11-1100/MATLAB Code for Non-Commutative Gaussian Elimination - Revision history

Tholden: MATLAB Code for Non-Commutative Gaussian Elimination moved to 11-1100/MATLAB Code for Non-Commutative Gaussian Elimination

2011-10-10T22:36:05Z

MATLAB Code for Non-Commutative Gaussian Elimination moved to 11-1100/MATLAB Code for Non-Commutative Gaussian Elimination

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Revision as of 18:36, 10 October 2011

Jmracek at 22:17, 9 October 2011

2011-10-09T22:17:33Z

← Older revision		Revision as of 18:17, 9 October 2011
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	[[Image:output.jpg]]		[[Image:output.jpg]]
	[[Image:pivot.jpg]]		[[Image:pivot.jpg]]

	The above plot is displaying the non-zero entries in the NCGE table. I calculate the order of the Pocket Cube group to be 3,674,160. I also ran the program using a restricted subset of the generators. Interestingly, the entire pocket cube group can be generated using only 3 of the face rotations!		The above plot is displaying the non-zero entries in the NCGE table. I calculate the order of the Pocket Cube group to be 3,674,160. I also ran the program using a restricted subset of the generators. Interestingly, the entire pocket cube group can be generated using only 3 of the face rotations!

Jmracek at 22:15, 9 October 2011

2011-10-09T22:15:37Z

← Older revision		Revision as of 18:15, 9 October 2011
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	At the MATLAB prompt, the following set of commands were used to run my program:		At the MATLAB prompt, the following set of commands were used to run my program:

			[[Image:output.jpg]]
	>> [Order, NCGE_Table] = PocketCube();
	>> Order
	Order =
	88179840
	>> Order/24
	ans =
	3674160
	>> spy(NCGE_Table)

	[[Image:pivot.jpg]]		[[Image:pivot.jpg]]
			The above plot is displaying the non-zero entries in the NCGE table. I calculate the order of the Pocket Cube group to be 3,674,160. I also ran the program using a restricted subset of the generators. Interestingly, the entire pocket cube group can be generated using only 3 of the face rotations!

Jmracek at 22:07, 9 October 2011

2011-10-09T22:07:50Z

← Older revision		Revision as of 18:07, 9 October 2011
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	3674160		3674160
	>> spy(NCGE_Table)		>> spy(NCGE_Table)

			[[Image:pivot.jpg]]

Jmracek at 21:59, 9 October 2011

2011-10-09T21:59:04Z

← Older revision		Revision as of 17:59, 9 October 2011
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	[[Image:Feed.jpg]]		[[Image:Feed.jpg]]
	[[Image:PocketCube.jpg]]		[[Image:PocketCube.jpg]]

			At the MATLAB prompt, the following set of commands were used to run my program:

			>> [Order, NCGE_Table] = PocketCube();
			>> Order
			Order =
			88179840
			>> Order/24
			ans =
			3674160
			>> spy(NCGE_Table)

Jmracek at 21:54, 9 October 2011

2011-10-09T21:54:05Z

← Older revision		Revision as of 17:54, 9 October 2011
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	First, I needed to define the problem for the computer. I picked an enumeration of the faces of the 2x2x2 cube, and wrote out the generators of the group based on this enumeration. The rotations of the faces are the generators for the group operations. Since the cube has six faces which can be rotated, there are six generators. The group I am dealing with is a subgroup of the permutations on 24 elements because a 2x2x2 cube has 6*4 = 24 faces. The following figure shows how I chose my enumeration of the cube faces:		First, I needed to define the problem for the computer. I picked an enumeration of the faces of the 2x2x2 cube, and wrote out the generators of the group based on this enumeration. The rotations of the faces are the generators for the group operations. Since the cube has six faces which can be rotated, there are six generators. The group I am dealing with is a subgroup of the permutations on 24 elements because a 2x2x2 cube has 6*4 = 24 faces. The following figure shows how I chose my enumeration of the cube faces:

	[[Image:cubelayout.jpg]]		[[Image:cubelayout.jpg]]
			(A) This sub-figure demonstrates the enumeration of the faces that I adopted. (B) An image of the Pocket Cube. (C) A list of the group generators (face rotations).

	Note that since the pocket cube has no faces which would fix its overall orientation, there are actually 24 equivalent cube configurations per permutation; thus, whatever answer is obtained at the end needs to be divided by 24 to give the actual group order.		Note that since the pocket cube has no faces which would fix its overall orientation, there are actually 24 equivalent cube configurations per permutation; thus, whatever answer is obtained at the end needs to be divided by 24 to give the actual group order.

Jmracek at 21:52, 9 October 2011

2011-10-09T21:52:04Z

← Older revision		Revision as of 17:52, 9 October 2011
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	From a computational standpoint this is kind of a bad idea, but I figured it was something different and no one else was likely to try this solution.		From a computational standpoint this is kind of a bad idea, but I figured it was something different and no one else was likely to try this solution.

			First, I needed to define the problem for the computer. I picked an enumeration of the faces of the 2x2x2 cube, and wrote out the generators of the group based on this enumeration. The rotations of the faces are the generators for the group operations. Since the cube has six faces which can be rotated, there are six generators. The group I am dealing with is a subgroup of the permutations on 24 elements because a 2x2x2 cube has 6*4 = 24 faces. The following figure shows how I chose my enumeration of the cube faces:
			[[Image:cubelayout.jpg]]
			Note that since the pocket cube has no faces which would fix its overall orientation, there are actually 24 equivalent cube configurations per permutation; thus, whatever answer is obtained at the end needs to be divided by 24 to give the actual group order.

	I split the program into 3 parts. BG2.m is a short script which builds the matrix representations of the generators. Feed.m is a script which carries out the non-commutative Gaussian elimination algorithm. Finally, PocketCube.m is a parent script in which the generators of the pocket cube problem are defined, the Feed algorithm is called for each generator, and once the lookup table is finally constructed, the order of the group is output.		I split the program into 3 parts. BG2.m is a short script which builds the matrix representations of the generators. Feed.m is a script which carries out the non-commutative Gaussian elimination algorithm. Finally, PocketCube.m is a parent script in which the generators of the pocket cube problem are defined, the Feed algorithm is called for each generator, and once the lookup table is finally constructed, the order of the group is output.

Jmracek at 21:12, 9 October 2011

2011-10-09T21:12:28Z

← Older revision		Revision as of 17:12, 9 October 2011
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	[[Image:Matrix.jpg]]		[[Image:Matrix.jpg]]

	From a computational standpoint this is kind of a bad idea, but I figured it was something different and no one else was likely to try this.		From a computational standpoint this is kind of a bad idea, but I figured it was something different and no one else was likely to try this solution.

	I split the program into 3 parts. BG2.m is a short script which builds the matrix representations of the generators. Feed.m is a script which carries out the non-commutative Gaussian elimination algorithm. Finally, PocketCube.m is a parent script in which the generators of the pocket cube problem are defined, the Feed algorithm is called for each generator, and once the lookup table is finally constructed, the order of the group is output.		I split the program into 3 parts. BG2.m is a short script which builds the matrix representations of the generators. Feed.m is a script which carries out the non-commutative Gaussian elimination algorithm. Finally, PocketCube.m is a parent script in which the generators of the pocket cube problem are defined, the Feed algorithm is called for each generator, and once the lookup table is finally constructed, the order of the group is output.

Jmracek at 21:11, 9 October 2011

2011-10-09T21:11:55Z

← Older revision		Revision as of 17:11, 9 October 2011
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	[[Image:BG2.jpg]]		[[Image:BG2.jpg]]
	[[Image:Feed.jpg]]		[[Image:Feed.jpg]]
	[[Image:PocketCube.jpg]]~~</math>~~		[[Image:PocketCube.jpg]]

Jmracek at 21:11, 9 October 2011

2011-10-09T21:11:08Z

← Older revision		Revision as of 17:11, 9 October 2011
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	From a computational standpoint this is kind of a bad idea, but I figured it was something different and no one else was likely to try this.		From a computational standpoint this is kind of a bad idea, but I figured it was something different and no one else was likely to try this.

			I split the program into 3 parts. BG2.m is a short script which builds the matrix representations of the generators. Feed.m is a script which carries out the non-commutative Gaussian elimination algorithm. Finally, PocketCube.m is a parent script in which the generators of the pocket cube problem are defined, the Feed algorithm is called for each generator, and once the lookup table is finally constructed, the order of the group is output.
	I split the program into 3 parts: BG2.m is a short script which

	[[Image:BG2.jpg]]		[[Image:BG2.jpg]]