Non-isomorphic cubes

[guzman]

How many non isomorphic cube stickerings do you get
if you're allowed to sticker a 2x2x2 cube in any way you want ?

Just to be really clear:
Definition of isomorphic stickerings:
Two cubes A and B (both nxnxn) are said to be isomorphic
if there exists a bijection f
between the cubies of A and the cubies of B
such that, for any cubies a and a' of A we have:
1) the type of f(a) is the same type of a
(for instance, f maps corners to corners and edges to edges)
2) f(a) and f(a') are indistinguishable (that is they have the same stickerings) if and only if, a and a' are indistinguishable.
3) a and f(a) have the same number of orientations.

The question (in the 2x2x2 case) is not difficult to answer but I think some of you may enjoy solving the problem.

More tedious is the case of 3x3x3 non-isomorphic stickerings.
I didn't solve it precisely. My rough estimate is that there are more or less 38.000.000 non-isomorphic stickerings of the 3x3x3 cube.

[We could also think about modifying the definition in order to define when f is a stickering omomorphism and says therefore when A is a sort of subcube of B ... I dont know if this leads somewhere but it seems to me quite a natural topic ... ]

PS: Observe that with this defintion it is as if the cubes are considered disassembled; in other words the way the cubes are assembled doesn't play any role.

Edit: I substituted "cubes" with "cube stickrings" which is more clear

 

[theace]

dude... O_o

 

[Rinfiyks]

I don't understand this at all :confused: I'll list what I don't understand.

You say "how many non-isomorphic cubes" as if "isomorphic" is a property of a certain scramble. For example: "this scramble makes an isomorphic cube."
But in the definition of isomorphic (which I don't fully understand), it looks like isomorphic is not a property of a cube, but two cubes (e.g., cubes A and B are isomorphic to each other).
What is a bijection?
What is the "type", when you say f(a) and a?

I might understand with examples. Here's some scrambles, are cubes A and B isomorphic?

Spoiler

1.
A: U
B: U'

2.
A: U
B: F

3.
A: U R
B: U' R'

4.
A: U R
B: R U

5.
A: U R
B: U R'

Are 1,2,3,4 isomorphic?

 

[irontwig]

Yeah, A and B are isomorphic in example 1-4, but not 5.

 

[Rinfiyks]

So when guz says "How many non isomorphic cubes do you get", non-isomorphic to what? Any of the other cubes in the non-isomorphic cube set?

On a side note, where do you guys learn all of this crazy group theory and maths? I never understand half of it :/

 

[LewisJ]

So when guz says "How many non isomorphic cubes do you get", non-isomorphic to what? Any of the other cubes in the non-isomorphic cube set?

On a side note, where do you guys learn all of this crazy group theory and maths? I never understand half of it :/

Yes; in other words, how many uniquely distinguishable stickerings of the 2x2 exist?

I guess it's kind of cheating to say I learned about this stuff from Mathcamp. Really though, I probably learned enough to understand this post just from reading wikipedia.

 

[guzman]

Sorry guys,
my english is not that fluent so I explained the concept in math terms to avoid ambiguities in english, but apparently it didn't work.
I'm however convinced that the math definition is clear to those used to read math texts.

Anyway, I'll try to be more clear and avoid math technicalities.

First of all, isomorphic applies to 2 cubes
(not to scrambles).
So, consider two 3x3x3 cubes stickered in any way you want.

Now consider the cubies of the first cube.
You can subdivide them in sets of undistinguishable cubies
(for example, there might be 4 undistinguishable corners in one set,
3 undistinguishable corners in another set, and 1 corner alone different from all the rest, ... then there are sets of undistinguishable edges, ... ).
Our first request is that the sets you obtain in the first cube (according to its stickering) are the same sets (in quantity and number of elements) you obtain in the second cube (according to its stickering).

Then consider the orientations of the cubies.
The cubies in a set S of undistinguishable cubies of the first cube
have all the same number of orientations (since they are undistinguishable) ...
our request is that the cubies in the corresponding set of the second cube have the same number of orientations of the cubies in S.

I dont know if this clarifies the definition.
I'm writing one more post with an example ...

ps: by type of a cubie a mean: edge, corner, center, ...

 

[guzman]

Consider a 3x3x3 cube that has
4 undistinguishable corners (according to stickers) with 1 orientation each,
3 undistinguishable corners with 3 orientations each,
1 corner different from all other corners and with 3 orientations,

4 undistinguishable edges with 1 orientation,
4 undistinguishable edges (different from the ones above) with 1 orientation,
4 undistinguishable edges with 2 orientations each,

6 undistinguishable centers with 1 orientation each,

well,
such a cube is "isomorphic" to any other 3x3x3 cube stickered
in such a way to have the same groups of cubies.

The standard rubik cube has
all different corners, all with 3 orientations,
all different edges, all with 2 orientations,
all different centers, all with 1 orientation.

Scambles, moves, and ways to assemble the cube don't play any role so far.

My question is:
in how many different ways can you sticker a 2x2x2 cube,

(the math stuff only served the porpouse of defining what different means).

hope this helps a bit ...

Edit:

Yes; in other words, how many uniquely distinguishable stickerings of the 2x2 exist?

yes, exactly,
I've put the definition of "uniquely distinguishable stickerings" in my post so as to avoid ambiguities but the language I used created even more ...

 

[Rinfiyks]

I understand what you mean now. When you can sticker it in any way you want, I could put 3 of the same stickers on one corner?

 

[guzman]

I understand what you mean now. When you can sticker it in any way you want, I could put 3 of the same stickers on one corner?

Yes, you can do anything you want.

 

[guzman]

solution 2x2x2 case

The problem of counting the number of 2x2x2 non-isomorphic stikcerings is strictly related to the partitons of number 8.
We have 8 corners and we have first of all to divide them in sets of undistinguishable cubies.
There are various ways to that ....


the numbers indicate the number of elements in each set of undistinguishable corners:

Spoiler

8

7 1

6 2
6 1 1


5 3
5 2 1
5 1 1 1


4 4
4 3 1
4 2 2
4 2 1 1
4 1 1 1 1


3 3 2
3 3 1 1
3 2 2 1
3 2 1 1 1
3 1 1 1 1 1


2 2 2 2
2 2 2 1 1
2 2 1 1 1 1
2 1 1 1 1 1 1


1 1 1 1 1 1 1 1

Once you have decided the way to partition the corners
you have to decide the orientations to give to each set ...

in the first case of the list above (all 8 corner undistinguishable)
you have 2 choices, they all have 1 orientation
or they all have 3 orientations,

in the case of two sets, each with 4 undistinguishable corners
you have 3 choices: both sets contain corners with 1 orientation
(which I denote by 1,1), both sets contain corners with 3 orientations
(which I denote by 3,3), or one set contains non-orientable corners and the other set contains orientable corners (which I denote by 1,3).
Observe that the case 3,1 is just the same case since both sets contain 4 elements and play therefore the same role.

So, here are all the cases, the second column indicates the number of ways you can assign orientations to the specified partition:

Spoiler

Number of equal cubies
-> cases considering orientations

8 -> 2 = 2

7 1 -> 2*2 = 4

6 2 -> 2*2 = 4
6 1 1 -> 2*3 = 6

5 3 -> 2*2 = 4
5 2 1 -> 2*2*2 = 8
5 1 1 1 -> 2*4 = 8

4 4 -> 3 = 3
4 3 1 -> 2*2*2 = 8
4 2 2 -> 2*3 = 6
4 2 1 1 -> 2*2*3 = 12
4 1 1 1 1 -> 2*5 = 10

3 3 2 -> 3*2 = 6
3 3 1 1 -> 3*3 = 9
3 2 2 1 -> 2*3*2 = 12
3 2 1 1 1 -> 2*2*4 = 16
3 1 1 1 1 1 -> 2*6 = 12

2 2 2 2 -> 5 = 5
2 2 2 1 1 -> 4*3 = 12
2 2 1 1 1 1 -> 3*5 = 15
2 1 1 1 1 1 1 -> 2*7 = 14

1 1 1 1 1 1 1 1 -> 9 = 9

185

this gives a total of 185 non-isomorphic stickerings for the 2x2x2 cube.

The case of the 3x3x3 cube is a bit more tedious
(I have the list if anyone is interested) and we have:

185 stickerings for corners,
1165 stickerings for edges,
224 stickerings for centers (if you are allowed to get centers caps off and swap them)

this gives 48277600
non-isomorphic stickerings for the 3x3x3 cube.

Edit:
PS: the 180+ cubes you find here have all non-isomorphic stickerings.

One more question is: in how many ways can you assemble a cube with a given stikering. If we are talking about a 3x3x3 cube in most of the cases you can assemble it in 12 ways (if centeres are fixed) but there are other stikerings that dont give to the cube 12 ways to assemble it. For instance the completely white cube can be assembled in just 1 way.

Moreover: there are a few cases of non-isomorphic stickerings that give rise to isomorphic cubes. For instance the totally white cube is isomorphic to a cube with white cubes except for one completely red cube. None of these cubes can be scrambled.

guzman.