CubeTwister

Superset ENG 3x3

Superset ENG 3x3 is a notation for Rubik's Cubes with 3 layers. The notation is based on the English language. It supports 90° and 180° twists of the faces of the cube. A whole cube can be rotated along its face axes. Permutation and orientation changes of individual cube parts can be specified using permutation cycles.

The notation supports the following high-level constructs:
- grouping,
- repetition,
- inversion,
- reflection,
- conjugation,
- commutation, and
- rotation,
- single-line and multiple-line comments.


Single moves
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Face twists: R, R', R2
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Clockwise 90° twists of the six faces of the cube are described by the capital letters: «R», «U», «F», «L», «D» and «B», (right, up, front, left, down, back).

Counterclockwise twists and rotations are described by appending the inversion operator «'» to a move: «R'», «U'», «F'», … .

In environments, where the the «'» character can not be used, the character «-» can be used as the inversion operator.

180° twists and rotations are described by appending the number «2» to a move: «R2», «U2», «F2», … .


Cube rotations: CR, CR', CR2
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Rotations of the whole cube are described by prepending the letter «C» to a face twist.

For example, «CR» rotates the whole cube in clockwise direction as seen from the right face.


Tier twists: TR
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Twists which go two layers deep (tier twists) are described by prepending the letter «T» to a face twist: «TR», «TU'», «TF2».


Mid-layer twists: MR
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Twists of the layer in the middle (mid-layer twists) are described by prepending the letter «M» to a face twist: «MR», «MU'», «MF2».


Slice twists: SR
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Twists of opposing faces (slice twists) are described by prepending the letter «S» to a face twist. For example, «SR» turns the right face and the opposing left face in clockwise direction as seen from the right face.


Constructs
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Sequences of moves
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Sequences of moves are specified by appending moves, for example «R2L2UD'F2B2UD'».

To improve readability, whitespace can be used to separate moves: «R2 L2 U D' F2 B2 U D'».

Sequences of moves can be separated using a middle dot character «·». In environments where this character is not available, the full stop character can be used «.». The following sequences are equivalent: «R2 L2 · U D' · F2 B2 · U D'» and «R2 L2 . U D' . F2 B2 . U D'».


Grouping
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Sequences of moves can be grouped using brackets: «(R U F)».


Repetition
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Moves and groups can be repeated multiple times by appending the number of repetitions to the closing bracket: «(R U F)3».


Inversion
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Moves and groups can be inverted by appending the invertor operator «'» to the closing bracket: «(R U F)'».


Reflection
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Moves and groups can be reflected using the reflector operator «*»

Conjugation
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Conjugations of type A B A' are described by enclosing the conjugator A with angle brackets and prepending it to the conjugee B: «<A>B».

Commutation
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Commutations of type A B A' B' are described by enclosing the commutator A and the commutee B with square brackets and delimiting A and B by a comma: «[A,B]».

Rotation
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Rotations of type A' B A are described by enclosing the rotator A with angle brackets and appending a invertor operator. The rotator is then prepended to rotatee B: «<A>' B».


Permutation cycles
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A permutation of the cube is described by a sequence of parenthesized lists for each cycle of the cube parts which are permuted.

The location of a cube part is described by the faces on which its stickers can be seen. A face of the cubes is described by the lower-case letters: «r», «u», «f», «l», «d» and «b», (right, up, front, left, down, back).

e. g. A corner part having its stickers at the upper front right corner is described by the three letter sequence: «ufr».
e. g. An edge part having its stickers at the front right corner is described by the two letter sequence: «fr».
e. g. A side part having its sticker at the right face of the cube is described by the letter: «r».

Each cycle contains a list of either corner parts, edge parts or side parts which have been permuted.

The face names in a cycle are used to describe orientation changes of the parts. The first face name in a location description denotes where the first sticker goes. The second face name in a location description denotes where the second sticker goes, and so on.

e.g. «(ufl,rfu)» describes, that the first sticker on face «u» at corner location «uf» goes to face «r» at corner location «rfu», the second sticker on face «f» at corner location «ufl» goes to face «f» at «rfu», and the third sticker on «l» at «ufl» goes to «u» at «rfu».

If the cube parts change orientation after going through a single cycle, then a prefix is added to the first location description of a sequence to denote this change in orientation.

If an edge part is flipped, a «+» prefix is added. If a corner part is twisted clockwise a «+» prefix is added, and if it is twisted counterclockwise a «–» prefix is added. If a side part is twisted 90° clockwise a «+» prefix is added, if it is twisted by 180° a «++» prefix is added and if it is twisted 90° counterclockwise a «-» prefix is added.

e. g. the move R is described by the following permutation: «(urf,bru,drb,frd) (ur,br,dr,fr) (+r)»


Comments
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Single line comments start with two slash characters: «// this is a comment».

Multiline comments starts with a slash and a star characer and end with star and a slash character: «/* this is a comment */».

Werner Randelshofer, Walter Randelshofer 2010-03-27