Difference between revisions of "Permutations Notation"

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In order to study cubes larger than the 34, it will be useful to develop a better system for naming the
+
<br>
pieces under consideration than simply referring to the number of facelets a piece has. This is because
+
Following is a generalized notation system which assigns names to the pieces of every
on larger cubes, there are often multiple groups of pieces with the same number of facelets.
+
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
cube throughout all possible configurations. For example, on a 54 cube, there are two families of
+
permutation counts found in [[Permutations_Index]].
3-colored pieces and three families of 2-colored pieces.
+
<br> <br />
The notation system that follows is useful in four dimensions, but is even more valuable in dimensions
+
We shall denote by the term family a complete group of pieces that can occupy the same
five and above. The basic idea is to classify families of pieces by the dimension of the section they are
+
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. If the Rubik's Cube were
+
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 54 cube, a (1D) region is
+
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 cube, which seems
+
(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 3, we continue to break
+
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 2. For example, consider a 54 cube. Its (3D) pieces on
+
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. (Note that the term (3D) pieces will often be excluded when referring
+
the term region will refer to a section that contains connected pieces of a specific
to types of pieces in general, as it represents multiple piece types.) The center piece will be referred to
+
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 n4 cubes, when n 6, all of the the (3D) pieces other than the
+
<br> <br />
center are broken down into either (3D)(0D), (3D)(1D), or (3D)(0D) pieces regardless of the depth of
+
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 74 cube. The (3D) pieces in the inner 3×3×3 section will
+
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 54 cube in the example above.
+
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 74 cube. A single face has one (3D) region,
+
(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 nd cubes when
+
<br> <br />
d 5. For example, if d = 6, we can have (5D)(4D)(2D) pieces, (4D)(1D) pieces, or even
+
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
They are the pieces that do not lie on the main diagonals of the (2D) or (3D)(2D) region or (in the case
+
completely clear, consider a 7^4 cube. A single face has one (3D) region, which is the
of an odd cube) on the lines that divide the region into four equal square quadrants. Pieces that are
+
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 (0D) and (3D)(0D) pieces, which can simply
+
(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), (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

Revision as of 05:08, 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), (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