In the New York Times: a piece in praise of recreational mathematics.

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## “The Importance of Recreational Math”

## Big natural ship in B358/S23

## Big and natural and (5,2)c/190

## Rich’s p18

## Rich’s p16

## Gun show (part 4)

## Gun show (part 3)

In the New York Times: a piece in praise of recreational mathematics.

Here’s another big spaceship evolving from a soup. The rule here is B358/S23, and the soup has D2_+1 symmetry.

x = 143, y = 41, rule = B358/S23 47b2o$49bo$51b2o$45bo7bo32bo$b2o21bo20bob3o36bo35b3o$o2bo19bobo24bo3bo 30bobo34bo14b2o$bobo18bo3bo14bobo5b2o32bo5bo32bo4bo12b2o$5bob3o12bo3bo 17b3obo2b4o2bo24bo7bo17bobo12bo12b2obo2bo$7bo2bo12bobo15bo3b2ob2o3bo3b o25bobobobo18bobo14bo2bo4bo3bob2obo$3b5ob2o13bo15bo2b3o11b2o25bo3bo35b 2o2bo4bo3bo2b3o$2bo2bo36bo3b2o3b2o4bobo19b2o34b2o6bo4bo4bo$2bo2b2o5b2o 30bob2o8bo2bo19b2o4bobo20bobo4b2o6bo4b2o4bo2bo$6bo6b2o2bo22b3ob2o2bo7b o27b2ob2o19bobo13bo4bo6bo$4bob2o4b2o2bo3bo23b2o2bo7bobo26b3o37bo$2bobo 12bobobo24bobobo6bo28bo39bo$2bobo4bo7bob4o25b2o$4bo3bo3b2o4b2obo26b2o$ 8b3ob2o3b2obo$18bo4$18bo$8b3ob2o3b2obo$4bo3bo3b2o4b2obo26b2o$2bobo4bo 7bob4o25b2o$2bobo12bobobo24bobobo6bo28bo39bo$4bob2o4b2o2bo3bo23b2o2bo 7bobo26b3o37bo$6bo6b2o2bo22b3ob2o2bo7bo27b2ob2o19bobo13bo4bo6bo$2bo2b 2o5b2o30bob2o8bo2bo19b2o4bobo20bobo4b2o6bo4b2o4bo2bo$2bo2bo36bo3b2o3b 2o4bobo19b2o34b2o6bo4bo4bo$3b5ob2o13bo15bo2b3o11b2o25bo3bo35b2o2bo4bo 3bo2b3o$7bo2bo12bobo15bo3b2ob2o3bo3bo25bobobobo18bobo14bo2bo4bo3bob2ob o$5bob3o12bo3bo17b3obo2b4o2bo24bo7bo17bobo12bo12b2obo2bo$bobo18bo3bo 14bobo5b2o32bo5bo32bo4bo12b2o$o2bo19bobo24bo3bo30bobo34bo14b2o$b2o21bo 20bob3o36bo35b3o$45bo7bo32bo$51b2o$49bo$47b2o!

It goes left to right (right to left in the original soup) at speed 36c/72.

Again, the way this comes about is through development of a small seed. In this case at generation 83 you get a couple of these objectswhich in four generations recur, inverted, but with some debris.By itself, this seed becomes a 36c/72 puffer.But two of them, mirror images at just the right separation, have their smoke trails interact in such a way as to extinguish them, and the result is a spaceship. If you start with this pair

x = 9, y = 29, rule = B358/S23 $2bo2b3o$2bob2obo$b2obo2bo$5b2o$2b2o18$2b2o$5b2o$b2obo2bo$2bob2obo$2bo 2b3o!

you end up with a ship, plus a couple of blinkers.

These puffer seeds crop up fairly regularly — they develop into puffers 214 times out of 5,274,253,500 C1 soups — though this looks like the first time two of them have produced a spaceship. This symmetric object isn’t the only way to kill the puffer smoke, though. Here are six other ships produced by placing two puffers at various relative positions and phases. Undoubtedly there are lots more.

x = 363, y = 80, rule = B358/S23 37bo$11b2o9bo13b4o264bo3bo38bo$10bobo11b2o10b2ob2o262bo2bo2bo14b4obo2b obo9b3obo$12bo8bo3bo7b2o5b2o147b3o131bo3bo2bobob2o7b3o2b2o$13bo11bo6b 4o3b2o148bo14b2o117bob3o5b2o7b2ob4o$2b2o8b3o10bo7b4o152bo4bo12b2o90b2o 3bobobo15bo7bobo2bo4bo2b3o$2b2o6b2o2bo7bobo9b3o138bobo12bo12b2obo2bo 89b2o3bo3bo20bobo4bo$9b2ob2obo7bo151bobo14bo2bo4bo3bob2obo95bobo12b2o 12b3o$11b2o2bo175b2o2bo4bo3bo2b3o96bo3b3o6bo14bo$16bo165b2o6bo4bo4bo 119b2o$15bo159bobo4b2o6bo4b2o4bo2bo102b3o$11bo163bobo13bo4bo6bo102bo2b o$11b3o178bo113bo$193bo112b4o$182b3o123b2o31bo$35bo146bo14b2o107b2o31b o2bo$9b2o9bo13b4o111bo32bo4bo12b2o104b2o3bo26bo$8bobo11b2o10b2ob2o108b obo33bo12b2obo2bo108bo11b2o2b2o3b2o8bo2b3o$10bo8bo3bo7b2o5b2o106bo38bo 2bo4bo3bob2obo104b2o2bo10bobob3o2b3o8bob2obo$11bo11bo6b4o3b2o108b3o34b 2o2bo4bo3bo2b3o98b3o3b2ob2o10b3o2bobo5bo2bo2b2obo2bo$2o8b3o10bo7b4o 112bob2o32bo4bo4bo104b2o2bo2bo2b3o17b3ob3o2bo7b2o$2o6b2o2bo7bobo9b3o 113b2o33bo4b2o4bo2bo100b2o3b2obob2o19b2o3b2o6b2o$7b2ob2obo7bo126b2o34b o4bo6bo122bo12b3o$9b2o2bo171bo119bobo11b2o$14bo134b2o35bo119bo12b2o$ 13bo137bo156b2o$9bo297b4o$9b3o295b4o$309bo25$5bo3bo38bo$4bo2bo2bo14b4o bo2bobo9b3obo$24bo3bo2bobob2o7b3o2b2o251b3o52b2o$24bob3o5b2o7b2ob4o 139b2o108bob2o57b2o$2o3bobobo15bo7bobo2bo4bo2b3o106bo34bob2o9bo6b3o85b 3obobo29b5o8bobo6b2obo2bo$2o3bo3bo20bobo4bo116b2o34bob2o8b2o4bo2bob2o 83b2o3bo29bob2ob2o7b2obo2bo3bob2obo$6bobo12b2o12b3o115b4o5b3o26b3o3b2o 4bo3bo6bo121b2o10bob2obo3bo2b3o$7bo12bo14bo119bo3bo2b3o32b2o2b3o3bo6bo 112b2o20bobobo$21b2o125b2o4b2o3bo3bo20b2o10b3o3b2o3b4o2bo113b2o7bo12b 2o3bo2bo$9bo3b2o133b2o3bobo3bo24b2o10bo5bo8bo123bo20bo$9bo4b2o136bo39b obo138b5o$9bo149bo32b3o137b3o$49bo101bo3bob3o33bo138b3o2bobo$11b2o34bo 2bo99b2o180b3o3bobo$46bo103bob2o178b2o5bo$31b2o2b2o3b2o8bo2b3o95b2o4b 3o173b2o$14b2o14bobob3o2b3o8bob2obo104b2o32b2o160bo$13bo2bo13b3o2bobo 5bo2bo2b2obo2bo99bo4b2o30b5o13bo123b2o9b3o4b4o2bo$6b2o3bobo2bo19b3ob3o 2bo7b2o99bobo3b2o30b2ob2o11bo2bo131b2o7bo6bo$6b2o3b5o21b2o3b2o6b2o105b o2bo32b2o2bo9bo127b7obo5b2obo6bo$27bo12b3o110bo3bo2bo34b5o6bo4bo2b3o 118b5o2b5o6bo2bob2o$27b2o123b3obo39bob3o3b2o5bob2obo110b2o6bo12bobo4b 3o$27b2o123b3o42bo2b2o2b2obo2b2obo2bo110b2o12bo7bo$16b2o169b2o9b2o5b2o 7b2o119b2o$15b4o168b2o8b2ob2o9b2o121b2obo$15b4o176b3o137bobo$17bo178bo 139bo!

Generally speaking the larger a Life object is, the less likely it is to arise from a random soup. Going by the current Catagolue census, for instance, gliders arise in Life 684 times as often as lightweight spaceships, which are seen 3.8 times as often as middleweight spaceships, which turn up 5.8 times as often as heavyweight spaceships. Or look at the statistics page: All of the still lifes of size up to 13 have arisen, and 616 of the 619 size 14 still lifes, but only 1256 out of 1353 size 15, 2484 out of 3286 size 16, 4199 out of 7773 size 17 and so on… to only 7769 out of 4,051,711 still lifes of size 24.

Now, the smallest known Life spaceship that isn’t a glider, a *WSS, or a flotilla of *WSSs is the loafer, which has population 20 in a 9 by 9 bounding box. For comparison the HWSS is 13 cells in a 7 by 4 bounding box. There are 2^81 possible states for a 9 by 9 box versus 2^28 for a 7 by 4, or 2^53 times as many — about 9 quadrillion. From that point of view it’s not too surprising no loafer has evolved naturally from a soup so far. Only 111 trillion objects have been seen so far, after all.

So what are the odds of natural occurrence of a population 49 spaceship in a 47 by 17 bounding box? Incomprehensibly tiny, you would think — never in many times the lifetime of the universe would it happen.

Well, so you might think, anyway. Evidently that thinking’s not entirely correct:Because that pattern evolved, not in Life but in the Life-like B38/S23, from a random D2_+2 soup, on my computer in the past few hours. It may not look like much… *but it’s a spaceship*. A spaceship which in 190 generations travels obliquely, 5 cells up and 2 cells to the left.

I was pretty excited by this discovery, until I checked the census for B38/S23 C1 soups, and saw that a bunch of p190 ships have been found already, the first by David S. Miller last April. Then I found out, well, re-found out these ships had been discussed extensively in a forum thread shortly after that. A thread which I read. And forgot about.

All these ships are based on the same fundamental engine. Take a look at the part on the right of the above ship. Run just that for 190 generations and you get this:Three of the pieces of the original pattern come back, shifted by (5, 2). The fourth piece gets changed. So this is a near spaceship by itself.

Now if you look at the part on the left and run that 52 generations you get:The same thing as the right half at generation zero, minus the boat. So the ship consists basically of two out of phase copies of a single engine, plus a boat, evolving in such a way that the interaction between them makes up for the lack of a boat for the left engine, and changes the evolution of the 7-bit piece in both engines to make it recur in 190 generations.

Another way to look at it: Start with an R pentomino and a boat:After 192 generations you get this:And if you add a second R pentomino in just the right place at just the right phase, it’ll react with the first R and boat in just such a way as to make a spaceship. Seems kind of miraculous, but in fact there several ways to accomplish it. According to David S. Miller, at least 692 ways. Of which, as of today, apparently 11 have turned up in soup searches. There are also another 120 combinations of two Rs and a boat that produce puffers, rakes, and so on.

So a 47 by 17 spaceship evolving naturally? Not quite as astronomically unlikely as it looks. A remarkable system, though, and there’s nothing like it known in Life. Yet.

Hot on the heels of Rich’s p16, here’s a period 18 oscillator, once again found using apgsearch. It even bears a family resemblance to the p16: D2_+1 symmetry and shuttle behavior. But… it doesn’t work in Life (B3/S23). It works in B357/S23.

RLE:

x = 13, y = 5, rule = B357/S23 b2ob2ob2ob2ob$o4bobo4bo$5bobo5b$4bo3bo4b$3b3ob3o3b!

Here’s a new period 16 oscillator:

The stubby wiki page says I discovered it, which is silly of course: all I did was install apgsearch and run it looking at D2_+1 soups in standard Life (B3/S23). I was asleep when it found this and woke up to find it’d been tweeted, retweeted, reported on the forum, used to make a smaller p48 gun, deemed awesome, and written up on the wiki.

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.

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.

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