaction, a backwards glider can be made to do useful things. Added to a period 45 backwards glider rake, this period 15 spaceship can obviously create boats with period 45. And then the boats can be used to turn other gliders, so for example, turning backwards gliders into forwards gliders.

But, the best thing occurs with the double-glider output of the c/3 p45 backwards rake. The two gliders cleanly delete the boat and each other in a cloud of smoke:

This smoke can be perturbed in several ways to do useful things. For example, adding an edge spark spaceship creates a new small c/3 period 45 forwards rake. I am so happy that this works!:

Another perturbation of the smoke results in a pre-block, allowing the easy generation of backwards MWSS's:

Hitting the smoke with a glider can also produce useful results. There are many reactions in which an input glider deletes the boat before it is hit by the second glider, thereby releasing the second glider. So this allows easy glider turning and glider reflections with many different phases and paths.

But more interesting are these two reactions in which an input glider is duplicated. In the first one, two very close backwards gliders are produced:

And in this one, a forwards and backwards glider are produced:

I haven't found a duplicating reaction in which both output gliders are forwards gliders. That would obviously be most useful.

The forwards glider that is released is also released from other reactions. Input gliders can leave blinkers, blocks, boats, and so on, in addition to the output forwards glider.

Finally, the new p15 spaceship can be used to shrink my old period 90 c/3 rakes, since the engine can be placed closer to the action. Here is the new c/3 p90 backwards rake:

And here is the new c/3 period 90 forwards rake:

I shall send updated large period spaceships using the new p45 forwards rake in another mail.

BCNU, -dbell-

As promised, here is the new version of my odd period spaceships. The p45 forwards rakes have been replaced, and everything has been moved in as far as it will go. The leading spaceship is out front because it no longer has room to fit between the other ships in all locations. So if you want to construct one of the few earlier ships in the series, you might have to move the rakes further out again.

This uses a a base period of 2610 and has two LWSS+boats in the loop. That makes it period 1305:

BCNU, -dbell-

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