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April 18th, 2016

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.

Read the whole story at mathematrec.wordpress.com

April 17th, 2016

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.

Read the whole story at mathematrec.wordpress.com

April 16th, 2016

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.

Read the whole story at mathematrec.wordpress.com

April 15th, 2016

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.

Read the whole story at mathematrec.wordpress.com

April 10th, 2016

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:

Read the whole story at mathematrec.wordpress.com

March 11th, 2016
Reposted with permission from Alexey Nigin's blog:

The day before yesterday (March 6, 2016) ConwayLife.com forums saw a new member named zdr. When we the lifenthusiasts meet a newcomer, we expect to see things like “brand new” 30-cell 700-gen methuselah and then have to explain why it is not notable. However, what zdr showed us made our jaws drop.

It was a 28-cell c/10 orthogonal spaceship:

Read the whole story at b3s23life.blogspot.com

March 5th, 2016

Newly discovered c/10 orthogonal Life spaceship… Minimum population 28 and size 6 by 12!! And the slowest known orthogonal. Reported by “zdr” on the conwaylife.com forum.

Read the whole story at mathematrec.wordpress.com

November 6th, 2015

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.

Read the whole story at mathematrec.wordpress.com

November 5th, 2015

A little more searching turned up this paper by I. Korec called “Real time generation of primes by a one dimensional cellular automaton with 9 states”, which is pretty much what it says on the box. He shows a 1D CA where the cell at the origin is in a particular state (“1”) in generation $t$ if and only if $t$ is prime.

The rule list is rather lengthy and he doesn’t explicitly give it (nor did I find it elsewhere). Instead he shows the history for the first 99 generations and says you can infer most of the rules from that; the rules that don’t enter in until after that point he lists. So I cut and pasted his history and wrote a little Python script to extract the rules list, from which I created a Golly .rules file. I guess it would’ve been possible to make the states look like Korec’s symbols but I didn’t bother; I just used different colors. Korec’s states “.”, “/”, “0”, “1”, “L”, “R”, “r”, “V”, and “v” are states 0 through 8, respectively.

The initial state for the CA is just a “0” (state #2 in this rules file) in one cell. The CA will be built toward the right and the first cell will be “0” in non prime generations, “1” (state #3) otherwise. It looks like this at generation 30:At generations 3, 11, 113, 307, 311, and 1103 the leftmost cell is green, so I’m thinking it works.

Read the whole story at mathematrec.wordpress.com

November 3rd, 2015

Today, Nov 3 (11/3 or 11/03 or 3/11), is the 307th day of 2015. And 3, 11, 113, 1103, 311, and 307 are all prime. Ignore Amazon, THIS is PrimeDay. (But next year it’s March 11, because 308 is not prime, but 71 is.)

So since I’ve been playing with cellular automata lately, here’s some prime CA material.

Dean Hickerson came up with primer, a Life pattern that generates prime numbers, back in 1991. Nathaniel Johnston wrote about it in a 2009 blog post, mentioning twin and Fermat prime variants devised by Hickerson and Jason Summers, and then going on to present patterns that generate Mersenne primes, prime quadruplets, cousin primes, and sexy primes.

Read the whole story at mathematrec.wordpress.com