# Mathematics in cubing



## shadowslice e (Nov 23, 2015)

I'm asking here because the people who frequent this sub-forum seem to be the sort of people who would know the answer to the question which is;

What mathematical concepts would be useful when looking at the behaviours of the cube?

So far I only think of the obvious such as group theory, matricies and fractal geometry (perhaps also n-dimensional geometry).

Are there any other fields which may be useful and are there any good resources (I don't mind having to look up more basic stuff first) which are worth a read?

Thanks,

SSE


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## Lucas Garron (Nov 23, 2015)

Combinatorics (some of it related to group theory)
 Algebra (but mostly group theory)
 Algorithms (esp. search algorithms)

But mostly just lots of group theory.

Where does fractal geometry fit in?


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## JustinTimeCuber (Nov 23, 2015)

this gave me an idea that probably wouldn't do very much for cubing but I thought it would be interesting to play around with:
3 dimensional matrices (edit: woah this actually exists)
4 dimensional matrices *mind blown*


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## cuBerBruce (Nov 24, 2015)

Permutation theory (I think this is basically part of combinatorics, mentioned by Lucas)
 Graph theory (closely connected with group theory and combinatorics)

I'll also mention that within group theory, there are lots of topics that would seem to have no direct connection to understanding the cube. But of course, several basic group theory concepts are highly connected - subgroups, cosets, commutators and conjugation, and homomorphisms/isomorphisms/automorphisms to name some.


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## shadowslice e (Nov 24, 2015)

Lucas Garron said:


> Where does fractal geometry fit in?



I included it because I was just thinking that if you had a specific set of generators (probably not the R/R' etc) and ploted each cube in N-dimensional space you might find fractal patterns when looking at the cubes using the standard generators.

Possible maybe but extremely difficult to work out.


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## bobthegiraffemonkey (Nov 24, 2015)

Graph theory (as previously mentioned by Bruce) is particularly useful for the sq-1 since you can't describe all the states as a group, but you can look at all the shapes as a graph. I've been doing some work with this to find some interesting trivia, but I've been too busy to work on it recently. I'm hoping to get some time over Christmas to finish up and post.

Anyway, some interesting stuff in graph theory, though I'm usually dealing with the related area of network theory.


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## Cubo largo (Nov 25, 2015)

bobthegiraffemonkey said:


> Graph theory (as previously mentioned by Bruce) is particularly useful for the sq-1 since you can't describe all the states as a group, but you can look at all the shapes as a graph. I've been doing some work with this to find some interesting trivia, but I've been too busy to work on it recently. I'm hoping to get some time over Christmas to finish up and post.
> 
> Anyway, some interesting stuff in graph theory, though I'm usually dealing with the related area of network theory.


Must be extremely interesting. Please keep us informed! Would you mind quoting this message in the case or sending me a PM? 

Sent from my tostapane using Tapatalk


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## shadowslice e (Nov 30, 2015)

Just as a sort of reading list, does anyone have any recommendations for group theory and graph theory?


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## shadowslice e (Dec 3, 2015)

Is it possible to derive a series of matrices which have specific stationary points without having to work out each matrix?

That is in terms of a 2x2x2 cube with coordinates representing the corners {-1<x<1} where each of the components are part of the set Z (fancy font)?

Or at least would all the matrices have common immediately obvious properties other than the same stationary point?

As well as this, is it worth defining specific groups or sets from their stationary points? If it is could you derive methods in order to work out the other generators for a case in the group?

I'm just asking because I was just wondering if this was at all a fruitful track to go down at a glance. Perhaps this could also be linked to graph theory where a specific set of stationary points have a specific "shape". Or maybe if plotted in N dimensions a cube would pass through various fractal shaped patterns which denote specific stationary points.


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## JustinTimeCuber (Dec 3, 2015)

someone should make some cool math notation specifically for cubes, something like this:
Rubiks[000000000,111111111,222222222,333333333,444444444,555555555] = solved cube
Rubiks[0000,1111,2222,3333,4444,5555,6666] = solved 2x2
+ = apply move
' = inverted position (basically obtained by solving cube and then applying the solution again, the solution to that would be the inverse of the origional

I have no idea what makes the most sense, these notations would probably not work very well.


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## shadowslice e (Dec 3, 2015)

JustinTimeCuber said:


> someone should make some cool math notation specifically for cubes, something like this:
> Rubiks[000000000,111111111,222222222,333333333,444444444,555555555] = solved cube
> Rubiks[0000,1111,2222,3333,4444,5555,6666] = solved 2x2
> + = apply move
> ...


This is almost what in asking about. I just think that using matrices would be clearer.


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## Lucas Garron (Dec 3, 2015)

JustinTimeCuber said:


> someone should make some cool math notation specifically for cubes



How's this permutation-based generator definition?



shadowslice e said:


> As well as this, is it worth defining specific groups or sets from their stationary points?



As far as I know, this would only be useful as a convenience to generate the permutations (and maybe for graphics, but that's not really the point of this thread). I don't know of anything interesting you can learn about the cube this way, though.

However, you might want to look at group representation theory.


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## shadowslice e (Dec 3, 2015)

Lucas Garron said:


> As far as I know, this would only be useful as a convenience to generate the permutations (and maybe for graphics, but that's not really the point of this thread). I don't know of anything interesting you can learn about the cube this way, though.
> 
> However, you might want to look at group representation theory.



Ok thanks. I'll try to find some stuff on it.

The reason I thought that defining groups by stationary points is that you may be able to generated essentially optimal permutation solutions for corners by working out generators needed to solve by working with the stationary points if all the permutations with that have patterns with the moves needed to solve.

Is there any way to represent orientation as a matrix efficiently and the transformation created by the generators? That is at the same time as permutation?


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## shadowslice e (Dec 7, 2015)

Does the inverse of a matrix have the same stationary points as the matrix itself (I'm guessing yes but just to check)?


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## unsolved (Dec 21, 2015)

JustinTimeCuber said:


> someone should make some cool math notation specifically for cubes, something like this:
> Rubiks[000000000,111111111,222222222,333333333,444444444,555555555] = solved cube
> Rubiks[0000,1111,2222,3333,4444,5555,6666] ....



I've done something very similar with my 4x4x4 and 5x5x5 brute force solvers. In the case of the 4x4x4 cube, you have 64 bits represent each face of the cube, so 4 bits per square. Computers are very fast at 64-bit math. So if the top face is "1", then to see if it is solved you only need one test. Is it 00010001000100010001000100010001...


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