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 a piece has. This is because on larger cubes, there are often multiple groups of pieces with the same number of facelets. We shall denote by the term family a complete group of pieces that can occupy the same positions on a cube throughout all possible configurations. For example, on a 54 cube, there are two families of 3-colored pieces and three families of 2-colored pieces. The notation system that follows is useful in four dimensions, but is even more valuable in dimensions five and above. The basic idea is to classify families of pieces by the dimension of the section they are located in. To make this more explicit, consider the 4-colored corner pieces. If the Rubik's 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. Also, the term region will refer to a section that contains connected pieces of a specific type. For example, on a 54 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 cube, 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 54 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. (Note that the term (3D) pieces will often be excluded when referring to types of pieces in general, as it represents multiple piece types.) 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 n4 cubes, when n � 6, all of the the (3D) pieces other than the center are broken down into either (3D)(0D), (3D)(1D), or (3D)(0D) pieces regardless of the depth of each piece within the face. For example, take a 74 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 54 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), 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 74 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 nd 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) or (3D)(2D) region. They are the pieces that do not lie on the main diagonals of the (2D) or (3D)(2D) region or (in the case of an odd cube) 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) and (3D)(0D) pieces, which can simply be referred to as pieces without confusion.