2D circular or spiral motion

[jeremyrutman]

All of the 'gliders' that I've seen seem to go in straight lines indefinitely. The only forum posts I've seen referring to circular or spiral motion are for 3D automata - are there any 2D gliders that exhibit circular or spiral motion?

[confocaloid]

I believe one of best known examples of (approximately) circular motion is the following "soldier bug" in a Larger than Life rule. This pattern is included in Golly's pattern collection as "Patterns/Larger-than-Life/SoldierBugs.rle":

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#
# Dean Hickerson discovered the "soldier bug" in 2001.
# A single soldier bug moves around in a circle, with period 552.
# The following pattern by Dave Greene shows how twelve soldier bugs
# can be arranged to create a period 46 oscillator.
#
x = 113, y = 113, rule = R7,C0,M1,S65..114,B65..95,NM
49b4o$47b7o$46b10o$45b12o$45b13o19b2o$44b3o5b6o17b6o$22b3o19b3o5b6o16b
9o$20b7o17b3o5b6o15b11o$19b9o16b3o5b6o14b3ob9o$18b12o15b3o2b8o14b3o4b
7o$18b13o14b12o14b3o6b6o$17b5o3b6o15b10o15b3o6b6o$17b2o7b5o16b7o17b3o
6b6o$17b2o7b5o18b3o20b3obo2b7o$17b3o5b6o41b13o$17b3o2b9o42b11o$18b3ob
8o44b8o$19b10o46b6o16b5o$20b7o50b2o17b8o$21b5o69b4o2b4o$94b3o4b5o$93b
3o5b5o$93b5o4b5o$93b5o4b5o$93b5o4b5o$93b6o2b5o$94b12o$9b5o81b10o$8b7o
80b9o$7b9o80b8o$6b10o81b6o$6b11o$5b13o$5b5o3b5o$4b5o5b5o$4b5o5b5o$5b4o
4b5o$5b3o6b4o$6b4o3b4o$7b9o$8b7o$10b3o3$104b4o$102b8o$101b10o$100b4o4b
4o$100b3o5b4o$99b4o5b5o$99b5o4b5o$99b5o4b5o$100b13o$100b12o$101b10o$4b
6o91b10o$3b8o91b8o$2b10o91b6o$2b10o$b12o$13o$5o4b5o$5o4b5o$5o5b4o$b4o
5b3o$b4o4b4o$2b10o$3b8o$5b4o3$100b3o$98b7o$97b9o$96b4o3b4o$95b4o6b3o$
95b5o4b4o$94b5o5b5o$94b5o5b5o$95b5o3b5o$95b13o$96b11o$10b6o81b10o$9b8o
80b9o$9b9o80b7o$8b10o81b5o$7b12o$7b5o2b6o$6b5o4b5o$6b5o4b5o$6b5o4b5o$
7b5o5b3o$7b5o4b3o$8b4o2b4o69b5o$9b8o17b2o50b7o$11b5o16b6o46b10o$31b8o
44b8ob3o$29b11o42b9o2b3o$28b13o41b6o5b3o$27b7o2bob3o20b3o18b5o7b2o$27b
6o6b3o17b7o16b5o7b2o$27b6o6b3o15b10o15b6o3b5o$27b6o6b3o14b12o14b13o$
27b7o4b3o14b8o2b3o15b12o$28b9ob3o14b6o5b3o16b9o$29b11o15b6o5b3o17b7o$
30b9o16b6o5b3o19b3o$32b6o17b6o5b3o$34b2o19b13o$56b12o$57b10o$59b7o$60b
4o!

It also works with some bugs removed:

Code: Select all

x = 113, y = 25, rule = R7,C0,M1,S65..114,B65..95,NM
104b4o$102b8o$101b10o$100b4o4b4o$100b3o5b4o$99b4o5b5o$99b5o4b5o$99b5o
4b5o$100b13o$100b12o$101b10o$4b6o91b10o$3b8o91b8o$2b10o91b6o$2b10o$b
12o$13o$5o4b5o$5o4b5o$5o5b4o$b4o5b3o$b4o4b4o$2b10o$3b8o$5b4o!
#C [[ ZOOM 4 ]]

There are also reflectorless rotating oscillators in other rules which behave in a similar fashion, for example these:

Code: Select all

x = 13, y = 13, rule = B2ce3a4ajnr5ry6ci/S2aik3ejnr4ajr5i7e8
11bo$obo7bo$b2o7b2o8$b2o7b2o$2bo7bobo$bo!

Code: Select all

x = 14, y = 14, rule = B2c3aenq4aijryz5cikqr6ac8/S1e2cik3ejqry4anrwz5a6k
b2o8b2o$4o8b2o$o9bob2o$2b2o6bobo7$bobo6b2o$2obo9bo$2o8b4o$b2o8b2o!

A hexagonal RRO:

Code: Select all

x = 48, y = 9, rule = B2o3o4m5/S2H
bo19bo19bo$3o17b3o17b3o3$46bo$41bo4bo$41b2o3b2o$24b3o16bo$25bo!

All of these patterns are counted as oscillators (rather than spaceships); even though they appear to move, they return back to the starting place.

[jeremyrutman]

Fantastic! The compact rule specification and inline viewer are really great. If I understand right there is no known standard GOL (which again iiuc is specified by R1,C2,S2-3,B3) that shows circular/spiral motion but only those variants you showed - so what is it about the different rules, that allow circular motion - and is there a way to predict from a rule, whether circular motion can occur? Maybe a discrete version of the rotor aka curl operator of vector calculus can be recruited.
Anyway thanks for this response , very interesting

[dvgrn]

Fantastic! The compact rule specification and inline viewer are really great. If I understand right there is no known standard GOL [pattern] ... that shows circular/spiral motion but only those variants you showed ...

That's sort of true and sort of not true. Standard Conway's Life rules are usually notated in outer-totalistic notation -- "B3/S23" -- rather than in Larger than Life notation, since CGoL doesn't actually need a range larger than R1.

Check the second paragraph of the reflectorless rotating oscillator link for an engineered example of an RRO in Conway's Life. It's not exactly "circular motion", though, it's definitely diamond-shaped.

It would be fairly easy to build an engineered RRO in Life that was octagonal, and it's technically possible to approximate a circle as closely as you might want, but the engineering would get really annoying once you try to improve on an octagon.

[azulavoir]

So what is it about the different rules, that allow circular motion - and is there a way to predict from a rule, whether circular motion can occur?

Generally, I think: The higher the neighborhood range of a given CA rule, the more likely it will be to have the capacity for a relatively simple pattern to move in a circle. Since the neighborhood of a range-1 rule only contains 8 cells, it ends up far more likely for small patterns to travel in one of eight directions. Rules can be constructed to make a given small pattern move in just about any direction - see the Smallest Spaceships Supporting Specific Speeds project for more on that - but a grid can never really make perfect circles.

If you pop over to this thread you can find a (very large) pattern in basic Life that develops in a (square) spiral, so it's not impossible, just needs some manual construction most of the time.

[confocaloid]

Another example of approximately circular movement (in a rule on the hexagonal grid) is in the quote below.
Here is the "really neat pattern" p594, with the rule definition translated into a weighted neighbourhood rulestring syntax which works in Golly/LifeViewer:

Code: Select all

x = 158, y = 19, rule = R1,C0,S2-3,6-9,11-15,B5,10-11,14-15,NW410104041H
151bo2$149bobobobo$152b2o$70bobobo76b5o$obo68b2ob2o77b2o$2o70b5o74bobo
bobo$3o65bo$obob2o63b2o4b2o68b3o7bo$ob3o68bo70b3o$3bo69bobo70b3o$2bo
67bo4bo70b2o$70b2o76bo$69bo77b4o$146b6o$145b7o$149bob4o$149bobo2bo$
150bo!

Lets kick things off with HexInverseFire, a rule by Ben Schaeffer. HexInverseFire is a hexagonal rule, in which the NE and SW corners are weighted to zero.

NW4,NN1,NE0,WW1,ME0,EE4,SW0,SS4,SE1,HI0,
RS2,RS3,RS6,RS7,RS8,RS9,RS11,RS12,RS13,RS14,RS15,
RB5,RB10,RB11,RB14,RB15

A good introduction to HexInverseFire is to simply fill in a large rectangle and watch what happens.

But beyond that, here's a really neat pattern:

Code: Select all

x = 158, y = 19, rule = HexInverseFire
151bo2$149bobobobo$152b2o$70bobobo76b5o$obo68b2ob2o77b2o$2o70b5o74bobo
bobo$3o65bo$obob2o63b2o4b2o68b3o7bo$ob3o68bo70b3o$3bo69bobo70b3o$2bo
67bo4bo70b2o$70b2o76bo$69bo77b4o$146b6o$145b7o$149bob4o$149bobo2bo$
150bo!

It would be accurate to describe it as a p594 oscillator, but it seems more descriptive to describe it as a spaceship that travels in a circle. I'm including three of them -- one traveling clockwise, one traveling counter-clockwise, and one next to a p2 oscillator. If you run this pattern at just the right speed, it looks like a spaceship orbiting a star.
[...]

[alexm3945]

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