Fractal.RUFF.Mandelbrot.Address

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.

Documentation

type Angle = Rational

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

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

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

prettyAngle :: Angle -> String

Convert to human readable form.

prettyAngles :: [Angle] -> String

Convert to human readable form.

angles :: Angle -> [Angle]

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

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

Binary representation of a (pre-)periodic angle.

binary :: Angle -> BinAngle

Convert an angle to binary representation.

unbinary :: BinAngle -> Angle

Convert an angle from binary representation.

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

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

prettyBinAngle :: BinAngle -> String

Convert to human readable form.

parseBinAngle :: String -> Maybe BinAngle

Convert from human readable form.

bperiod :: BinAngle -> Int

Period under angle doubling.

bpreperiod :: BinAngle -> Int

Preperiod under angle doubling.

bangles :: BinAngle -> [BinAngle]

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

data Knead

Elements of kneading sequences.

Constructors

Zero

One

Star

Instances

Bounded Knead

Enum Knead

Eq Knead

Data Knead

Ord Knead

Read Knead

Show Knead

Typeable * Knead

kneadChar :: Knead -> Char

Knead character representation.

data Kneading

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

Constructors

Aperiodic [Knead]

PrePeriodic [Knead] [Knead]

StarPeriodic [Knead]

Periodic [Knead]

Instances

Eq Kneading

Data Kneading

Ord Kneading

Read Kneading

Show Kneading

Typeable * Kneading

prettyKneading :: Kneading -> String

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

kneading :: Angle -> Kneading

The kneading sequence for an external angle.

period :: Kneading -> Maybe Int

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

unwrap :: Kneading -> [Knead]

Unwrap a kneading sequence to an infinite list.

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

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

upper :: Kneading -> Maybe Kneading

The upper associated kneading sequence.

lower :: Kneading -> Maybe Kneading

The lower associated kneading sequence.

data InternalAddress

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

Constructors

InternalAddress [Int]

Instances

Eq InternalAddress

Data InternalAddress

Ord InternalAddress

Read InternalAddress

Show InternalAddress

Typeable * InternalAddress

prettyInternalAddress :: InternalAddress -> String

Internal address as a string.

internalAddress :: Kneading -> Maybe InternalAddress

Construct an InternalAddress from a kneading sequence.

internalFromList :: [Int] -> Maybe InternalAddress

Construct a valid InternalAddress, checking the precondition.

internalToList :: InternalAddress -> [Int]

Extract the sequence of integers.

data AngledInternalAddress

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

Constructors

Unangled Int

Angled Int Angle AngledInternalAddress

Instances

Eq AngledInternalAddress

Data AngledInternalAddress

Ord AngledInternalAddress

Read AngledInternalAddress

Show AngledInternalAddress

Typeable * AngledInternalAddress

prettyAngledInternalAddress :: AngledInternalAddress -> String

Angled internal address as a string.

angledInternalAddress :: Angle -> Maybe AngledInternalAddress

The angled internal address corresponding to an external angle.

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

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)]

Convert an AngledInternalAddress to a list.

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

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

stripAngles :: AngledInternalAddress -> InternalAddress

Discard angle information from an internal address.

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

Split an angled internal address at the last island.

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

The inverse of splitAddress.

addressPeriod :: AngledInternalAddress -> Int

The period of an angled internal address.

parseAngle :: String -> Maybe Angle

Parse an angle.

parseAngles :: String -> Maybe [Angle]

Parse a list of angles.

parseKnead :: String -> Maybe Knead

Parse a kneading element.

parseKneading :: String -> Maybe Kneading

Parse a non-aperiodic kneading sequence.

parseInternalAddress :: String -> Maybe InternalAddress

Parse an internal address.

parseAngledInternalAddress :: String -> Maybe AngledInternalAddress

Parse an angled internal address, accepting some unambiguous abbreviations.