Oscillators on Rubik's Cube

[rutabaga]

This might be a dumb idea but I was thinking about the Rubik's cube and realized that because every sequence of turns eventually repeats itself, it's like an Oscillator from CA. For instance, a single clockwise turn of the top face (denoted as U) would have a period of 4 because that's how long it takes to repeat itself. An example of a still life would be F' F (that means a clockwise front turn followed by a counterclockwise front turn) because it ends up where it started after just one instance of that algorithm. As for the highest period oscillator, I have no clue, but it's at least 64. As for the smallest impossible period, I have no clue either. Thanks for coming to my TED-Talk!

[confocaloid]

As for the highest period oscillator, I have no clue, but it's at least 64.

Should be at least 105, which is the order of (U R). Doing it seven times restores the edges; doing it 15 times restores the corners; LCM(7,15) = 105.
Since only two faces are used, probably this can be combined with an order-2 change in the unaffected region to get LCM(105, 2) = 210.

As for the smallest impossible period, I have no clue either.

Orders 1, 2, 4 possible by a single face turn. Order 6 possible by (U R' U' R). Orders 3, 5, 7, 15, 21, 35 possible because those divide 105 = the order of (U R). Orders 9 and 63 are possible, because the order of (U' R) is 63.
10, 14, 30, 42, 70 are likely possible, because 210 is likely possible.
Are 8, 11, 12, 13, 16, 17, 18, 19 possible?

[H. H. P. M. P. Cole]

Are 8, 11, 12, 13, 16, 17, 18, 19 possible?

This problem is isomorphic to finding all the cyclic subgroups of the Rubik's cube group. Does this site help? http://mzrg.com/rubik/orders.shtml

Further reading about 4-dimensional Rubiks' Cubes:
https://superliminal.com/cube/2x2x2x2/
https://superliminal.com/cube/

(I think this thread is off-topic)

[confocaloid]

[...] finding all the cyclic subgroups of the Rubik's cube group [...]

An interesting detail that might be lost here, is that the group is given by a set of generators (face turns). It seems desirable to find examples that are short, and preferably using only two or three different faces.
The same questions could be asked for subgroups. What is the highest period possible using only three faces U, R, F? What is the lowest impossible period using the same faces?

(I think this thread is off-topic)

It's in Sandbox, and seems more interesting than many other Sandbox threads.

To make it more CA-related, the title of the thread (and the questions asked) can be easily reinterpreted. There are 54 stickers on the Rubik's cube (9 on each of six faces). Define a two-state outer-totalistic cellular automaton with a range-1 neighbourhood on these 54 stickers. (There can be from 7 to 8 neighbours, depending on the location.) What is the highest possible period? What is the lowest impossible period?

For people who are into higher dimensions, the title of the thread and the questions asked can be reinterpreted again. Consider the graph of all possible configurations, with two configurations adjacent iff there's a single face turn transforming one into another. Run a two-state outer-totalistic cellular automaton with range-1 neighbourhood on that graph. (There are always 18 neighbours.) What is the highest possible period? What is the lowest impossible period?

edit:

Are 8, 11, 12, 13, 16, 17, 18, 19 possible?

This problem is isomorphic to finding all the cyclic subgroups of the Rubik's cube group. Does this site help? http://mzrg.com/rubik/orders.shtml

Looks like that page allows rotations of the whole cube in space, in addition to face turns. Consequently, some examples that are given there don't work with restriction "face turns only". However, according to the page, R U F has order 80, which should provide (long) solutions for periods 5, 8, 10, 16, 20, 40. Likewise, according to the page, F U R has order 84 and R L U has order 90, which should solve periods 12, 14, 18, 28, 42, 45.

[confocaloid]

[...] Looks like that page allows rotations of the whole cube in space, in addition to face turns. [...]

When restricted to face turns (without whole cube rotations), the following sequences should work. (incomplete, likely can be reduced)

• Order 2: R2
• Order 3: U2 R2 U2 R2
• Order 4: R
• Order 5: U R' U R
• Order 6: U2 R2
• Order 7: U' R U F
• Order 8: R D2 L2 U2
• Order 9: F2 U' R'
• Order 10: U' R U F2
• Order 11: F D' U L2 R F D' U L2 R
• Order 12: F U2 R2
• Order 15: U2 R U2 R
• Order 18: R' U' R U F
• Order 20: D2 R' U R
• Order 21: F R2 U R2
• Order 22: F D' U L2 R
• Order 24: R L2 D2 U2
• Order 30: U2 R
• Order 33: F' U L R'
• Order 35: D' R2 U R2
• Order 36: U R' U R U2 R2
• Order 44: F' U L' R
• Order 55: D' B' F2 U' L2 R
• Order 60: F' R' U
• Order 63: U' R
• Order 77: F' L U R
• Order 80: F U R
• Order 84: R U F
• Order 90: U L R
• Order 105: U R
• Order 168: U L2 R
• Order 180: U L' R
• Order 210: L' D R' U
• Order 315: F L B R
• Order 360: R' U F
• Order 420: L' D R U
• Order 495: U2 F' U L2 R2

[squareroot12621]

I think these will fill some of the gaps:
14: R U F R U F R U F R U F R U F R U F (R U F ×6)
16: F U R F U R F U R F U R F U R (F U R ×5)
28: R U F R U F R U F (R U F ×3)
40: F U R F U R (F U R ×2)
42: R U F R U F (R U F ×2)
45: U L R U L R (U L R ×2)
56: U L2 R U L2 R U L2 R (U L2 R ×3)
70: L' D R' U L' D R' U L' D R' U (L' D R' U ×3)
72: R' U F R' U F R' U F R' U F R' U F (R' U F ×5)
99: U2 F' U L2 R2 U2 F' U L2 R2 U2 F' U L2 R2 U2 F' U L2 R2 U2 F' U L2 R2 (U2 F' U L2 R2 ×5)
120: R' U F R' U F R' U F (R' U F ×3)
140: L' D R U L' D R U L' D R U (L' D R U ×3)
165: U2 F' U L2 R2 U2 F' U L2 R2 U2 F' U L2 R2 (U2 F' U L2 R2 ×3)

[tommyaweosme]

MISSING PERIODS (1-57 listed):
PRIME (level hard):
13,17,19,23,29,31,37,41,43,47,53
COPRIME (level medium):
25,26,27,34,38,39,46,49,51,57
COMPOSITE (level easy):
32,48,50,52,54

PERIODS:
2: R2
3: U2 R2 U2 R2
4: R
5: U R' U R
6: U2 R2
7: U' R U F
8: R D2 L2 U2
9: F2 U' R'
10: U' R U F2
11: F D' U L2 R F D' U L2 R
12: F U2 R2
14: R U F R U F R U F R U F R U F R U F (R U F ×6)
15: U2 R U2 R
16: F U R F U R F U R F U R F U R (F U R ×5)
18: R' U' R U F
20: D2 R' U R
21: F R2 U R2
22: F D' U L2 R
24: R L2 D2 U2
28: R U F R U F R U F (R U F ×3)
30: U2 R
33: F' U L R'
35: D' R2 U R2
36: U R' U R U2 R2
40: F U R F U R (F U R ×2)
42: R U F R U F (R U F ×2)
44: F' U L' R
45: U L R U L R (U L R ×2)
55: D' B' F2 U' L2 R
56: U L2 R U L2 R U L2 R (U L2 R ×3)
60: F' R' U
63: U' R
70: L' D R' U L' D R' U L' D R' U (L' D R' U ×3)
72: R' U F R' U F R' U F R' U F R' U F (R' U F ×5)
77: F' L U R
80: F U R
84: R U F
90: U L R
99: U2 F' U L2 R2 U2 F' U L2 R2 U2 F' U L2 R2 U2 F' U L2 R2 U2 F' U L2 R2 (U2 F' U L2 R2 ×5)
105: U R
120: R' U F R' U F R' U F (R' U F ×3)
140: L' D R U L' D R U L' D R U (L' D R U ×3)
165: U2 F' U L2 R2 U2 F' U L2 R2 U2 F' U L2 R2 (U2 F' U L2 R2 ×3)
168: U L2 R
180: U L' R
210: L' D R' U
315: F L B R
360: R' U F
420: L' D R U
495: U2 F' U L2 R2