Copyright	(c) Claude Heiland-Allen 201020112015
License	BSD3
Maintainer	claude@mathr.co.uk
Stability	unstable
Portability	portable
Safe Haskell	None
Language	Haskell98

Fractal.RUFF.Mandelbrot.Address

Description

External angles give rise to kneading sequences under the angle doubling map. Internal addresses encode kneading sequences in human-readable form, when extended to angled internal addresses they distinguish hyperbolic components in a concise and meaningful way.

The algorithms are mostly based on Dierk Schleicher's papers Internal Addresses Of The Mandelbrot Set And Galois Groups Of Polynomials (version of February 5, 2008) http://arxiv.org/abs/math/9411238v2 and Rational parameter rays of the Mandelbrot set (version of August 11, 1998) http://arxiv.org/abs/math/9711213v2.

Synopsis

Documentation

type Angle = Rational Source #

Angle as a fraction of a turn, usually in [0, 1).

tune :: Angle -> (Angle, Angle) -> Angle Source #

Tuning transformation for angles. Probably only valid for angle pairs representing hyperbolic components.

prettyAngle :: Angle -> String Source #

Convert to human readable form.

prettyAngles :: [Angle] -> String Source #

Convert to human readable form.

angles :: Angle -> [Angle] Source #

All external angles landing at the same location as the given external angle.

type BinAngle = ([Bool], [Bool]) Source #

Binary representation of a (pre-)periodic angle.

binary :: Angle -> BinAngle Source #

Convert an angle to binary representation.

unbinary :: BinAngle -> Angle Source #

Convert an angle from binary representation.

btune :: BinAngle -> (BinAngle, BinAngle) -> BinAngle Source #

Tuning transformation for binary represented periodic angles. Probably only valid for angle pairs representing hyperbolic components.

prettyBinAngle :: BinAngle -> String Source #

Convert to human readable form.

parseBinAngle :: String -> Maybe BinAngle Source #

Convert from human readable form.

bperiod :: BinAngle -> Int Source #

Period under angle doubling.

bpreperiod :: BinAngle -> Int Source #

Preperiod under angle doubling.

bangles :: BinAngle -> [BinAngle] Source #

All external angles landing at the same location as the given external angle (binary angle variant).

data Knead Source #

Elements of kneading sequences.

Constructors

Zero
One
Star

Instances

Bounded Knead Source #
Methods minBound :: Knead # maxBound :: Knead #
Enum Knead Source #
Methods succ :: Knead -> Knead # pred :: Knead -> Knead # toEnum :: Int -> Knead # fromEnum :: Knead -> Int # enumFrom :: Knead -> [Knead] # enumFromThen :: Knead -> Knead -> [Knead] # enumFromTo :: Knead -> Knead -> [Knead] # enumFromThenTo :: Knead -> Knead -> Knead -> [Knead] #
Eq Knead Source #
Methods (==) :: Knead -> Knead -> Bool # (/=) :: Knead -> Knead -> Bool #
Data Knead Source #
Methods gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Knead -> c Knead # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c Knead # toConstr :: Knead -> Constr # dataTypeOf :: Knead -> DataType # dataCast1 :: Typeable (* -> ) t => (forall d. Data d => c (t d)) -> Maybe (c Knead) # dataCast2 :: Typeable ( -> * -> *) t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c Knead) # gmapT :: (forall b. Data b => b -> b) -> Knead -> Knead # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Knead -> r # gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Knead -> r # gmapQ :: (forall d. Data d => d -> u) -> Knead -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> Knead -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> Knead -> m Knead # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Knead -> m Knead # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Knead -> m Knead #
Ord Knead Source #
Methods compare :: Knead -> Knead -> Ordering # (<) :: Knead -> Knead -> Bool # (<=) :: Knead -> Knead -> Bool # (>) :: Knead -> Knead -> Bool # (>=) :: Knead -> Knead -> Bool # max :: Knead -> Knead -> Knead # min :: Knead -> Knead -> Knead #
Read Knead Source #
Methods readsPrec :: Int -> ReadS Knead # readList :: ReadS [Knead] # readPrec :: ReadPrec Knead # readListPrec :: ReadPrec [Knead] #
Show Knead Source #
Methods showsPrec :: Int -> Knead -> ShowS # show :: Knead -> String # showList :: [Knead] -> ShowS #

kneadChar :: Knead -> Char Source #

Knead character representation.

data Kneading Source #

Kneading sequences. Note that the Aperiodic case has an infinite list.

Constructors

Aperiodic [Knead]
PrePeriodic [Knead] [Knead]
StarPeriodic [Knead]
Periodic [Knead]

Instances

Eq Kneading Source #
Methods (==) :: Kneading -> Kneading -> Bool # (/=) :: Kneading -> Kneading -> Bool #
Data Kneading Source #
Methods gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Kneading -> c Kneading # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c Kneading # toConstr :: Kneading -> Constr # dataTypeOf :: Kneading -> DataType # dataCast1 :: Typeable (* -> ) t => (forall d. Data d => c (t d)) -> Maybe (c Kneading) # dataCast2 :: Typeable ( -> * -> *) t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c Kneading) # gmapT :: (forall b. Data b => b -> b) -> Kneading -> Kneading # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Kneading -> r # gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Kneading -> r # gmapQ :: (forall d. Data d => d -> u) -> Kneading -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> Kneading -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> Kneading -> m Kneading # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Kneading -> m Kneading # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Kneading -> m Kneading #
Ord Kneading Source #
Methods compare :: Kneading -> Kneading -> Ordering # (<) :: Kneading -> Kneading -> Bool # (<=) :: Kneading -> Kneading -> Bool # (>) :: Kneading -> Kneading -> Bool # (>=) :: Kneading -> Kneading -> Bool # max :: Kneading -> Kneading -> Kneading # min :: Kneading -> Kneading -> Kneading #
Read Kneading Source #
Methods readsPrec :: Int -> ReadS Kneading # readList :: ReadS [Kneading] # readPrec :: ReadPrec Kneading # readListPrec :: ReadPrec [Kneading] #
Show Kneading Source #
Methods showsPrec :: Int -> Kneading -> ShowS # show :: Kneading -> String # showList :: [Kneading] -> ShowS #

prettyKneading :: Kneading -> String Source #

Kneading sequence as a string. The Aperiodic case is truncated arbitrarily.

kneading :: Angle -> Kneading Source #

The kneading sequence for an external angle.

period :: Kneading -> Maybe Int Source #

The period of a kneading sequence, or Nothing when it isn't periodic.

unwrap :: Kneading -> [Knead] Source #

Unwrap a kneading sequence to an infinite list.

associated :: Kneading -> Maybe (Kneading, Kneading) Source #

A star-periodic kneading sequence's upper and lower associated kneading sequences.

upper :: Kneading -> Maybe Kneading Source #

The upper associated kneading sequence.

lower :: Kneading -> Maybe Kneading Source #

The lower associated kneading sequence.

data InternalAddress Source #

Internal addresses are a non-empty sequence of strictly increasing integers beginning with '1'.

Constructors

InternalAddress [Int]

Instances

Eq InternalAddress Source #
Methods (==) :: InternalAddress -> InternalAddress -> Bool # (/=) :: InternalAddress -> InternalAddress -> Bool #
Data InternalAddress Source #
Methods gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> InternalAddress -> c InternalAddress # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c InternalAddress # toConstr :: InternalAddress -> Constr # dataTypeOf :: InternalAddress -> DataType # dataCast1 :: Typeable (* -> ) t => (forall d. Data d => c (t d)) -> Maybe (c InternalAddress) # dataCast2 :: Typeable ( -> * -> *) t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c InternalAddress) # gmapT :: (forall b. Data b => b -> b) -> InternalAddress -> InternalAddress # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> InternalAddress -> r # gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> InternalAddress -> r # gmapQ :: (forall d. Data d => d -> u) -> InternalAddress -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> InternalAddress -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> InternalAddress -> m InternalAddress # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> InternalAddress -> m InternalAddress # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> InternalAddress -> m InternalAddress #
Ord InternalAddress Source #
Methods compare :: InternalAddress -> InternalAddress -> Ordering # (<) :: InternalAddress -> InternalAddress -> Bool # (<=) :: InternalAddress -> InternalAddress -> Bool # (>) :: InternalAddress -> InternalAddress -> Bool # (>=) :: InternalAddress -> InternalAddress -> Bool # max :: InternalAddress -> InternalAddress -> InternalAddress # min :: InternalAddress -> InternalAddress -> InternalAddress #
Read InternalAddress Source #
Methods readsPrec :: Int -> ReadS InternalAddress # readList :: ReadS [InternalAddress] # readPrec :: ReadPrec InternalAddress # readListPrec :: ReadPrec [InternalAddress] #
Show InternalAddress Source #
Methods showsPrec :: Int -> InternalAddress -> ShowS # show :: InternalAddress -> String # showList :: [InternalAddress] -> ShowS #

prettyInternalAddress :: InternalAddress -> String Source #

Internal address as a string.

internalAddress :: Kneading -> Maybe InternalAddress Source #

Construct an InternalAddress from a kneading sequence.

internalFromList :: [Int] -> Maybe InternalAddress Source #

Construct a valid InternalAddress, checking the precondition.

internalToList :: InternalAddress -> [Int] Source #

Extract the sequence of integers.

data AngledInternalAddress Source #

Angled internal addresses have angles between each integer in an internal address.

Constructors

Unangled Int
Angled Int Angle AngledInternalAddress

Instances

Eq AngledInternalAddress Source #
Methods (==) :: AngledInternalAddress -> AngledInternalAddress -> Bool # (/=) :: AngledInternalAddress -> AngledInternalAddress -> Bool #
Data AngledInternalAddress Source #
Methods gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> AngledInternalAddress -> c AngledInternalAddress # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c AngledInternalAddress # toConstr :: AngledInternalAddress -> Constr # dataTypeOf :: AngledInternalAddress -> DataType # dataCast1 :: Typeable (* -> ) t => (forall d. Data d => c (t d)) -> Maybe (c AngledInternalAddress) # dataCast2 :: Typeable ( -> * -> *) t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c AngledInternalAddress) # gmapT :: (forall b. Data b => b -> b) -> AngledInternalAddress -> AngledInternalAddress # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> AngledInternalAddress -> r # gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> AngledInternalAddress -> r # gmapQ :: (forall d. Data d => d -> u) -> AngledInternalAddress -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> AngledInternalAddress -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> AngledInternalAddress -> m AngledInternalAddress # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> AngledInternalAddress -> m AngledInternalAddress # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> AngledInternalAddress -> m AngledInternalAddress #
Ord AngledInternalAddress Source #
Methods compare :: AngledInternalAddress -> AngledInternalAddress -> Ordering # (<) :: AngledInternalAddress -> AngledInternalAddress -> Bool # (<=) :: AngledInternalAddress -> AngledInternalAddress -> Bool # (>) :: AngledInternalAddress -> AngledInternalAddress -> Bool # (>=) :: AngledInternalAddress -> AngledInternalAddress -> Bool # max :: AngledInternalAddress -> AngledInternalAddress -> AngledInternalAddress # min :: AngledInternalAddress -> AngledInternalAddress -> AngledInternalAddress #
Read AngledInternalAddress Source #
Methods readsPrec :: Int -> ReadS AngledInternalAddress # readList :: ReadS [AngledInternalAddress] # readPrec :: ReadPrec AngledInternalAddress # readListPrec :: ReadPrec [AngledInternalAddress] #
Show AngledInternalAddress Source #
Methods showsPrec :: Int -> AngledInternalAddress -> ShowS # show :: AngledInternalAddress -> String # showList :: [AngledInternalAddress] -> ShowS #

prettyAngledInternalAddress :: AngledInternalAddress -> String Source #

Angled internal address as a string.

angledInternalAddress :: Angle -> Maybe AngledInternalAddress Source #

The angled internal address corresponding to an external angle.

angledFromList :: [(Int, Maybe Angle)] -> Maybe AngledInternalAddress Source #

Builds a valid AngledInternalAddress from a list, checking the precondition that only the last 'Maybe Angle' should be Nothing, and the Integer must be strictly increasing.

angledToList :: AngledInternalAddress -> [(Int, Maybe Angle)] Source #

Convert an AngledInternalAddress to a list.

externalAngles :: AngledInternalAddress -> Maybe (Angle, Angle) Source #

The pair of external angles whose rays land at the root of the hyperbolic component described by the angled internal address.

stripAngles :: AngledInternalAddress -> InternalAddress Source #

Discard angle information from an internal address.

splitAddress :: AngledInternalAddress -> (AngledInternalAddress, [Angle]) Source #

Split an angled internal address at the last island.

joinAddress :: AngledInternalAddress -> [Angle] -> AngledInternalAddress Source #

The inverse of splitAddress.

addressPeriod :: AngledInternalAddress -> Int Source #

The period of an angled internal address.