## Gun show (part 4)

The p61 gun is quite different, though it too makes use of herschel tracks. To get a better picture of what’s going on, here it is with history turned on: the blue cells are ones that were live at some point: To start with let’s zoom in to the upper right corner. You see a couple of lightweight spaceships moving west to east, and the spark on the one near the center is about to perturb a southwest-going glider: 39 generations later, and several cells to the south, this becomes an r pentomino: And another 48 generations later, quite a bit further south, it becomes a herschel.That herschel gets sucked up into a downward conduit (purple line below). It gets converted into two parallel southwest-going gliders. One of these (red line) gets bounced off a series of 90° reflectors, snarks again like the ones we saw in the p58 gun, ending up at the top where it becomes (a later version of) the glider we saw at the start, getting converted to an r pentomino. The other one (yellow line) gets kicked right by an interaction with a herschel loop (orange line). I presume this very complicated reflector is used because it can reflect one glider without messing up the parallel stream (and I’m guessing a similar loop can’t be made to work at p58, hence the different solution used in that gun?). Not quite sure. Anyway, it then gets bounced a couple more times before ending up at the top of another section of the gun, where it’ll share the other glider’s fate: getting converted by a lightweight spaceship into an r pentomino, then a herschel, to feed another herschel track.

Here’s the middle stage:Again a downward track (purple) produces two parallel gliders (red and yellow). Again the yellow one gets bounced by a herschel loop to the top of a third stage for yet another r pentomino conversion. As for the red one, it bounces a bunch of times up to the top left where it runs into… something.

The third stage yet again has a downward track producing two gliders, one bounced off a loop and the other just kicked around with snark reflectors.

Both of these gliders arrive at the proper phase, spacing, and direction to interact with each other and with the red glider from the second stage to produce, of course, a lightweight spaceship. And the spaceship travels east, perturbing three gliders as it goes but remaining unscathed itself. If this were all, you’d have a p61 lightweight spaceship gun, but instead there are a few more still lifes at the right edge which convert the lightweight spaceships into gliders. And there you are. A p61 glider gun.

## Gun show (part 3)

Next (in reverse chronological order, but it makes sense to me) the p58 gun. I think “AbhpzTa”‘s version is pretty much the same thing as “Thunk”‘s (based on Matthias’s component), but in such a compact form it’s harder to see what’s going on. Here’s “Thunk”‘s:What we have here is not one but two herschel loops, both period 58. The top one is connected to the bottom one by another herschel track, and there’s a reaction that duplicates the herschels in the top track, sending one on its way around the loop again and another down toward the bottom track. But this doesn’t happen without input: it needs a period 58 glider stream. Where does it get one? Patience…

Where the cross track feeds into the bottom loop, the two herschels collide and out of the collision come not one but two gliders every 58 generations, heading southeast. They’re pretty close together. Too close, in fact, because we want to reflect one stream 90°, and that can’t be done without messing up, and getting messed up by, the other stream. So we use this cute reaction:

Two perpendicular glider streams go in, two go out. Same directions, but displaced. Meanwhile the parallel glider stream just squeaks by. That puts the two streams further apart, but not by enough, so we do the same thing again. Now they’re separated by enough.

(But wait, that reaction needs a second glider stream, going northeast, to work. Two of them to make it work twice. Where do we get two? Patience…)

One of the two not-so-close-together parallel streams gets kicked to the right, and the other to the left, with this apparatus. It’s called a snark, and it’s by far the smallest and fastest stable glider reflector known. Here you can see a glider coming in from the northwest and another on its way out to the northeast.

The stream that gets kicked to the left gets kicked left again, using a different, larger, oscillatory object, I think in order to get the correct glider phase or position for the outgoing stream. It’s now heading northwest, back toward the herschel loops — in particular, toward the intersection of the upper loop with the downward connector. That’s right, it becomes the glider stream needed to make the herschel duplicator work.

The other stream gets kicked to the right three times — now it’s heading northeast, crossing perpendicularly the two parallel streams, and it runs into a block at just the right time and phase to make the stream displacer work. Then it gets bent to the right four more times, putting it perpendicular to the two parallel streams again, so it can make the other stream displacer work. We didn’t need two new streams after all for the displacers, or even one… the displaced stream and both of the auxiliary streams are in fact all the same stream! Reminds me of a Heinlein story for some reason.

Finally, in the version “Thunk” posted, there’s one more kick to the right sending this stream off to the southeast to become the gun’s output, but there’s no need to do that; it could just continue to the northeast. And that’s the gun.

Unlike, say, the Gosper glider gun, which just needs two queen bees and two blocks to get started, this one relies on glider streams to work; it regenerates those streams itself, but it has to be built in the first place with glider steams to get started with. What happens, I wondered, if you erase one of the gliders heading into the herschel duplicator? Does it just create a gap in the output glider streams, or does something more serious occur? Something more serious, it turns out.

## Gun show (part 2)

For me the easiest of these guns to comprehend is the p57 one, so let’s work our way up to that.

Start by considering the heptomino that has acquired the somewhat erroneous name of herschel. It arises, along with some debris, early in the evolution of the r pentomino and spits out a glider, which is how the glider was discovered back in 1970. Without the r pentomino’s debris, the herschel stabilizes in 128 generations leaving two blocks, two glders, and a ship. But a notable thing about the herschel is that its evolution isn’t centered around its original position; most of the action happens to one side. Here’s a herschel (in red) and its stable state (in green), with the cells that otherwise were live in blue:Notice how, aside from the gliders, most of the action happened off to the left of the initial state.

So you can use a hershel over here to make something happen over there. In particular, you can imagine setting up some still lifes that will interact with the herschel in such a way as to make another herschel happen over there — while preserving the still lifes. Like this. Start with this state: and 117 generations later you have this state:plus a glider off to the southwest, which can be disposed of with another eater if you want. The eaters and snake perturb the herschel without getting injured; the block gets destroyed but is then remade in the same place.

So that’s very cool. It’s called a conduit. You could put a second conduit to the right of the first, positioned so the herschel output by the first conduit is in just the right place to be input to the second one, and at generation 234 the herschel will appear to the right of the second conduit. And of course you could put a third conduit, and a fourth, and a fifth, and so on, and transport that herschel as far as you like in a straight line, popping up every 117 generations.

What’s a little less obvious is that if you can contrive a way to feed such a track of conduits a periodic series of herschels, they’ll get transported just fine even if they turn up more often than every 117 generations. In fact, this conduit is ready to accept its next herschel as frequently as every 63 generations.

Furthermore, you’re not limited to straight lines. There are other conduits that will transport a herschel around a corner. Some conduits do a mirror flip of the herschel, some don’t. Some even manage to send a herschel backwards.

So with some ingenuity, you can set up a track of conduits that goes around four corners and connects back on itself in a loop! All of this was pioneered by David Buckingham in the 1990s, and his period 61 loop was the first period 61 Life oscillator discovered.

Since then there’s been lots of herschel conduit exploration going on, involving discovery of both new conduits and new ways to make use of them. And that’s what’s going on in the p57 gun; you have a loop, built of conduits that can accept herschels every 57 generations. Unfortunately you can’t just leave out one of the tub-with-tail still lifes that eats the gliders emitted by the herschels without breaking the conduit, but hanging off the bottom of the loop is a crazy lump of a period 3 oscillator. It hassles the nearby conduit into spitting out a glider while preserving the conduit action, and, boom, p57 gun.

## Gun show (part 1)

I’ve dabbled intermittently with Conway’s Game of Life — strong emphasis on both “dabbled” and “intermittently” — for more than 45 years now. In fact I think I read Martin Gardner’s classic article on the subject in the October 1970 Scientific American when it was hot off the press (in my high school library), a month or so before William Gosper found the first glider gun. That gun bounces two queen bee shuttles off one another; the mechanism repeats itself every 30 generations, producing a glider each time, so it’s a period 30 gun. The following year Gosper found another glider gun, with period 46.

You can perhaps imagine a gun like one of these, which emits a glider, for instance, every 50 generations, but whose mechanism repeats itself at a multiple of that period — every 100 generations, say. In that case one says the gun has a true period of 100 and a pseudo period of 50. (Despite the pejorative connotations of “pseudo”, though, if you’re using a gun to build something, it’s probably the pseudo period that’s of more interest to you.)

The shortest (pseudo) period a glider gun can have is 14. If you try to make a glider stream with shorter period, it doesn’t work: the gliders interact with each other and die. There are guns known with all pseudo periods from 14 on up.

For the true periods, though, there are holes. All periods from 62 up exist, but the smallest true period known is 20, and there are quite a few periods in between with no known gun.

But fewer such holes than there were a couple days ago.

On Wednesday, “AbhpzTa”, a newcomer to the conwaylife.com forums, posted a p61 gun:

Some discussion ensued in which, yesterday, Matthias Merzenich suggested a component which “thunk” used to make a p58 gun:

and “AbhpzTa” compactified it this morning:

Just an hour later, Matthias posted a p57 gun:

Three new true periods found in less than 48 hours!

How do these things work? Like I said, I dabble, and there’s a lot of arcane Life knowledge out there I’m not up on. But I think I get at least some of the ideas at work here, and I’ll write them up for my own benefit if no one else’s soon.

## How slow do you want it?

Another interesting Life development. Michael Simkin has found an orthogonal c/8 spaceship, the first of that speed. Or maybe better to say he’s built one, since it’s not an elementary spaceship discovered by a search program but a large engineered object. Furthermore the technology used, called a caterloopillar, can in principle be modified to produce spaceships — or, with trivial modifications, puffers or rakes — of any speed slower than c/4.

I said large. How large? Simkin says:

It’s pretty big. Some numbers:

cell count:
minimal – 232,815
maximal – 239,370

bounding box ~ 734 X 500K

Note, not 734K but 734 by 500K. Loaded into Golly and zoomed to fit it looks like this:

No really. That’s a spaceship. Zoomed in you can see it’s mostly periodic in structure.

If you look here you can see a big GIF showing some of the glider and standard orthogonal spaceship action going on within the ship.

## fCAtorization

I came up with a cellular automaton for factoring numbers. Fairly ironheaded and I’m sure not novel but a fun exercise for me. Here’s the Golly rules file. Yeah, there’s 13 states. I did say ironheaded.

In the initial state there are two cells on, one in state 1 (blue) at $(0, n)$ and one in state 2 (dark red) at $(n, n)$ (with $n>2$). Here’s $n = 12$:The CA starts building some infrastructure: a top axis ($y = n$) in dark red, a vertical axis ($x=0$) in blue, a bottom axis ($y = 0$) in light green, and a main diagonal ($y=x$) in dark green. The bottom axis is where results will be shown: after $3n-3$ generations, cell $(m,0)$ will be red if and only if $m$ is a factor of $n$ (for $0\leq m\leq n$).But in that picture the top and left axes and the main diagonal are only half finished, and there’s other stuff going on. What’s that? It’s already testing numbers to see if they’re factors. The infrastructure’s not complete but there’s enough there to get started.

Here the top and left axes and the main diagonal are completed, and the bottom axis is under way. On the right is a grey glider heading south, which will tell the bottom axis where to stop. And here the bottom axis is done, in time for a blue south-going glider to mark 6 as a factor. 1 is already marked as a factor: the regular CA mechanism won’t work for $m=1$, but it’s kind of safe to assume $1$ is a factor of $n$, so it’s marked as such immediately. So is $n$. Here the process is nearly finished. Four factors are marked:After 33 generations, action stops, with factors at 1, 2, 3, 4, 6, and 12 marked.It works as follows: For each $m$ from $n-1$ down to $2$, at the top axis a slow south-going (S) glider, with speed $c/2$ where $c$ means one cell per generation, and a SW-going glider with speed $c$ are dropped. When the SW glider hits the vertical axis it bounces off as an E glider (again speed $c$). After $2m$ generations the two gliders will collide, unless they hit the main diagonal first, in which case they’re deleted. If they collide before hitting the main diagonal the E glider bounces back SW, hits the vertical axis, and bounces back E for another potential collision after $4m$ generations. This continues, with the SW/E glider bouncing off the S glider and hitting the vertical axis at intervals of $m$ cells, until they collide with the main diagonal and are destroyed… or they collide with each other on the main diagonal. If that happens then obviously a bounce to the SW would send that glider to the origin, and that means $n$ (the length of the vertical axis) is evenly divisible by $m$. But instead of bouncing SW in that case, a new fast (speed $c$) S glider is dropped to the bottom axis, where it marks cell $m$ as a factor.

What’s tricky here is that this whole process is launched for $m-1$ one generation after it’s started for $m$! So a whole bunch of potential factors are being tested at any given time. It’s not obvious this can be made to work, especially since various SW and S gliders are launched cheek by jowl, and the slow S gliders are two cells wide. But it does work.

Here’s a couple videos: Testing $n=60$:

giving results 1, 2, 3, 4, 5, 10, 15, 20, 30, and 60, and testing $n=403$:

$403 = 13\times 31$, so there’s not much action on the bottom axis until just moments before the end.