Difference between revisions of "Permutations Notation"
(Created page with "In order to study cubes larger than the 34, it will be useful to develop a better system for naming the pieces under consideration than simply referring to the number of facelets...") |
|||
| (One intermediate revision by the same user not shown) | |||
| Line 1: | Line 1: | ||
| − | + | <br> | |
| − | + | Following is a generalized notation system which assigns names to the pieces of every | |
| − | + | puzzle and most likely every future puzzle created by the members of this community in | |
| − | We shall denote by the term family a complete group of pieces that can occupy the same positions on a | + | a straightforward and precise manner. It will be used in the derivations of the |
| − | + | permutation counts found in [[Permutations_Index]]. | |
| − | 3-colored pieces and three families of 2-colored pieces. | + | <br> <br /> |
| − | + | We shall denote by the term family a complete group of pieces that can occupy the same | |
| − | + | positions on a puzzle throughout all possible configurations. For example, on a 5^4 | |
| − | located in. To make this more explicit, consider the 4-colored corner pieces | + | cube, there are two families of 3-colored pieces and three families of 2-colored pieces. |
| − | replaced by an actual tesseract, a corner would be a point, or dimension zero. So, we will refer to a | + | <br> <br /> |
| − | corner piece as a (0D) piece. 3-colored pieces will be called (1D) pieces, 2-colored pieces will be called | + | The basic idea is to classify families of pieces by the dimension of the section they |
| − | (2D) pieces, and 1-colored pieces will be called (3D) pieces. This classification of a piece by the | + | are located in, rather than by the number of facelets they have. To make this more |
| − | dimension of the section it resides in will be called the type of that piece. Also, the term region will refer | + | explicit, consider the 4-colored corner pieces of the 3^4 Rubik's cube. If the cube |
| − | to a section that contains connected pieces of a specific type. For example, on a | + | were replaced by an actual tesseract, a corner would be a point, or dimension zero. |
| − | a 3×1 section that holds three (1D) pieces. | + | So, we will refer to a corner piece as a (0D) piece. 3-colored pieces will be called |
| − | So far, this only appears to be a relabeling of the pieces based on their location on the | + | (1D) pieces, 2-colored pieces will be called (2D) pieces, and 1-colored pieces will be |
| − | redundant due to the fact that the number of facelets on a piece is itself determined by the location of | + | called (3D) pieces. This classification of a piece by the dimension of the section it |
| − | that piece on the cube. However, the key idea is this: For (nD) pieces, where n | + | resides in will be called the type of that piece. It is easy to see how this generalizes |
| − | down the location of the piece until we have n | + | to any other puzzle. For MagicTile, consider the order-5 Klein's Quartic. The 3-colored |
| − | a single face form a 3×3×3 cube. The pieces on the corners of that cube will be called (3D)(0D) pieces, | + | pieces are (0D) pieces, the 2-colored pieces are (1D) pieces, and the 1-colored pieces |
| − | the edges, (3D)(1D) pieces, and the faces, (3D)(2D) pieces, while the term (3D) pieces will continue to | + | are (2D) pieces, of which there are two families, not counting the fixed centers. Also, |
| − | denote the entire group of pieces. | + | the term region will refer to a section that contains connected pieces of a specific |
| − | + | type. For example, on a 5^4 cube, a (1D) region is a 3×1 section that holds three (1D) | |
| − | as a (3D) center, the term center always being reserved for a piece that lies at the center of the region it is | + | pieces. |
| − | located in. It should also be noted that on | + | <br> <br /> |
| − | center are broken down into either (3D)(0D), (3D)(1D), or (3D)( | + | So far, this only appears to be a relabeling of the pieces based on their location on |
| − | each piece within the face. For example, take a | + | the puzzle, which seems redundant due to the fact that the number of facelets on a piece |
| − | be broken down in the same manner as if they were the (3D) pieces in the | + | is itself determined by the location of that piece on the cube. However, the key idea |
| − | There is another observation to be made regarding the concept of regions. When considering regions for | + | is this: For (nD) pieces, where n >= 3, we continue to break down the location of the |
| − | a type of piece that is subclassified (i.e. (3D)(1D) and (3D)(2D) regions for 4-dimensional cubes), a | + | piece until we have n <= 2. For example, consider a 5^4 cube. Its (3D) pieces on a |
| − | group of pieces of such a type will be broken into separate regions according to the dimension of the | + | single face form a 3×3×3 cube. The pieces on the corners of that cube will be called |
| − | subclassification. To make this completely clear, consider a | + | (3D)(0D) pieces, the edges, (3D)(1D) pieces, and the faces, (3D)(2D) pieces, while the |
| − | which is the complete 5×5×5 section of (3D) pieces on that face, as expected. Also, that face has twelve | + | term (3D) pieces will continue to denote the entire group of pieces. The center piece |
| − | (3D)(2D) regions: Six of them are 3×3 sections of (3D)(2D) pieces (in the outer layer of the (3D) | + | will be referred to as a (3D) center, the term center always being reserved for a piece |
| − | pieces), and the other six are 1×1 sections of (3D)(2D) pieces (in the inner layer of the (3D) pieces). We | + | that lies at the center of the region it is located in. It should also be noted that on |
| − | count these regions as separate, so that each region is a 2-dimensional group of pieces, even though each | + | n^4 cubes for example, when n >= 6, all of the the (3D) pieces other than the center are |
| − | 3×3 region is connected to the piece of the corresponding 1×1 region. We can also see that there are | + | broken down into either (3D)(0D), (3D)(1D), or (3D)(2D) pieces regardless of the depth of |
| − | twelve (3D)(1D) regions, six of which are 3×1 sections of (3D)(1D) pieces, and the remaining six 1×1 | + | each piece within the face. For example, take a 7^4 cube. The (3D) pieces in the inner |
| − | sections of (3D)(1D) pieces. It should now be clear what is meant by a (3D)(2D) and (3D)(1D) center | + | 3×3×3 section will be broken down in the same manner as if they were the (3D) pieces in |
| − | on a cube of any size. | + | the 5^4 cube in the example above. |
| − | This notation system, or one which uses similar concepts, is essential when considering | + | <br> <br /> |
| − | d | + | There is another observation to be made regarding the concept of regions. When considering |
| − | (5D)(4D)(3D)(0D) pieces! | + | regions for a type of piece that is subclassified (i.e. (3D)(1D) and (3D)(2D) regions for |
| − | There is one more distinction to be made. We have already defined what we mean by a center piece. | + | 4-dimensional cubes, for example), a group of pieces of such a type will be broken into |
| − | The rest are either wings or normals. Wings are pieces that only occur on a (2D) or (3D)(2D) region. | + | separate regions according to the dimension of the subclassification. To make this |
| − | + | completely clear, consider a 7^4 cube. A single face has one (3D) region, which is the | |
| − | + | complete 5×5×5 section of (3D) pieces on that face, as expected. Also, that face has twelve | |
| − | neither wings nor centers will be called normals, except for | + | (3D)(2D) regions: Six of them are 3×3 sections of (3D)(2D) pieces (in the outer layer of the |
| − | be referred to as pieces without confusion. | + | (3D) pieces), and the other six are 1×1 sections of (3D)(2D) pieces (in the inner layer of |
| + | the (3D) pieces). We count these regions as separate, so that each region is a 2-dimensional | ||
| + | group of pieces, even though each 3×3 region is connected to the piece of the corresponding | ||
| + | 1×1 region. We can also see that there are twelve (3D)(1D) regions, six of which are 3×1 | ||
| + | sections of (3D)(1D) pieces, and the remaining six 1×1 sections of (3D)(1D) pieces. It | ||
| + | should now be clear what is meant by a (3D)(2D) and (3D)(1D) center on a cube of any size. | ||
| + | <br> <br /> | ||
| + | This notation system, or one which uses similar concepts, is essential when considering | ||
| + | n^d cubes when d >= 5. For example, if d = 6, we can have (5D)(4D)(2D) pieces, (4D)(1D) | ||
| + | pieces, or even (5D)(4D)(3D)(0D) pieces! | ||
| + | <br> <br /> | ||
| + | There is one more distinction to be made. We have already defined what we mean by a center | ||
| + | piece. The rest are either wings or normals. Wings are pieces that only occur on a ..(2D) | ||
| + | (e.g., (2D), or (3D)(2D), or (4D)(2D), or (4D)(3D)(2D), etc.) region. Consider that region | ||
| + | isolated from the rest of the puzzle, and can only rotate about its center in the plane it | ||
| + | resides in. A piece is considered a wing if it belongs to a family (as defined under this | ||
| + | restriction) to which there corresponds a separate family which are mirror images of each | ||
| + | other. For example, on a square n×n ..(2D) region, wings are the pieces that do not lie | ||
| + | on the main diagonals of the region or, when n is odd, additionally on the lines that divide | ||
| + | the region into four equal square quadrants. Pieces that are neither wings nor centers will | ||
| + | be called normals, except for ..(0D) pieces, which can simply be referred to as pieces without | ||
| + | confusion. It is important to distinguish between wings and normals because the permutations | ||
| + | and orientations of each always need to be calculated differently. | ||
| + | <br> <br /> | ||
| + | To conclude, here is a list of all possible types of pieces on a 4-dimensional cube: | ||
| + | <br> <br /> | ||
| + | (0D) pieces (corners) | ||
| + | <br> | ||
| + | (1D) centers | ||
| + | <br> | ||
| + | (1D) normals | ||
| + | <br> | ||
| + | (2D) centers | ||
| + | <br> | ||
| + | (2D) normals | ||
| + | <br> | ||
| + | (2D) wings | ||
| + | <br> | ||
| + | (3D) centers | ||
| + | <br> | ||
| + | (3D)(0D) pieces | ||
| + | <br> | ||
| + | (3D)(1D) centers | ||
| + | <br> | ||
| + | (3D)(1D) normals | ||
| + | <br> | ||
| + | (3D)(2D) centers | ||
| + | <br> | ||
| + | (3D)(2D) normals | ||
| + | <br> | ||
| + | (3D)(2D) wings | ||
Latest revision as of 05:10, 3 May 2011
Following is a generalized notation system which assigns names to the pieces of every
puzzle and most likely every future puzzle created by the members of this community in
a straightforward and precise manner. It will be used in the derivations of the
permutation counts found in Permutations_Index.
We shall denote by the term family a complete group of pieces that can occupy the same
positions on a puzzle throughout all possible configurations. For example, on a 5^4
cube, there are two families of 3-colored pieces and three families of 2-colored pieces.
The basic idea is to classify families of pieces by the dimension of the section they
are located in, rather than by the number of facelets they have. To make this more
explicit, consider the 4-colored corner pieces of the 3^4 Rubik's cube. If the cube
were replaced by an actual tesseract, a corner would be a point, or dimension zero.
So, we will refer to a corner piece as a (0D) piece. 3-colored pieces will be called
(1D) pieces, 2-colored pieces will be called (2D) pieces, and 1-colored pieces will be
called (3D) pieces. This classification of a piece by the dimension of the section it
resides in will be called the type of that piece. It is easy to see how this generalizes
to any other puzzle. For MagicTile, consider the order-5 Klein's Quartic. The 3-colored
pieces are (0D) pieces, the 2-colored pieces are (1D) pieces, and the 1-colored pieces
are (2D) pieces, of which there are two families, not counting the fixed centers. Also,
the term region will refer to a section that contains connected pieces of a specific
type. For example, on a 5^4 cube, a (1D) region is a 3×1 section that holds three (1D)
pieces.
So far, this only appears to be a relabeling of the pieces based on their location on
the puzzle, which seems redundant due to the fact that the number of facelets on a piece
is itself determined by the location of that piece on the cube. However, the key idea
is this: For (nD) pieces, where n >= 3, we continue to break down the location of the
piece until we have n <= 2. For example, consider a 5^4 cube. Its (3D) pieces on a
single face form a 3×3×3 cube. The pieces on the corners of that cube will be called
(3D)(0D) pieces, the edges, (3D)(1D) pieces, and the faces, (3D)(2D) pieces, while the
term (3D) pieces will continue to denote the entire group of pieces. The center piece
will be referred to as a (3D) center, the term center always being reserved for a piece
that lies at the center of the region it is located in. It should also be noted that on
n^4 cubes for example, when n >= 6, all of the the (3D) pieces other than the center are
broken down into either (3D)(0D), (3D)(1D), or (3D)(2D) pieces regardless of the depth of
each piece within the face. For example, take a 7^4 cube. The (3D) pieces in the inner
3×3×3 section will be broken down in the same manner as if they were the (3D) pieces in
the 5^4 cube in the example above.
There is another observation to be made regarding the concept of regions. When considering
regions for a type of piece that is subclassified (i.e. (3D)(1D) and (3D)(2D) regions for
4-dimensional cubes, for example), a group of pieces of such a type will be broken into
separate regions according to the dimension of the subclassification. To make this
completely clear, consider a 7^4 cube. A single face has one (3D) region, which is the
complete 5×5×5 section of (3D) pieces on that face, as expected. Also, that face has twelve
(3D)(2D) regions: Six of them are 3×3 sections of (3D)(2D) pieces (in the outer layer of the
(3D) pieces), and the other six are 1×1 sections of (3D)(2D) pieces (in the inner layer of
the (3D) pieces). We count these regions as separate, so that each region is a 2-dimensional
group of pieces, even though each 3×3 region is connected to the piece of the corresponding
1×1 region. We can also see that there are twelve (3D)(1D) regions, six of which are 3×1
sections of (3D)(1D) pieces, and the remaining six 1×1 sections of (3D)(1D) pieces. It
should now be clear what is meant by a (3D)(2D) and (3D)(1D) center on a cube of any size.
This notation system, or one which uses similar concepts, is essential when considering
n^d cubes when d >= 5. For example, if d = 6, we can have (5D)(4D)(2D) pieces, (4D)(1D)
pieces, or even (5D)(4D)(3D)(0D) pieces!
There is one more distinction to be made. We have already defined what we mean by a center
piece. The rest are either wings or normals. Wings are pieces that only occur on a ..(2D)
(e.g., (2D), or (3D)(2D), or (4D)(2D), or (4D)(3D)(2D), etc.) region. Consider that region
isolated from the rest of the puzzle, and can only rotate about its center in the plane it
resides in. A piece is considered a wing if it belongs to a family (as defined under this
restriction) to which there corresponds a separate family which are mirror images of each
other. For example, on a square n×n ..(2D) region, wings are the pieces that do not lie
on the main diagonals of the region or, when n is odd, additionally on the lines that divide
the region into four equal square quadrants. Pieces that are neither wings nor centers will
be called normals, except for ..(0D) pieces, which can simply be referred to as pieces without
confusion. It is important to distinguish between wings and normals because the permutations
and orientations of each always need to be calculated differently.
To conclude, here is a list of all possible types of pieces on a 4-dimensional cube:
(0D) pieces (corners)
(1D) centers
(1D) normals
(2D) centers
(2D) normals
(2D) wings
(3D) centers
(3D)(0D) pieces
(3D)(1D) centers
(3D)(1D) normals
(3D)(2D) centers
(3D)(2D) normals
(3D)(2D) wings
