Rubik.Symmetry

• Tables of symmetry classes

Documentation

type SymRepr a = RawCoord a

Smallest representative of a symmetry class. (An element of the symClasses table)

type SymClass' = Int

newtype SymClass symType a

Symmetry class. (Index of the smallest representative in the symClasses table)

Constructors

SymClass

Fields

unSymClass :: SymClass'

Instances

Eq (SymClass symType a)

Ord (SymClass symType a)

Show (SymClass symType a)

type SymCoord sym a = (SymClass sym a, SymCode sym)

type SymCoord' = Int

An Int representing a pair (Repr, Sym).

If x = symClass * symOrder + symCode, where symClass :: SymClass is the index of the symmetry class with smallest representative r :: SymRepr (for an arbitrary order relation), symOrder is the size of the symmetry group, symCode :: Sym is the index of a symmetry s; then s^(-1) <> r <> s is the value represented by x.

type SymOrder' = Int

newtype Action s a

Constructors

Action [a -> a]

newtype SymClassTable s a

Constructors

SymClassTable

Fields

unSymClassTable :: Vector RawCoord'

Instances

Eq (SymClassTable s a)

Ord (SymClassTable s a)

Show (SymClassTable s a)

NFData (SymClassTable s a)

Binary (SymClassTable s a)

newtype SymReprTable s a

Constructors

SymReprTable

Fields

unSymReprTable :: Vector Int

Instances

Eq (SymReprTable s a)

Ord (SymReprTable s a)

Show (SymReprTable s a)

NFData (SymReprTable s a)

Binary (SymReprTable s a)

newtype SymMove s a

Constructors

SymMove (Vector SymCoord')

Instances

Eq (SymMove s a)

Ord (SymMove s a)

Show (SymMove s a)

NFData (SymMove s a)

Binary (SymMove s a)

type Symmetries sym a = MoveTag sym (Vector (RawMove a))

symClasses

Arguments

:: RawEncodable a

=> Action s a

Symmetry group, including the identity, - represented by its action on a

-> SymClassTable s a

Smallest representative

Compute the table of smallest representatives for all symmetry classes. The RawCoord' coordinate of that representative is a Repr. The table is sorted in increasing order.

symClasses' :: forall a s. RawEncodable a => Action s a -> [RawCoord a]

symClassTable :: Int -> SymReprTable s a -> SymClassTable s a

symReprTable

Arguments

:: (RawEncodable a, Foldable t)

=> Int

Number of symmetries nSym

-> (RawCoord a -> t (RawCoord a))

-> SymReprTable s a

symReprTable'

Arguments

:: Foldable t

=> Int

Number of elements n

-> Int

Number of symmetries nSym

-> (Int -> t Int)

f x, symmetrical elements to x, including itself

-> Vector Int

v, where (y, i) = (v ! x) divMod nSym gives the representative y of the symmetry class of x and the index of one symmetry mapping x to y:

f x !! i == y.

symMoveTable

Arguments

:: RawEncodable a

=> Action s a

Symmetry group

-> SymClassTable s a

(Sorted) table of representatives

-> (a -> a)

Endofunction to encode

-> SymMove s a

symMoveTable'

Arguments

:: RawEncodable a

=> Int

Symmetry group order

-> SymReprTable s a

-> SymClassTable s a

-> (a -> a)

-> SymMove s a

symMove :: SymOrder' -> SymMove s a -> SymClass s a -> SymCoord s a

symMove' :: SymOrder' -> SymMove sym a -> (SymClass sym a, SymCode sym) -> (SymClass sym a, SymCode sym)

reprToClass :: SymClassTable s a -> RawCoord a -> SymClass s a

symCoord :: RawEncodable a => Action s a -> SymClassTable s a -> a -> SymCoord s a

Find the representative as the one corresponding to the smallest coordinate

symCoord' :: Int -> SymReprTable s a -> SymClassTable s a -> RawCoord a -> SymCoord s a

symToRaw :: SymClassTable s a -> (RawCoord a -> SymCode s -> RawCoord a) -> SymCoord s a -> RawCoord a

sym :: Symmetries s a -> RawCoord a -> SymCode s -> RawCoord a