Cube definitions

The definitions provided in the following table relate to Rubik's family cubes and conform to what is used in this Unravel Online Help. In most cases they are in agreement with common usage but there may be some variations.

Item Description
Cube size The standard Rubik's cube is often referred to as a 3x3x3 cube. That cube will be referred to as a size 3 cube and in general an n x n x n cube will be referred to as a size n cube.
Rubik cube family Cubes that have similar rotational properties to the standard Rubik's size 3 cube and obey generalized rules for a size n cube are considered to be members of the Rubik cube family. Cubes of size 2 and above are available in unit size steps in the Unravel program. The program limit is size 99 but such a large size or large sizes in general may be too difficult for many users to handle.
Cubie Individual cube elements will be referred to as cubies (others sometimes refer to them as "cubelets"). There are three types of cubies: corner cubies (three coloured surfaces), edge cubies (two coloured surfaces) and centre cubies (one coloured surface). The absolute centre cubies for odd size cubes sit on the central axes of the six faces and their relative positions never change.
Cubicle A cubicle is the compartment in which a cubie resides. For a permutation (defined below), cubicles are considered to occupy fixed positions relative to the cube object but their contents (cubies) may change.
Facelet A facelet is a visible coloured surface of a cubie (corner cubies have three facelets, edge cubies have two and centre cubies have one).
Cube state A particular arrangement of the cubies will be referred to as a cube state. What looks the same is considered to be the same (unless specific mention to the contrary is made). Each state has equal probability of being produced after a genuine random scrambling sequence. A rotation of the whole cube does not change the state considered herein. In other texts the various states are often referred to as permutations or arrangements.
Cube layer A cube layer is a one cubie width slice of the cube perpendicular to its axis of rotation. Outer layers contain more cubies than inner layers. For a cube of size n there will be n layers along any given axis.
Cube face The meaning of a cube face depends on the context in which it is used. It usually means one of the six three-dimensional outer layers but can also refer to just the outside layer's surface which is perpendicular to its axis of rotation. The faces are designated as up (U), down (D), front (F), back (B), left (L) and right (R).
Set state The set (or solved) state of the cube is one for which a uniform colour appears on each of the six faces. For cubes with marked centres the set state also includes a unique arrangement of all centre cubies.
Scrambled state The scrambled state is the starting point for unscrambling the cube. It arises when a cube in the set or any other state is subject to a large number of randomly chosen layer rotations.
"Fixed-in-space" axes of rotation There are three mutually perpendicular axes of rotation for the cube. One set of axes defined in terms of D, U, B, F, L and R terms can be considered to have a fixed orientation in space. Think of these axes as belonging to a cube-shaped container in which the cube object can be positioned in any of 24 orientations. One axis can be drawn through the centres of the D and U faces (the DU axis). The others are the BF and LR axes. The main display of the cube for the Unravel program uses these axes.
"Cube object" axes of rotation Another set of axes, can be defined for the cube object itself. These axes relate to the face colours, which are off-white, red, light-orange, green, light-blue and violet for the Unravel's default colour set. The axes are white-blue, red-violet and orange-green. For odd size cubes these axes are always fixed relative to the internal frame of the cube object via the absolute centre cubies. For even size cubes these axes remain fixed relative to the internal frame of the cube object after initial selections. The origin for the axes is the centre of the cube object. It is likely that most users would make these axes coincide with the previous fixed-in-space set as the final solution is approached.
Layer rotation The only way that cube state can be changed is by rotations of cube layers about their axes of rotation. All changes of state involve rotation steps that can be considered as a sequence of single layer quarter turns.
Orbit For a basic quarter turn of a cube layer for cubes of all sizes, sets-of-four cubies move in separate four-cubicle trajectories. When all the possible trajectories for a given cubie type are considered for the whole cube we will refer to all the possible movement positions as being in a given orbit. We consider that the size 3 cube has two orbits, one in which the eight corner cubies are constrained to move and one in which the 12 edge cubies are constrained to move. Transfer of cubies between these orbits is impossible.

For cubes of size 4 and above we will also define an edge cubie orbit as comprising 12 cubies but will use the term complementary orbit to describe a pair of orbits between which edge cubies can move. A pair of complementary edge cubie orbits contains a total of 24 cubies. Cubes of size 4 and above include centre cubie orbits that contain 24 cubies. Transfer of cubies between one such orbit and another is not possible (applies to cubes of size 5 and above).
Move A move is a quarter turn rotation of a layer or a sequence of such quarter turns that a person would apply as a single step.
Move notation A clockwise quarter turn of an outer layer is usually expressed as U, D, F, B, L or R. The Unravel program uses a prefix to define a single inner layer or multiple layer rotation.
Algorithm An algorithm defines a sequence of layer rotations to transform a given state to another (usually less scrambled) state. Usually an algorithm is expressed as a printable character sequence according to some move notation. An algorithm can be considered to be a "smart" move. All algorithms are moves but only some moves are considered to be algorithms.
Permutation A permutation of the cube usually means the act of permuting (i.e. rearranging or changing) the positions of cubies. A permutation is an all-inclusive term which includes all moves (and algorithms) and more. Even the solving of the cube from a scrambled state represents a permutation. The term "permutation" is used extensively by mathematicians who use Group Theory to quantify the process involved in a rearrangement of cubies.

The term "permutation" is also often used to mean the state of the cube that results after it is rearranged but that meaning will not be used herein. In such cases the term "cube state" will be used. That allows the term "permutation" to be used in situations where the permutation results in no change of state - a specific area of interest for permutations.
Parity A cube permutation can be represented by a number of swaps of two cubies. If that number is even the permutation has even parity, and if the number is odd the permutation has odd parity.